Abstract

Synthetic dimensions provide a promising platform for photonic quantum simulations. Manipulating the flow of photons in these dimensions requires an electric field. However, photons do not have charge and do not directly interact with electric fields. Therefore, alternative approaches are needed to realize electric fields in photonics. One approach is to use engineered gauge fields that can mimic the effect of electric fields and produce the same dynamical behavior. Here, we demonstrate such an electric field for photons propagating in a two-dimensional synthetic space. Generation of electric fields in a two-dimensional synthetic lattice provides the possibility to guide photons and to trap them through the creation of quantum confined structures. We achieve this using a linearly time-varying gauge field generated by direction-dependent phase modulations. We show that the generated electric field leads to Bloch oscillations and the revival of the state after a certain number of steps dependent on the field strength. We measure the probability of the revival and demonstrate a good agreement between the observed values and the theoretically predicted results. Furthermore, by applying a nonuniform electric field, we show the possibility of waveguiding photons. Ultimately, our results open up new opportunities for manipulating the propagation of photons with potential applications in photonic quantum simulations.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  1. C. Sparrow, E. Martín-López, N. Maraviglia, A. Neville, C. Harrold, J. Carolan, Y. N. Joglekar, T. Hashimoto, N. Matsuda, J. L. O’Brien, D. P. Tew, and A. Laing, “Simulating the vibrational quantum dynamics of molecules using photonics,” Nature 557, 660–667 (2018).
    [Crossref]
  2. X.-S. Ma, B. Dakic, W. Naylor, A. Zeilinger, and P. Walther, “Quantum simulation of the wavefunction to probe frustrated Heisenberg spin systems,” Nat. Phys. 7, 399–405 (2011).
    [Crossref]
  3. B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nat. Chem. 2, 106–111 (2010).
    [Crossref]
  4. A. Aspuru-Guzik and P. Walther, “Photonic quantum simulators,” Nat. Phys. 8, 285–291 (2012).
    [Crossref]
  5. C. Navarrete-Benlloch, A. Pérez, and E. Roldán, “Nonlinear optical Galton board,” Phys. Rev. A 75, 062333 (2007).
    [Crossref]
  6. D. Bouwmeester, I. Marzoli, G. P. Karman, W. Schleich, and J. P. Woerdman, “Optical Galton board,” Phys. Rev. A 61, 013410 (1999).
    [Crossref]
  7. B. A. Bell, K. Wang, A. S. Solntsev, D. N. Neshev, A. A. Sukhorukov, and B. J. Eggleton, “Spectral photonic lattices with complex long-range coupling,” Optica 4, 1433–1436 (2017).
    [Crossref]
  8. C. Qin, F. Zhou, Y. Peng, D. Sounas, X. Zhu, B. Wang, J. Dong, X. Zhang, A. Alù, and P. Lu, “Spectrum control through discrete frequency diffraction in the presence of photonic gauge potentials,” Phys. Rev. Lett. 120, 133901 (2018).
    [Crossref]
  9. A. Dutt, M. Minkov, Q. Lin, L. Yuan, D. A. B. Miller, and S. Fan, “Experimental band structure spectroscopy along a synthetic dimension,” Nat. Commun. 10, 3122 (2019).
    [Crossref]
  10. A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167–171 (2012).
    [Crossref]
  11. E. Lustig, S. Weimann, Y. Plotnik, Y. Lumer, M. A. Bandres, A. Szameit, and M. Segev, “Photonic topological insulator in synthetic dimensions,” Nature 567, 356–360 (2019).
    [Crossref]
  12. Q. Lin, M. Xiao, L. Yuan, and S. Fan, “Photonic Weyl point in a two-dimensional resonator lattice with a synthetic frequency dimension,” Nat. Commun. 7, 13731 (2016).
    [Crossref]
  13. Q. Lin, X.-Q. Sun, M. Xiao, S.-C. Zhang, and S. Fan, “A three-dimensional photonic topological insulator using a two-dimensional ring resonator lattice with a synthetic frequency dimension,” Sci. Adv. 4, eaat2774 (2018).
    [Crossref]
  14. L. Yuan, M. Xiao, Q. Lin, and S. Fan, “Synthetic space with arbitrary dimensions in a few rings undergoing dynamic modulation,” Phys. Rev. B 97, 104105 (2018).
    [Crossref]
  15. S. K. Goyal, F. S. Roux, A. Forbes, and T. Konrad, “Implementing quantum walks using orbital angular momentum of classical light,” Phys. Rev. Lett. 110, 263602 (2013).
    [Crossref]
  16. F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, “Quantum walks and wavepacket dynamics on a lattice with twisted photons,” Sci. Adv. 1, e1500087 (2015).
    [Crossref]
  17. F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7, 11439 (2016).
    [Crossref]
  18. F. Cardano, A. D’Errico, A. Dauphin, M. Maffei, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, L. Marrucci, M. Lewenstein, and P. Massignan, “Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons,” Nat. Commun. 8, 15516 (2017).
    [Crossref]
  19. A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
    [Crossref]
  20. A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, I. Jex, and C. Silberhorn, “Decoherence and disorder in quantum walks: from ballistic spread to localization,” Phys. Rev. Lett. 106, 180403 (2011).
    [Crossref]
  21. A. Schreiber, A. Gabris, P. P. Rohde, K. Laiho, M. Stefanak, V. Potocek, C. Hamilton, I. Jex, and C. Silberhorn, “A 2D quantum walk simulation of two-particle dynamics,” Science 336, 55–58 (2012).
    [Crossref]
  22. T. Nitsche, F. Elster, J. Novotný, A. Gábris, I. Jex, S. Barkhofen, and C. Silberhorn, “Quantum walks with dynamical control: graph engineering, initial state preparation and state transfer,” New J. Phys. 18, 063017 (2016).
    [Crossref]
  23. S. Barkhofen, T. Nitsche, F. Elster, L. Lorz, A. Gábris, I. Jex, and C. Silberhorn, “Measuring topological invariants in disordered discrete-time quantum walks,” Phys. Rev. A 96, 033846 (2017).
    [Crossref]
  24. C. Chen, X. Ding, J. Qin, Y. He, Y.-H. Luo, M.-C. Chen, C. Liu, X.-L. Wang, W.-J. Zhang, H. Li, L.-X. You, Z. Wang, D.-W. Wang, B. C. Sanders, C.-Y. Lu, and J.-W. Pan, “Observation of topologically protected edge states in a photonic two-dimensional quantum walk,” Phys. Rev. Lett. 121, 100502 (2018).
    [Crossref]
  25. 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]
  26. S. Mittal, J. Fan, S. Faez, A. Migdall, J. M. Taylor, and M. Hafezi, “Topologically robust transport of photons in a synthetic gauge field,” Phys. Rev. Lett. 113, 087403 (2014).
    [Crossref]
  27. 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]
  28. 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]
  29. T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91, 015006 (2019).
    [Crossref]
  30. W. Hu, J. C. Pillay, K. Wu, M. Pasek, P. P. Shum, and Y. D. Chong, “Measurement of a topological edge invariant in a microwave network,” Phys. Rev. X 5, 011012 (2015).
    [Crossref]
  31. A. D’Errico, F. Cardano, M. Maffei, A. Dauphin, R. Barboza, C. Esposito, B. Piccirillo, M. Lewenstein, P. Massignan, and L. Marrucci, “Two-dimensional topological quantum walks in the momentum space of structured light,” Optica 7, 108–114 (2018).
    [Crossref]
  32. S. Mittal, S. Ganeshan, J. Fan, A. Vaezi, and M. Hafezi, “Measurement of topological invariants in a 2D photonic system,” Nat. Photonics 10, 180–183 (2016).
    [Crossref]
  33. Y. Bromberg, Y. Lahini, and Y. Silberberg, “Bloch oscillations of path-entangled photons,” Phys. Rev. Lett. 105, 263604 (2010).
    [Crossref]
  34. G. Corrielli, A. Crespi, G. Della Valle, S. Longhi, and R. Osellame, “Fractional Bloch oscillations in photonic lattices,” Nat. Commun. 4, 1555 (2013).
    [Crossref]
  35. D. Witthaut, F. Keck, H. J. Korsch, and S. Mossmann, “Bloch oscillations in two-dimensional lattices,” New J. Phys. 6, 41 (2004).
    [Crossref]
  36. F. Bloch, “Über die Quantenmechanik der Elektronen in Kristallgittern,” Z. Phys. 52, 555–600 (1929).
    [Crossref]
  37. E. Haller, R. Hart, M. J. Mark, J. G. Danzl, L. Reichsöllner, and H.-C. Nägerl, “Inducing transport in a dissipation-free lattice with super Bloch oscillations,” Phys. Rev. Lett. 104, 200403 (2010).
    [Crossref]
  38. M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, “Bloch oscillations of atoms in an optical potential,” Phys. Rev. Lett. 76, 4508–4511 (1996).
    [Crossref]
  39. T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
    [Crossref]
  40. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
    [Crossref]
  41. G. Lenz, I. Talanina, and C. M. de Sterke, “Bloch Oscillations in an array of curved optical waveguides,” Phys. Rev. Lett. 83, 963–966 (1999).
    [Crossref]
  42. H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
    [Crossref]
  43. S. Longhi, “Optical Zener-Bloch oscillations in binary waveguide arrays,” Europhys. Lett. 76, 416–421 (2006).
    [Crossref]
  44. F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, “Bloch-Zener oscillations in binary superlattices,” Phys. Rev. Lett. 102, 076802 (2009).
    [Crossref]
  45. M. Levy and P. Kumar, “Nonreciprocal Bloch oscillations in magneto-optic waveguide arrays,” Opt. Lett. 35, 3147–3149 (2010).
    [Crossref]
  46. P. Xue, R. Zhang, H. Qin, X. Zhan, Z. H. Bian, J. Li, and B. C. Sanders, “Experimental quantum-walk revival with a time-dependent coin,” Phys. Rev. Lett. 114, 140502 (2015).
    [Crossref]
  47. C. Cedzich and R. F. Werner, “Revivals in quantum walks with a quasiperiodically-time-dependent coin,” Phys. Rev. A 93, 032329 (2016).
    [Crossref]
  48. U. Peschel, C. Bersch, and G. Onishchukov, “Discreteness in time,” Open Phys. 6, 619–627 (2008).
    [Crossref]
  49. C. Bersch, G. Onishchukov, and U. Peschel, “Experimental observation of spectral Bloch oscillations,” Opt. Lett. 34, 2372–2374 (2009).
    [Crossref]
  50. L. Yuan and S. Fan, “Bloch oscillation and unidirectional translation of frequency in a dynamically modulated ring resonator,” Optica 3, 1014–1018 (2016).
    [Crossref]
  51. C. Cedzich, T. Rybár, A. H. Werner, A. Alberti, M. Genske, and R. F. Werner, “Propagation of quantum walks in electric fields,” Phys. Rev. Lett. 111, 160601 (2013).
    [Crossref]
  52. M. Genske, W. Alt, A. Steffen, A. H. Werner, R. F. Werner, D. Meschede, and A. Alberti, “Electric quantum walks with individual atoms,” Phys. Rev. Lett. 110, 190601 (2013).
    [Crossref]
  53. L. A. Bru, M. Hinarejos, F. Silva, G. J. de Valcárcel, and E. Roldán, “Electric quantum walks in two dimensions,” Phys. Rev. A 93, 032333 (2016).
    [Crossref]
  54. C. Cedzich, T. Geib, A. H. Werner, and R. F. Werner, “Quantum walks in external gauge fields,” J. Math. Phys. 60, 012107 (2019).
    [Crossref]
  55. H. Chalabi, S. Barik, S. Mittal, T. E. Murphy, M. Hafezi, and E. Waks, “Synthetic gauge field for two-dimensional time-multiplexed quantum random walks,” Phys. Rev. Lett. 123, 150503 (2019).
    [Crossref]
  56. D. H. Dunlap and V. M. Kenkre, “Dynamic localization of a charged particle moving under the influence of an electric field,” Phys. Rev. B 34, 3625–3633 (1986).
    [Crossref]
  57. G. Lenz, R. Parker, M. Wanke, and C. de Sterke, “Dynamical localization and AC Bloch oscillations in periodic optical waveguide arrays,” Opt. Commun. 218, 87–92 (2003).
    [Crossref]
  58. S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
    [Crossref]
  59. R. Iyer, J. S. Aitchison, J. Wan, M. M. Dignam, and C. M. de Sterke, “Exact dynamic localization in curved AlGaAs optical waveguide arrays,” Opt. Express 15, 3212–3223 (2007).
    [Crossref]
  60. A. Joushaghani, R. Iyer, J. K. S. Poon, J. S. Aitchison, C. M. de Sterke, J. Wan, and M. M. Dignam, “Quasi-Bloch oscillations in curved coupled optical waveguides,” Phys. Rev. Lett. 103, 143903 (2009).
    [Crossref]
  61. A. Joushaghani, R. Iyer, J. K. S. Poon, J. S. Aitchison, C. M. de Sterke, J. Wan, and M. M. Dignam, “Generalized exact dynamic localization in curved coupled optical waveguide arrays,” Phys. Rev. Lett. 109, 103901 (2012).
    [Crossref]
  62. L. Yuan and S. Fan, “Three-dimensional dynamic localization of light from a time-dependent effective gauge field for photons,” Phys. Rev. Lett. 114, 243901 (2015).
    [Crossref]
  63. V. V. Orre, E. A. Goldschmidt, A. Deshpande, A. V. Gorshkov, V. Tamma, M. Hafezi, and S. Mittal, “Interference of temporally distinguishable photons using frequency-resolved detection,” Phys. Rev. Lett. 123, 123603 (2019).
    [Crossref]
  64. H. Pichler, S. Choi, P. Zoller, and M. D. Lukin, “Universal photonic quantum computation via time-delayed feedback,” Proc. Natl. Acad. Sci. USA 114, 11362–11367 (2017).
    [Crossref]
  65. H. Chalabi and E. Waks, “Interaction of photons with a coupled atom-cavity system through a bidirectional time-delayed feedback,” Phys. Rev. A 98, 063832 (2018).
    [Crossref]

2019 (6)

A. Dutt, M. Minkov, Q. Lin, L. Yuan, D. A. B. Miller, and S. Fan, “Experimental band structure spectroscopy along a synthetic dimension,” Nat. Commun. 10, 3122 (2019).
[Crossref]

E. Lustig, S. Weimann, Y. Plotnik, Y. Lumer, M. A. Bandres, A. Szameit, and M. Segev, “Photonic topological insulator in synthetic dimensions,” Nature 567, 356–360 (2019).
[Crossref]

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91, 015006 (2019).
[Crossref]

C. Cedzich, T. Geib, A. H. Werner, and R. F. Werner, “Quantum walks in external gauge fields,” J. Math. Phys. 60, 012107 (2019).
[Crossref]

H. Chalabi, S. Barik, S. Mittal, T. E. Murphy, M. Hafezi, and E. Waks, “Synthetic gauge field for two-dimensional time-multiplexed quantum random walks,” Phys. Rev. Lett. 123, 150503 (2019).
[Crossref]

V. V. Orre, E. A. Goldschmidt, A. Deshpande, A. V. Gorshkov, V. Tamma, M. Hafezi, and S. Mittal, “Interference of temporally distinguishable photons using frequency-resolved detection,” Phys. Rev. Lett. 123, 123603 (2019).
[Crossref]

2018 (7)

H. Chalabi and E. Waks, “Interaction of photons with a coupled atom-cavity system through a bidirectional time-delayed feedback,” Phys. Rev. A 98, 063832 (2018).
[Crossref]

A. D’Errico, F. Cardano, M. Maffei, A. Dauphin, R. Barboza, C. Esposito, B. Piccirillo, M. Lewenstein, P. Massignan, and L. Marrucci, “Two-dimensional topological quantum walks in the momentum space of structured light,” Optica 7, 108–114 (2018).
[Crossref]

C. Chen, X. Ding, J. Qin, Y. He, Y.-H. Luo, M.-C. Chen, C. Liu, X.-L. Wang, W.-J. Zhang, H. Li, L.-X. You, Z. Wang, D.-W. Wang, B. C. Sanders, C.-Y. Lu, and J.-W. Pan, “Observation of topologically protected edge states in a photonic two-dimensional quantum walk,” Phys. Rev. Lett. 121, 100502 (2018).
[Crossref]

Q. Lin, X.-Q. Sun, M. Xiao, S.-C. Zhang, and S. Fan, “A three-dimensional photonic topological insulator using a two-dimensional ring resonator lattice with a synthetic frequency dimension,” Sci. Adv. 4, eaat2774 (2018).
[Crossref]

L. Yuan, M. Xiao, Q. Lin, and S. Fan, “Synthetic space with arbitrary dimensions in a few rings undergoing dynamic modulation,” Phys. Rev. B 97, 104105 (2018).
[Crossref]

C. Qin, F. Zhou, Y. Peng, D. Sounas, X. Zhu, B. Wang, J. Dong, X. Zhang, A. Alù, and P. Lu, “Spectrum control through discrete frequency diffraction in the presence of photonic gauge potentials,” Phys. Rev. Lett. 120, 133901 (2018).
[Crossref]

C. Sparrow, E. Martín-López, N. Maraviglia, A. Neville, C. Harrold, J. Carolan, Y. N. Joglekar, T. Hashimoto, N. Matsuda, J. L. O’Brien, D. P. Tew, and A. Laing, “Simulating the vibrational quantum dynamics of molecules using photonics,” Nature 557, 660–667 (2018).
[Crossref]

2017 (4)

B. A. Bell, K. Wang, A. S. Solntsev, D. N. Neshev, A. A. Sukhorukov, and B. J. Eggleton, “Spectral photonic lattices with complex long-range coupling,” Optica 4, 1433–1436 (2017).
[Crossref]

F. Cardano, A. D’Errico, A. Dauphin, M. Maffei, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, L. Marrucci, M. Lewenstein, and P. Massignan, “Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons,” Nat. Commun. 8, 15516 (2017).
[Crossref]

S. Barkhofen, T. Nitsche, F. Elster, L. Lorz, A. Gábris, I. Jex, and C. Silberhorn, “Measuring topological invariants in disordered discrete-time quantum walks,” Phys. Rev. A 96, 033846 (2017).
[Crossref]

H. Pichler, S. Choi, P. Zoller, and M. D. Lukin, “Universal photonic quantum computation via time-delayed feedback,” Proc. Natl. Acad. Sci. USA 114, 11362–11367 (2017).
[Crossref]

2016 (7)

L. A. Bru, M. Hinarejos, F. Silva, G. J. de Valcárcel, and E. Roldán, “Electric quantum walks in two dimensions,” Phys. Rev. A 93, 032333 (2016).
[Crossref]

L. Yuan and S. Fan, “Bloch oscillation and unidirectional translation of frequency in a dynamically modulated ring resonator,” Optica 3, 1014–1018 (2016).
[Crossref]

C. Cedzich and R. F. Werner, “Revivals in quantum walks with a quasiperiodically-time-dependent coin,” Phys. Rev. A 93, 032329 (2016).
[Crossref]

T. Nitsche, F. Elster, J. Novotný, A. Gábris, I. Jex, S. Barkhofen, and C. Silberhorn, “Quantum walks with dynamical control: graph engineering, initial state preparation and state transfer,” New J. Phys. 18, 063017 (2016).
[Crossref]

S. Mittal, S. Ganeshan, J. Fan, A. Vaezi, and M. Hafezi, “Measurement of topological invariants in a 2D photonic system,” Nat. Photonics 10, 180–183 (2016).
[Crossref]

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7, 11439 (2016).
[Crossref]

Q. Lin, M. Xiao, L. Yuan, and S. Fan, “Photonic Weyl point in a two-dimensional resonator lattice with a synthetic frequency dimension,” Nat. Commun. 7, 13731 (2016).
[Crossref]

2015 (4)

F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, “Quantum walks and wavepacket dynamics on a lattice with twisted photons,” Sci. Adv. 1, e1500087 (2015).
[Crossref]

W. Hu, J. C. Pillay, K. Wu, M. Pasek, P. P. Shum, and Y. D. Chong, “Measurement of a topological edge invariant in a microwave network,” Phys. Rev. X 5, 011012 (2015).
[Crossref]

P. Xue, R. Zhang, H. Qin, X. Zhan, Z. H. Bian, J. Li, and B. C. Sanders, “Experimental quantum-walk revival with a time-dependent coin,” Phys. Rev. Lett. 114, 140502 (2015).
[Crossref]

L. Yuan and S. Fan, “Three-dimensional dynamic localization of light from a time-dependent effective gauge field for photons,” Phys. Rev. Lett. 114, 243901 (2015).
[Crossref]

2014 (1)

S. Mittal, J. Fan, S. Faez, A. Migdall, J. M. Taylor, and M. Hafezi, “Topologically robust transport of photons in a synthetic gauge field,” Phys. Rev. Lett. 113, 087403 (2014).
[Crossref]

2013 (5)

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. Corrielli, A. Crespi, G. Della Valle, S. Longhi, and R. Osellame, “Fractional Bloch oscillations in photonic lattices,” Nat. Commun. 4, 1555 (2013).
[Crossref]

S. K. Goyal, F. S. Roux, A. Forbes, and T. Konrad, “Implementing quantum walks using orbital angular momentum of classical light,” Phys. Rev. Lett. 110, 263602 (2013).
[Crossref]

C. Cedzich, T. Rybár, A. H. Werner, A. Alberti, M. Genske, and R. F. Werner, “Propagation of quantum walks in electric fields,” Phys. Rev. Lett. 111, 160601 (2013).
[Crossref]

M. Genske, W. Alt, A. Steffen, A. H. Werner, R. F. Werner, D. Meschede, and A. Alberti, “Electric quantum walks with individual atoms,” Phys. Rev. Lett. 110, 190601 (2013).
[Crossref]

2012 (5)

A. Joushaghani, R. Iyer, J. K. S. Poon, J. S. Aitchison, C. M. de Sterke, J. Wan, and M. M. Dignam, “Generalized exact dynamic localization in curved coupled optical waveguide arrays,” Phys. Rev. Lett. 109, 103901 (2012).
[Crossref]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref]

A. Aspuru-Guzik and P. Walther, “Photonic quantum simulators,” Nat. Phys. 8, 285–291 (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]

A. Schreiber, A. Gabris, P. P. Rohde, K. Laiho, M. Stefanak, V. Potocek, C. Hamilton, I. Jex, and C. Silberhorn, “A 2D quantum walk simulation of two-particle dynamics,” Science 336, 55–58 (2012).
[Crossref]

2011 (3)

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, I. Jex, and C. Silberhorn, “Decoherence and disorder in quantum walks: from ballistic spread to localization,” Phys. Rev. Lett. 106, 180403 (2011).
[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]

X.-S. Ma, B. Dakic, W. Naylor, A. Zeilinger, and P. Walther, “Quantum simulation of the wavefunction to probe frustrated Heisenberg spin systems,” Nat. Phys. 7, 399–405 (2011).
[Crossref]

2010 (5)

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nat. Chem. 2, 106–111 (2010).
[Crossref]

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref]

E. Haller, R. Hart, M. J. Mark, J. G. Danzl, L. Reichsöllner, and H.-C. Nägerl, “Inducing transport in a dissipation-free lattice with super Bloch oscillations,” Phys. Rev. Lett. 104, 200403 (2010).
[Crossref]

Y. Bromberg, Y. Lahini, and Y. Silberberg, “Bloch oscillations of path-entangled photons,” Phys. Rev. Lett. 105, 263604 (2010).
[Crossref]

M. Levy and P. Kumar, “Nonreciprocal Bloch oscillations in magneto-optic waveguide arrays,” Opt. Lett. 35, 3147–3149 (2010).
[Crossref]

2009 (3)

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, “Bloch-Zener oscillations in binary superlattices,” Phys. Rev. Lett. 102, 076802 (2009).
[Crossref]

C. Bersch, G. Onishchukov, and U. Peschel, “Experimental observation of spectral Bloch oscillations,” Opt. Lett. 34, 2372–2374 (2009).
[Crossref]

A. Joushaghani, R. Iyer, J. K. S. Poon, J. S. Aitchison, C. M. de Sterke, J. Wan, and M. M. Dignam, “Quasi-Bloch oscillations in curved coupled optical waveguides,” Phys. Rev. Lett. 103, 143903 (2009).
[Crossref]

2008 (1)

U. Peschel, C. Bersch, and G. Onishchukov, “Discreteness in time,” Open Phys. 6, 619–627 (2008).
[Crossref]

2007 (2)

2006 (3)

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[Crossref]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[Crossref]

S. Longhi, “Optical Zener-Bloch oscillations in binary waveguide arrays,” Europhys. Lett. 76, 416–421 (2006).
[Crossref]

2004 (1)

D. Witthaut, F. Keck, H. J. Korsch, and S. Mossmann, “Bloch oscillations in two-dimensional lattices,” New J. Phys. 6, 41 (2004).
[Crossref]

2003 (1)

G. Lenz, R. Parker, M. Wanke, and C. de Sterke, “Dynamical localization and AC Bloch oscillations in periodic optical waveguide arrays,” Opt. Commun. 218, 87–92 (2003).
[Crossref]

1999 (4)

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[Crossref]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[Crossref]

G. Lenz, I. Talanina, and C. M. de Sterke, “Bloch Oscillations in an array of curved optical waveguides,” Phys. Rev. Lett. 83, 963–966 (1999).
[Crossref]

D. Bouwmeester, I. Marzoli, G. P. Karman, W. Schleich, and J. P. Woerdman, “Optical Galton board,” Phys. Rev. A 61, 013410 (1999).
[Crossref]

1996 (1)

M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, “Bloch oscillations of atoms in an optical potential,” Phys. Rev. Lett. 76, 4508–4511 (1996).
[Crossref]

1986 (1)

D. H. Dunlap and V. M. Kenkre, “Dynamic localization of a charged particle moving under the influence of an electric field,” Phys. Rev. B 34, 3625–3633 (1986).
[Crossref]

1929 (1)

F. Bloch, “Über die Quantenmechanik der Elektronen in Kristallgittern,” Z. Phys. 52, 555–600 (1929).
[Crossref]

Aitchison, J. S.

A. Joushaghani, R. Iyer, J. K. S. Poon, J. S. Aitchison, C. M. de Sterke, J. Wan, and M. M. Dignam, “Generalized exact dynamic localization in curved coupled optical waveguide arrays,” Phys. Rev. Lett. 109, 103901 (2012).
[Crossref]

A. Joushaghani, R. Iyer, J. K. S. Poon, J. S. Aitchison, C. M. de Sterke, J. Wan, and M. M. Dignam, “Quasi-Bloch oscillations in curved coupled optical waveguides,” Phys. Rev. Lett. 103, 143903 (2009).
[Crossref]

R. Iyer, J. S. Aitchison, J. Wan, M. M. Dignam, and C. M. de Sterke, “Exact dynamic localization in curved AlGaAs optical waveguide arrays,” Opt. Express 15, 3212–3223 (2007).
[Crossref]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[Crossref]

Alberti, A.

C. Cedzich, T. Rybár, A. H. Werner, A. Alberti, M. Genske, and R. F. Werner, “Propagation of quantum walks in electric fields,” Phys. Rev. Lett. 111, 160601 (2013).
[Crossref]

M. Genske, W. Alt, A. Steffen, A. H. Werner, R. F. Werner, D. Meschede, and A. Alberti, “Electric quantum walks with individual atoms,” Phys. Rev. Lett. 110, 190601 (2013).
[Crossref]

Almeida, M. P.

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nat. Chem. 2, 106–111 (2010).
[Crossref]

Alt, W.

M. Genske, W. Alt, A. Steffen, A. H. Werner, R. F. Werner, D. Meschede, and A. Alberti, “Electric quantum walks with individual atoms,” Phys. Rev. Lett. 110, 190601 (2013).
[Crossref]

Alù, A.

C. Qin, F. Zhou, Y. Peng, D. Sounas, X. Zhu, B. Wang, J. Dong, X. Zhang, A. Alù, and P. Lu, “Spectrum control through discrete frequency diffraction in the presence of photonic gauge potentials,” Phys. Rev. Lett. 120, 133901 (2018).
[Crossref]

Amo, A.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91, 015006 (2019).
[Crossref]

Andersson, E.

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref]

Aspuru-Guzik, A.

A. Aspuru-Guzik and P. Walther, “Photonic quantum simulators,” Nat. Phys. 8, 285–291 (2012).
[Crossref]

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nat. Chem. 2, 106–111 (2010).
[Crossref]

Bandres, M. A.

E. Lustig, S. Weimann, Y. Plotnik, Y. Lumer, M. A. Bandres, A. Szameit, and M. Segev, “Photonic topological insulator in synthetic dimensions,” Nature 567, 356–360 (2019).
[Crossref]

Barbieri, M.

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nat. Chem. 2, 106–111 (2010).
[Crossref]

Barboza, R.

Barik, S.

H. Chalabi, S. Barik, S. Mittal, T. E. Murphy, M. Hafezi, and E. Waks, “Synthetic gauge field for two-dimensional time-multiplexed quantum random walks,” Phys. Rev. Lett. 123, 150503 (2019).
[Crossref]

Barkhofen, S.

S. Barkhofen, T. Nitsche, F. Elster, L. Lorz, A. Gábris, I. Jex, and C. Silberhorn, “Measuring topological invariants in disordered discrete-time quantum walks,” Phys. Rev. A 96, 033846 (2017).
[Crossref]

T. Nitsche, F. Elster, J. Novotný, A. Gábris, I. Jex, S. Barkhofen, and C. Silberhorn, “Quantum walks with dynamical control: graph engineering, initial state preparation and state transfer,” New J. Phys. 18, 063017 (2016).
[Crossref]

Bell, B. A.

Ben Dahan, M.

M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, “Bloch oscillations of atoms in an optical potential,” Phys. Rev. Lett. 76, 4508–4511 (1996).
[Crossref]

Bersch, C.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref]

C. Bersch, G. Onishchukov, and U. Peschel, “Experimental observation of spectral Bloch oscillations,” Opt. Lett. 34, 2372–2374 (2009).
[Crossref]

U. Peschel, C. Bersch, and G. Onishchukov, “Discreteness in time,” Open Phys. 6, 619–627 (2008).
[Crossref]

Biamonte, J. D.

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nat. Chem. 2, 106–111 (2010).
[Crossref]

Bian, Z. H.

P. Xue, R. Zhang, H. Qin, X. Zhan, Z. H. Bian, J. Li, and B. C. Sanders, “Experimental quantum-walk revival with a time-dependent coin,” Phys. Rev. Lett. 114, 140502 (2015).
[Crossref]

Bloch, F.

F. Bloch, “Über die Quantenmechanik der Elektronen in Kristallgittern,” Z. Phys. 52, 555–600 (1929).
[Crossref]

Bouwmeester, D.

D. Bouwmeester, I. Marzoli, G. P. Karman, W. Schleich, and J. P. Woerdman, “Optical Galton board,” Phys. Rev. A 61, 013410 (1999).
[Crossref]

Boyd, R. W.

F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, “Quantum walks and wavepacket dynamics on a lattice with twisted photons,” Sci. Adv. 1, e1500087 (2015).
[Crossref]

Bräuer, A.

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[Crossref]

Bromberg, Y.

Y. Bromberg, Y. Lahini, and Y. Silberberg, “Bloch oscillations of path-entangled photons,” Phys. Rev. Lett. 105, 263604 (2010).
[Crossref]

Bru, L. A.

L. A. Bru, M. Hinarejos, F. Silva, G. J. de Valcárcel, and E. Roldán, “Electric quantum walks in two dimensions,” Phys. Rev. A 93, 032333 (2016).
[Crossref]

Cardano, F.

A. D’Errico, F. Cardano, M. Maffei, A. Dauphin, R. Barboza, C. Esposito, B. Piccirillo, M. Lewenstein, P. Massignan, and L. Marrucci, “Two-dimensional topological quantum walks in the momentum space of structured light,” Optica 7, 108–114 (2018).
[Crossref]

F. Cardano, A. D’Errico, A. Dauphin, M. Maffei, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, L. Marrucci, M. Lewenstein, and P. Massignan, “Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons,” Nat. Commun. 8, 15516 (2017).
[Crossref]

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7, 11439 (2016).
[Crossref]

F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, “Quantum walks and wavepacket dynamics on a lattice with twisted photons,” Sci. Adv. 1, e1500087 (2015).
[Crossref]

Carolan, J.

C. Sparrow, E. Martín-López, N. Maraviglia, A. Neville, C. Harrold, J. Carolan, Y. N. Joglekar, T. Hashimoto, N. Matsuda, J. L. O’Brien, D. P. Tew, and A. Laing, “Simulating the vibrational quantum dynamics of molecules using photonics,” Nature 557, 660–667 (2018).
[Crossref]

Carusotto, I.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91, 015006 (2019).
[Crossref]

Cassemiro, K. N.

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, I. Jex, and C. Silberhorn, “Decoherence and disorder in quantum walks: from ballistic spread to localization,” Phys. Rev. Lett. 106, 180403 (2011).
[Crossref]

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref]

Castin, Y.

M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, “Bloch oscillations of atoms in an optical potential,” Phys. Rev. Lett. 76, 4508–4511 (1996).
[Crossref]

Cataudella, V.

F. Cardano, A. D’Errico, A. Dauphin, M. Maffei, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, L. Marrucci, M. Lewenstein, and P. Massignan, “Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons,” Nat. Commun. 8, 15516 (2017).
[Crossref]

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7, 11439 (2016).
[Crossref]

Cedzich, C.

C. Cedzich, T. Geib, A. H. Werner, and R. F. Werner, “Quantum walks in external gauge fields,” J. Math. Phys. 60, 012107 (2019).
[Crossref]

C. Cedzich and R. F. Werner, “Revivals in quantum walks with a quasiperiodically-time-dependent coin,” Phys. Rev. A 93, 032329 (2016).
[Crossref]

C. Cedzich, T. Rybár, A. H. Werner, A. Alberti, M. Genske, and R. F. Werner, “Propagation of quantum walks in electric fields,” Phys. Rev. Lett. 111, 160601 (2013).
[Crossref]

Chalabi, H.

H. Chalabi, S. Barik, S. Mittal, T. E. Murphy, M. Hafezi, and E. Waks, “Synthetic gauge field for two-dimensional time-multiplexed quantum random walks,” Phys. Rev. Lett. 123, 150503 (2019).
[Crossref]

H. Chalabi and E. Waks, “Interaction of photons with a coupled atom-cavity system through a bidirectional time-delayed feedback,” Phys. Rev. A 98, 063832 (2018).
[Crossref]

Chen, C.

C. Chen, X. Ding, J. Qin, Y. He, Y.-H. Luo, M.-C. Chen, C. Liu, X.-L. Wang, W.-J. Zhang, H. Li, L.-X. You, Z. Wang, D.-W. Wang, B. C. Sanders, C.-Y. Lu, and J.-W. Pan, “Observation of topologically protected edge states in a photonic two-dimensional quantum walk,” Phys. Rev. Lett. 121, 100502 (2018).
[Crossref]

Chen, M.-C.

C. Chen, X. Ding, J. Qin, Y. He, Y.-H. Luo, M.-C. Chen, C. Liu, X.-L. Wang, W.-J. Zhang, H. Li, L.-X. You, Z. Wang, D.-W. Wang, B. C. Sanders, C.-Y. Lu, and J.-W. Pan, “Observation of topologically protected edge states in a photonic two-dimensional quantum walk,” Phys. Rev. Lett. 121, 100502 (2018).
[Crossref]

Choi, S.

H. Pichler, S. Choi, P. Zoller, and M. D. Lukin, “Universal photonic quantum computation via time-delayed feedback,” Proc. Natl. Acad. Sci. USA 114, 11362–11367 (2017).
[Crossref]

Chong, Y. D.

W. Hu, J. C. Pillay, K. Wu, M. Pasek, P. P. Shum, and Y. D. Chong, “Measurement of a topological edge invariant in a microwave network,” Phys. Rev. X 5, 011012 (2015).
[Crossref]

Christodoulides, D. N.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref]

Cianci, E.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[Crossref]

Corrielli, G.

G. Corrielli, A. Crespi, G. Della Valle, S. Longhi, and R. Osellame, “Fractional Bloch oscillations in photonic lattices,” Nat. Commun. 4, 1555 (2013).
[Crossref]

Crespi, A.

G. Corrielli, A. Crespi, G. Della Valle, S. Longhi, and R. Osellame, “Fractional Bloch oscillations in photonic lattices,” Nat. Commun. 4, 1555 (2013).
[Crossref]

D’Errico, A.

A. D’Errico, F. Cardano, M. Maffei, A. Dauphin, R. Barboza, C. Esposito, B. Piccirillo, M. Lewenstein, P. Massignan, and L. Marrucci, “Two-dimensional topological quantum walks in the momentum space of structured light,” Optica 7, 108–114 (2018).
[Crossref]

F. Cardano, A. D’Errico, A. Dauphin, M. Maffei, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, L. Marrucci, M. Lewenstein, and P. Massignan, “Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons,” Nat. Commun. 8, 15516 (2017).
[Crossref]

Dakic, B.

X.-S. Ma, B. Dakic, W. Naylor, A. Zeilinger, and P. Walther, “Quantum simulation of the wavefunction to probe frustrated Heisenberg spin systems,” Nat. Phys. 7, 399–405 (2011).
[Crossref]

Dannberg, P.

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[Crossref]

Danzl, J. G.

E. Haller, R. Hart, M. J. Mark, J. G. Danzl, L. Reichsöllner, and H.-C. Nägerl, “Inducing transport in a dissipation-free lattice with super Bloch oscillations,” Phys. Rev. Lett. 104, 200403 (2010).
[Crossref]

Dauphin, A.

A. D’Errico, F. Cardano, M. Maffei, A. Dauphin, R. Barboza, C. Esposito, B. Piccirillo, M. Lewenstein, P. Massignan, and L. Marrucci, “Two-dimensional topological quantum walks in the momentum space of structured light,” Optica 7, 108–114 (2018).
[Crossref]

F. Cardano, A. D’Errico, A. Dauphin, M. Maffei, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, L. Marrucci, M. Lewenstein, and P. Massignan, “Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons,” Nat. Commun. 8, 15516 (2017).
[Crossref]

De Filippis, G.

F. Cardano, A. D’Errico, A. Dauphin, M. Maffei, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, L. Marrucci, M. Lewenstein, and P. Massignan, “Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons,” Nat. Commun. 8, 15516 (2017).
[Crossref]

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7, 11439 (2016).
[Crossref]

de Lisio, C.

F. Cardano, A. D’Errico, A. Dauphin, M. Maffei, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, L. Marrucci, M. Lewenstein, and P. Massignan, “Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons,” Nat. Commun. 8, 15516 (2017).
[Crossref]

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7, 11439 (2016).
[Crossref]

F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, “Quantum walks and wavepacket dynamics on a lattice with twisted photons,” Sci. Adv. 1, e1500087 (2015).
[Crossref]

de Sterke, C.

G. Lenz, R. Parker, M. Wanke, and C. de Sterke, “Dynamical localization and AC Bloch oscillations in periodic optical waveguide arrays,” Opt. Commun. 218, 87–92 (2003).
[Crossref]

de Sterke, C. M.

A. Joushaghani, R. Iyer, J. K. S. Poon, J. S. Aitchison, C. M. de Sterke, J. Wan, and M. M. Dignam, “Generalized exact dynamic localization in curved coupled optical waveguide arrays,” Phys. Rev. Lett. 109, 103901 (2012).
[Crossref]

A. Joushaghani, R. Iyer, J. K. S. Poon, J. S. Aitchison, C. M. de Sterke, J. Wan, and M. M. Dignam, “Quasi-Bloch oscillations in curved coupled optical waveguides,” Phys. Rev. Lett. 103, 143903 (2009).
[Crossref]

R. Iyer, J. S. Aitchison, J. Wan, M. M. Dignam, and C. M. de Sterke, “Exact dynamic localization in curved AlGaAs optical waveguide arrays,” Opt. Express 15, 3212–3223 (2007).
[Crossref]

G. Lenz, I. Talanina, and C. M. de Sterke, “Bloch Oscillations in an array of curved optical waveguides,” Phys. Rev. Lett. 83, 963–966 (1999).
[Crossref]

de Valcárcel, G. J.

L. A. Bru, M. Hinarejos, F. Silva, G. J. de Valcárcel, and E. Roldán, “Electric quantum walks in two dimensions,” Phys. Rev. A 93, 032333 (2016).
[Crossref]

Della Valle, G.

G. Corrielli, A. Crespi, G. Della Valle, S. Longhi, and R. Osellame, “Fractional Bloch oscillations in photonic lattices,” Nat. Commun. 4, 1555 (2013).
[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]

Deshpande, A.

V. V. Orre, E. A. Goldschmidt, A. Deshpande, A. V. Gorshkov, V. Tamma, M. Hafezi, and S. Mittal, “Interference of temporally distinguishable photons using frequency-resolved detection,” Phys. Rev. Lett. 123, 123603 (2019).
[Crossref]

Desyatnikov, A. S.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[Crossref]

Dignam, M. M.

A. Joushaghani, R. Iyer, J. K. S. Poon, J. S. Aitchison, C. M. de Sterke, J. Wan, and M. M. Dignam, “Generalized exact dynamic localization in curved coupled optical waveguide arrays,” Phys. Rev. Lett. 109, 103901 (2012).
[Crossref]

A. Joushaghani, R. Iyer, J. K. S. Poon, J. S. Aitchison, C. M. de Sterke, J. Wan, and M. M. Dignam, “Quasi-Bloch oscillations in curved coupled optical waveguides,” Phys. Rev. Lett. 103, 143903 (2009).
[Crossref]

R. Iyer, J. S. Aitchison, J. Wan, M. M. Dignam, and C. M. de Sterke, “Exact dynamic localization in curved AlGaAs optical waveguide arrays,” Opt. Express 15, 3212–3223 (2007).
[Crossref]

Ding, X.

C. Chen, X. Ding, J. Qin, Y. He, Y.-H. Luo, M.-C. Chen, C. Liu, X.-L. Wang, W.-J. Zhang, H. Li, L.-X. You, Z. Wang, D.-W. Wang, B. C. Sanders, C.-Y. Lu, and J.-W. Pan, “Observation of topologically protected edge states in a photonic two-dimensional quantum walk,” Phys. Rev. Lett. 121, 100502 (2018).
[Crossref]

Dong, J.

C. Qin, F. Zhou, Y. Peng, D. Sounas, X. Zhu, B. Wang, J. Dong, X. Zhang, A. Alù, and P. Lu, “Spectrum control through discrete frequency diffraction in the presence of photonic gauge potentials,” Phys. Rev. Lett. 120, 133901 (2018).
[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]

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, “Bloch-Zener oscillations in binary superlattices,” Phys. Rev. Lett. 102, 076802 (2009).
[Crossref]

Dunlap, D. H.

D. H. Dunlap and V. M. Kenkre, “Dynamic localization of a charged particle moving under the influence of an electric field,” Phys. Rev. B 34, 3625–3633 (1986).
[Crossref]

Dutt, A.

A. Dutt, M. Minkov, Q. Lin, L. Yuan, D. A. B. Miller, and S. Fan, “Experimental band structure spectroscopy along a synthetic dimension,” Nat. Commun. 10, 3122 (2019).
[Crossref]

Eggleton, B. J.

Eisenberg, H. S.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[Crossref]

Elflein, W.

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[Crossref]

Elster, F.

S. Barkhofen, T. Nitsche, F. Elster, L. Lorz, A. Gábris, I. Jex, and C. Silberhorn, “Measuring topological invariants in disordered discrete-time quantum walks,” Phys. Rev. A 96, 033846 (2017).
[Crossref]

T. Nitsche, F. Elster, J. Novotný, A. Gábris, I. Jex, S. Barkhofen, and C. Silberhorn, “Quantum walks with dynamical control: graph engineering, initial state preparation and state transfer,” New J. Phys. 18, 063017 (2016).
[Crossref]

Esposito, C.

Faez, S.

S. Mittal, J. Fan, S. Faez, A. Migdall, J. M. Taylor, and M. Hafezi, “Topologically robust transport of photons in a synthetic gauge field,” Phys. Rev. Lett. 113, 087403 (2014).
[Crossref]

Fan, J.

S. Mittal, S. Ganeshan, J. Fan, A. Vaezi, and M. Hafezi, “Measurement of topological invariants in a 2D photonic system,” Nat. Photonics 10, 180–183 (2016).
[Crossref]

S. Mittal, J. Fan, S. Faez, A. Migdall, J. M. Taylor, and M. Hafezi, “Topologically robust transport of photons in a synthetic gauge field,” Phys. Rev. Lett. 113, 087403 (2014).
[Crossref]

Fan, S.

A. Dutt, M. Minkov, Q. Lin, L. Yuan, D. A. B. Miller, and S. Fan, “Experimental band structure spectroscopy along a synthetic dimension,” Nat. Commun. 10, 3122 (2019).
[Crossref]

Q. Lin, X.-Q. Sun, M. Xiao, S.-C. Zhang, and S. Fan, “A three-dimensional photonic topological insulator using a two-dimensional ring resonator lattice with a synthetic frequency dimension,” Sci. Adv. 4, eaat2774 (2018).
[Crossref]

L. Yuan, M. Xiao, Q. Lin, and S. Fan, “Synthetic space with arbitrary dimensions in a few rings undergoing dynamic modulation,” Phys. Rev. B 97, 104105 (2018).
[Crossref]

Q. Lin, M. Xiao, L. Yuan, and S. Fan, “Photonic Weyl point in a two-dimensional resonator lattice with a synthetic frequency dimension,” Nat. Commun. 7, 13731 (2016).
[Crossref]

L. Yuan and S. Fan, “Bloch oscillation and unidirectional translation of frequency in a dynamically modulated ring resonator,” Optica 3, 1014–1018 (2016).
[Crossref]

L. Yuan and S. Fan, “Three-dimensional dynamic localization of light from a time-dependent effective gauge field for photons,” Phys. Rev. Lett. 114, 243901 (2015).
[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]

Foglietti, V.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[Crossref]

Forbes, A.

S. K. Goyal, F. S. Roux, A. Forbes, and T. Konrad, “Implementing quantum walks using orbital angular momentum of classical light,” Phys. Rev. Lett. 110, 263602 (2013).
[Crossref]

Gabris, A.

A. Schreiber, A. Gabris, P. P. Rohde, K. Laiho, M. Stefanak, V. Potocek, C. Hamilton, I. Jex, and C. Silberhorn, “A 2D quantum walk simulation of two-particle dynamics,” Science 336, 55–58 (2012).
[Crossref]

Gábris, A.

S. Barkhofen, T. Nitsche, F. Elster, L. Lorz, A. Gábris, I. Jex, and C. Silberhorn, “Measuring topological invariants in disordered discrete-time quantum walks,” Phys. Rev. A 96, 033846 (2017).
[Crossref]

T. Nitsche, F. Elster, J. Novotný, A. Gábris, I. Jex, S. Barkhofen, and C. Silberhorn, “Quantum walks with dynamical control: graph engineering, initial state preparation and state transfer,” New J. Phys. 18, 063017 (2016).
[Crossref]

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, I. Jex, and C. Silberhorn, “Decoherence and disorder in quantum walks: from ballistic spread to localization,” Phys. Rev. Lett. 106, 180403 (2011).
[Crossref]

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref]

Ganeshan, S.

S. Mittal, S. Ganeshan, J. Fan, A. Vaezi, and M. Hafezi, “Measurement of topological invariants in a 2D photonic system,” Nat. Photonics 10, 180–183 (2016).
[Crossref]

Geib, T.

C. Cedzich, T. Geib, A. H. Werner, and R. F. Werner, “Quantum walks in external gauge fields,” J. Math. Phys. 60, 012107 (2019).
[Crossref]

Genske, M.

M. Genske, W. Alt, A. Steffen, A. H. Werner, R. F. Werner, D. Meschede, and A. Alberti, “Electric quantum walks with individual atoms,” Phys. Rev. Lett. 110, 190601 (2013).
[Crossref]

C. Cedzich, T. Rybár, A. H. Werner, A. Alberti, M. Genske, and R. F. Werner, “Propagation of quantum walks in electric fields,” Phys. Rev. Lett. 111, 160601 (2013).
[Crossref]

Gillett, G. G.

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nat. Chem. 2, 106–111 (2010).
[Crossref]

Goggin, M. E.

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nat. Chem. 2, 106–111 (2010).
[Crossref]

Goldman, N.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91, 015006 (2019).
[Crossref]

Goldschmidt, E. A.

V. V. Orre, E. A. Goldschmidt, A. Deshpande, A. V. Gorshkov, V. Tamma, M. Hafezi, and S. Mittal, “Interference of temporally distinguishable photons using frequency-resolved detection,” Phys. Rev. Lett. 123, 123603 (2019).
[Crossref]

Gorshkov, A. V.

V. V. Orre, E. A. Goldschmidt, A. Deshpande, A. V. Gorshkov, V. Tamma, M. Hafezi, and S. Mittal, “Interference of temporally distinguishable photons using frequency-resolved detection,” Phys. Rev. Lett. 123, 123603 (2019).
[Crossref]

Goyal, S. K.

S. K. Goyal, F. S. Roux, A. Forbes, and T. Konrad, “Implementing quantum walks using orbital angular momentum of classical light,” Phys. Rev. Lett. 110, 263602 (2013).
[Crossref]

Hafezi, M.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91, 015006 (2019).
[Crossref]

V. V. Orre, E. A. Goldschmidt, A. Deshpande, A. V. Gorshkov, V. Tamma, M. Hafezi, and S. Mittal, “Interference of temporally distinguishable photons using frequency-resolved detection,” Phys. Rev. Lett. 123, 123603 (2019).
[Crossref]

H. Chalabi, S. Barik, S. Mittal, T. E. Murphy, M. Hafezi, and E. Waks, “Synthetic gauge field for two-dimensional time-multiplexed quantum random walks,” Phys. Rev. Lett. 123, 150503 (2019).
[Crossref]

S. Mittal, S. Ganeshan, J. Fan, A. Vaezi, and M. Hafezi, “Measurement of topological invariants in a 2D photonic system,” Nat. Photonics 10, 180–183 (2016).
[Crossref]

S. Mittal, J. Fan, S. Faez, A. Migdall, J. M. Taylor, and M. Hafezi, “Topologically robust transport of photons in a synthetic gauge field,” Phys. Rev. Lett. 113, 087403 (2014).
[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]

Haller, E.

E. Haller, R. Hart, M. J. Mark, J. G. Danzl, L. Reichsöllner, and H.-C. Nägerl, “Inducing transport in a dissipation-free lattice with super Bloch oscillations,” Phys. Rev. Lett. 104, 200403 (2010).
[Crossref]

Hamilton, C.

A. Schreiber, A. Gabris, P. P. Rohde, K. Laiho, M. Stefanak, V. Potocek, C. Hamilton, I. Jex, and C. Silberhorn, “A 2D quantum walk simulation of two-particle dynamics,” Science 336, 55–58 (2012).
[Crossref]

Harrold, C.

C. Sparrow, E. Martín-López, N. Maraviglia, A. Neville, C. Harrold, J. Carolan, Y. N. Joglekar, T. Hashimoto, N. Matsuda, J. L. O’Brien, D. P. Tew, and A. Laing, “Simulating the vibrational quantum dynamics of molecules using photonics,” Nature 557, 660–667 (2018).
[Crossref]

Hart, R.

E. Haller, R. Hart, M. J. Mark, J. G. Danzl, L. Reichsöllner, and H.-C. Nägerl, “Inducing transport in a dissipation-free lattice with super Bloch oscillations,” Phys. Rev. Lett. 104, 200403 (2010).
[Crossref]

Hashimoto, T.

C. Sparrow, E. Martín-López, N. Maraviglia, A. Neville, C. Harrold, J. Carolan, Y. N. Joglekar, T. Hashimoto, N. Matsuda, J. L. O’Brien, D. P. Tew, and A. Laing, “Simulating the vibrational quantum dynamics of molecules using photonics,” Nature 557, 660–667 (2018).
[Crossref]

He, Y.

C. Chen, X. Ding, J. Qin, Y. He, Y.-H. Luo, M.-C. Chen, C. Liu, X.-L. Wang, W.-J. Zhang, H. Li, L.-X. You, Z. Wang, D.-W. Wang, B. C. Sanders, C.-Y. Lu, and J.-W. Pan, “Observation of topologically protected edge states in a photonic two-dimensional quantum walk,” Phys. Rev. Lett. 121, 100502 (2018).
[Crossref]

Heinrich, M.

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, “Bloch-Zener oscillations in binary superlattices,” Phys. Rev. Lett. 102, 076802 (2009).
[Crossref]

Hinarejos, M.

L. A. Bru, M. Hinarejos, F. Silva, G. J. de Valcárcel, and E. Roldán, “Electric quantum walks in two dimensions,” Phys. Rev. A 93, 032333 (2016).
[Crossref]

Hu, W.

W. Hu, J. C. Pillay, K. Wu, M. Pasek, P. P. Shum, and Y. D. Chong, “Measurement of a topological edge invariant in a microwave network,” Phys. Rev. X 5, 011012 (2015).
[Crossref]

Iyer, R.

A. Joushaghani, R. Iyer, J. K. S. Poon, J. S. Aitchison, C. M. de Sterke, J. Wan, and M. M. Dignam, “Generalized exact dynamic localization in curved coupled optical waveguide arrays,” Phys. Rev. Lett. 109, 103901 (2012).
[Crossref]

A. Joushaghani, R. Iyer, J. K. S. Poon, J. S. Aitchison, C. M. de Sterke, J. Wan, and M. M. Dignam, “Quasi-Bloch oscillations in curved coupled optical waveguides,” Phys. Rev. Lett. 103, 143903 (2009).
[Crossref]

R. Iyer, J. S. Aitchison, J. Wan, M. M. Dignam, and C. M. de Sterke, “Exact dynamic localization in curved AlGaAs optical waveguide arrays,” Opt. Express 15, 3212–3223 (2007).
[Crossref]

Jex, I.

S. Barkhofen, T. Nitsche, F. Elster, L. Lorz, A. Gábris, I. Jex, and C. Silberhorn, “Measuring topological invariants in disordered discrete-time quantum walks,” Phys. Rev. A 96, 033846 (2017).
[Crossref]

T. Nitsche, F. Elster, J. Novotný, A. Gábris, I. Jex, S. Barkhofen, and C. Silberhorn, “Quantum walks with dynamical control: graph engineering, initial state preparation and state transfer,” New J. Phys. 18, 063017 (2016).
[Crossref]

A. Schreiber, A. Gabris, P. P. Rohde, K. Laiho, M. Stefanak, V. Potocek, C. Hamilton, I. Jex, and C. Silberhorn, “A 2D quantum walk simulation of two-particle dynamics,” Science 336, 55–58 (2012).
[Crossref]

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, I. Jex, and C. Silberhorn, “Decoherence and disorder in quantum walks: from ballistic spread to localization,” Phys. Rev. Lett. 106, 180403 (2011).
[Crossref]

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref]

Joglekar, Y. N.

C. Sparrow, E. Martín-López, N. Maraviglia, A. Neville, C. Harrold, J. Carolan, Y. N. Joglekar, T. Hashimoto, N. Matsuda, J. L. O’Brien, D. P. Tew, and A. Laing, “Simulating the vibrational quantum dynamics of molecules using photonics,” Nature 557, 660–667 (2018).
[Crossref]

Joushaghani, A.

A. Joushaghani, R. Iyer, J. K. S. Poon, J. S. Aitchison, C. M. de Sterke, J. Wan, and M. M. Dignam, “Generalized exact dynamic localization in curved coupled optical waveguide arrays,” Phys. Rev. Lett. 109, 103901 (2012).
[Crossref]

A. Joushaghani, R. Iyer, J. K. S. Poon, J. S. Aitchison, C. M. de Sterke, J. Wan, and M. M. Dignam, “Quasi-Bloch oscillations in curved coupled optical waveguides,” Phys. Rev. Lett. 103, 143903 (2009).
[Crossref]

Karimi, E.

F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, “Quantum walks and wavepacket dynamics on a lattice with twisted photons,” Sci. Adv. 1, e1500087 (2015).
[Crossref]

Karman, G. P.

D. Bouwmeester, I. Marzoli, G. P. Karman, W. Schleich, and J. P. Woerdman, “Optical Galton board,” Phys. Rev. A 61, 013410 (1999).
[Crossref]

Kassal, I.

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nat. Chem. 2, 106–111 (2010).
[Crossref]

Keck, F.

D. Witthaut, F. Keck, H. J. Korsch, and S. Mossmann, “Bloch oscillations in two-dimensional lattices,” New J. Phys. 6, 41 (2004).
[Crossref]

Kenkre, V. M.

D. H. Dunlap and V. M. Kenkre, “Dynamic localization of a charged particle moving under the influence of an electric field,” Phys. Rev. B 34, 3625–3633 (1986).
[Crossref]

Kivshar, Y. S.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[Crossref]

Konrad, T.

S. K. Goyal, F. S. Roux, A. Forbes, and T. Konrad, “Implementing quantum walks using orbital angular momentum of classical light,” Phys. Rev. Lett. 110, 263602 (2013).
[Crossref]

Korsch, H. J.

D. Witthaut, F. Keck, H. J. Korsch, and S. Mossmann, “Bloch oscillations in two-dimensional lattices,” New J. Phys. 6, 41 (2004).
[Crossref]

Krolikowski, W.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[Crossref]

Kumar, P.

Lahini, Y.

Y. Bromberg, Y. Lahini, and Y. Silberberg, “Bloch oscillations of path-entangled photons,” Phys. Rev. Lett. 105, 263604 (2010).
[Crossref]

Laiho, K.

A. Schreiber, A. Gabris, P. P. Rohde, K. Laiho, M. Stefanak, V. Potocek, C. Hamilton, I. Jex, and C. Silberhorn, “A 2D quantum walk simulation of two-particle dynamics,” Science 336, 55–58 (2012).
[Crossref]

Laing, A.

C. Sparrow, E. Martín-López, N. Maraviglia, A. Neville, C. Harrold, J. Carolan, Y. N. Joglekar, T. Hashimoto, N. Matsuda, J. L. O’Brien, D. P. Tew, and A. Laing, “Simulating the vibrational quantum dynamics of molecules using photonics,” Nature 557, 660–667 (2018).
[Crossref]

Lanyon, B. P.

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nat. Chem. 2, 106–111 (2010).
[Crossref]

Laporta, P.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[Crossref]

Lederer, F.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[Crossref]

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[Crossref]

Lenz, G.

G. Lenz, R. Parker, M. Wanke, and C. de Sterke, “Dynamical localization and AC Bloch oscillations in periodic optical waveguide arrays,” Opt. Commun. 218, 87–92 (2003).
[Crossref]

G. Lenz, I. Talanina, and C. M. de Sterke, “Bloch Oscillations in an array of curved optical waveguides,” Phys. Rev. Lett. 83, 963–966 (1999).
[Crossref]

Levy, M.

Lewenstein, M.

A. D’Errico, F. Cardano, M. Maffei, A. Dauphin, R. Barboza, C. Esposito, B. Piccirillo, M. Lewenstein, P. Massignan, and L. Marrucci, “Two-dimensional topological quantum walks in the momentum space of structured light,” Optica 7, 108–114 (2018).
[Crossref]

F. Cardano, A. D’Errico, A. Dauphin, M. Maffei, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, L. Marrucci, M. Lewenstein, and P. Massignan, “Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons,” Nat. Commun. 8, 15516 (2017).
[Crossref]

Li, H.

C. Chen, X. Ding, J. Qin, Y. He, Y.-H. Luo, M.-C. Chen, C. Liu, X.-L. Wang, W.-J. Zhang, H. Li, L.-X. You, Z. Wang, D.-W. Wang, B. C. Sanders, C.-Y. Lu, and J.-W. Pan, “Observation of topologically protected edge states in a photonic two-dimensional quantum walk,” Phys. Rev. Lett. 121, 100502 (2018).
[Crossref]

Li, J.

P. Xue, R. Zhang, H. Qin, X. Zhan, Z. H. Bian, J. Li, and B. C. Sanders, “Experimental quantum-walk revival with a time-dependent coin,” Phys. Rev. Lett. 114, 140502 (2015).
[Crossref]

Lin, Q.

A. Dutt, M. Minkov, Q. Lin, L. Yuan, D. A. B. Miller, and S. Fan, “Experimental band structure spectroscopy along a synthetic dimension,” Nat. Commun. 10, 3122 (2019).
[Crossref]

Q. Lin, X.-Q. Sun, M. Xiao, S.-C. Zhang, and S. Fan, “A three-dimensional photonic topological insulator using a two-dimensional ring resonator lattice with a synthetic frequency dimension,” Sci. Adv. 4, eaat2774 (2018).
[Crossref]

L. Yuan, M. Xiao, Q. Lin, and S. Fan, “Synthetic space with arbitrary dimensions in a few rings undergoing dynamic modulation,” Phys. Rev. B 97, 104105 (2018).
[Crossref]

Q. Lin, M. Xiao, L. Yuan, and S. Fan, “Photonic Weyl point in a two-dimensional resonator lattice with a synthetic frequency dimension,” Nat. Commun. 7, 13731 (2016).
[Crossref]

Liu, C.

C. Chen, X. Ding, J. Qin, Y. He, Y.-H. Luo, M.-C. Chen, C. Liu, X.-L. Wang, W.-J. Zhang, H. Li, L.-X. You, Z. Wang, D.-W. Wang, B. C. Sanders, C.-Y. Lu, and J.-W. Pan, “Observation of topologically protected edge states in a photonic two-dimensional quantum walk,” Phys. Rev. Lett. 121, 100502 (2018).
[Crossref]

Lobino, M.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[Crossref]

Longhi, S.

G. Corrielli, A. Crespi, G. Della Valle, S. Longhi, and R. Osellame, “Fractional Bloch oscillations in photonic lattices,” Nat. Commun. 4, 1555 (2013).
[Crossref]

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, “Bloch-Zener oscillations in binary superlattices,” Phys. Rev. Lett. 102, 076802 (2009).
[Crossref]

S. Longhi, “Optical Zener-Bloch oscillations in binary waveguide arrays,” Europhys. Lett. 76, 416–421 (2006).
[Crossref]

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[Crossref]

Lorz, L.

S. Barkhofen, T. Nitsche, F. Elster, L. Lorz, A. Gábris, I. Jex, and C. Silberhorn, “Measuring topological invariants in disordered discrete-time quantum walks,” Phys. Rev. A 96, 033846 (2017).
[Crossref]

Lu, C.-Y.

C. Chen, X. Ding, J. Qin, Y. He, Y.-H. Luo, M.-C. Chen, C. Liu, X.-L. Wang, W.-J. Zhang, H. Li, L.-X. You, Z. Wang, D.-W. Wang, B. C. Sanders, C.-Y. Lu, and J.-W. Pan, “Observation of topologically protected edge states in a photonic two-dimensional quantum walk,” Phys. Rev. Lett. 121, 100502 (2018).
[Crossref]

Lu, L.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91, 015006 (2019).
[Crossref]

Lu, P.

C. Qin, F. Zhou, Y. Peng, D. Sounas, X. Zhu, B. Wang, J. Dong, X. Zhang, A. Alù, and P. Lu, “Spectrum control through discrete frequency diffraction in the presence of photonic gauge potentials,” Phys. Rev. Lett. 120, 133901 (2018).
[Crossref]

Lukin, M. D.

H. Pichler, S. Choi, P. Zoller, and M. D. Lukin, “Universal photonic quantum computation via time-delayed feedback,” Proc. Natl. Acad. Sci. USA 114, 11362–11367 (2017).
[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]

Lumer, Y.

E. Lustig, S. Weimann, Y. Plotnik, Y. Lumer, M. A. Bandres, A. Szameit, and M. Segev, “Photonic topological insulator in synthetic dimensions,” Nature 567, 356–360 (2019).
[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]

Luo, Y.-H.

C. Chen, X. Ding, J. Qin, Y. He, Y.-H. Luo, M.-C. Chen, C. Liu, X.-L. Wang, W.-J. Zhang, H. Li, L.-X. You, Z. Wang, D.-W. Wang, B. C. Sanders, C.-Y. Lu, and J.-W. Pan, “Observation of topologically protected edge states in a photonic two-dimensional quantum walk,” Phys. Rev. Lett. 121, 100502 (2018).
[Crossref]

Lustig, E.

E. Lustig, S. Weimann, Y. Plotnik, Y. Lumer, M. A. Bandres, A. Szameit, and M. Segev, “Photonic topological insulator in synthetic dimensions,” Nature 567, 356–360 (2019).
[Crossref]

Ma, X.-S.

X.-S. Ma, B. Dakic, W. Naylor, A. Zeilinger, and P. Walther, “Quantum simulation of the wavefunction to probe frustrated Heisenberg spin systems,” Nat. Phys. 7, 399–405 (2011).
[Crossref]

Maffei, M.

A. D’Errico, F. Cardano, M. Maffei, A. Dauphin, R. Barboza, C. Esposito, B. Piccirillo, M. Lewenstein, P. Massignan, and L. Marrucci, “Two-dimensional topological quantum walks in the momentum space of structured light,” Optica 7, 108–114 (2018).
[Crossref]

F. Cardano, A. D’Errico, A. Dauphin, M. Maffei, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, L. Marrucci, M. Lewenstein, and P. Massignan, “Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons,” Nat. Commun. 8, 15516 (2017).
[Crossref]

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7, 11439 (2016).
[Crossref]

Marangoni, M.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[Crossref]

Maraviglia, N.

C. Sparrow, E. Martín-López, N. Maraviglia, A. Neville, C. Harrold, J. Carolan, Y. N. Joglekar, T. Hashimoto, N. Matsuda, J. L. O’Brien, D. P. Tew, and A. Laing, “Simulating the vibrational quantum dynamics of molecules using photonics,” Nature 557, 660–667 (2018).
[Crossref]

Mark, M. J.

E. Haller, R. Hart, M. J. Mark, J. G. Danzl, L. Reichsöllner, and H.-C. Nägerl, “Inducing transport in a dissipation-free lattice with super Bloch oscillations,” Phys. Rev. Lett. 104, 200403 (2010).
[Crossref]

Marrucci, L.

A. D’Errico, F. Cardano, M. Maffei, A. Dauphin, R. Barboza, C. Esposito, B. Piccirillo, M. Lewenstein, P. Massignan, and L. Marrucci, “Two-dimensional topological quantum walks in the momentum space of structured light,” Optica 7, 108–114 (2018).
[Crossref]

F. Cardano, A. D’Errico, A. Dauphin, M. Maffei, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, L. Marrucci, M. Lewenstein, and P. Massignan, “Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons,” Nat. Commun. 8, 15516 (2017).
[Crossref]

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7, 11439 (2016).
[Crossref]

F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, “Quantum walks and wavepacket dynamics on a lattice with twisted photons,” Sci. Adv. 1, e1500087 (2015).
[Crossref]

Martín-López, E.

C. Sparrow, E. Martín-López, N. Maraviglia, A. Neville, C. Harrold, J. Carolan, Y. N. Joglekar, T. Hashimoto, N. Matsuda, J. L. O’Brien, D. P. Tew, and A. Laing, “Simulating the vibrational quantum dynamics of molecules using photonics,” Nature 557, 660–667 (2018).
[Crossref]

Marzoli, I.

D. Bouwmeester, I. Marzoli, G. P. Karman, W. Schleich, and J. P. Woerdman, “Optical Galton board,” Phys. Rev. A 61, 013410 (1999).
[Crossref]

Massa, F.

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7, 11439 (2016).
[Crossref]

F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, “Quantum walks and wavepacket dynamics on a lattice with twisted photons,” Sci. Adv. 1, e1500087 (2015).
[Crossref]

Massignan, P.

A. D’Errico, F. Cardano, M. Maffei, A. Dauphin, R. Barboza, C. Esposito, B. Piccirillo, M. Lewenstein, P. Massignan, and L. Marrucci, “Two-dimensional topological quantum walks in the momentum space of structured light,” Optica 7, 108–114 (2018).
[Crossref]

F. Cardano, A. D’Errico, A. Dauphin, M. Maffei, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, L. Marrucci, M. Lewenstein, and P. Massignan, “Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons,” Nat. Commun. 8, 15516 (2017).
[Crossref]

Matsuda, N.

C. Sparrow, E. Martín-López, N. Maraviglia, A. Neville, C. Harrold, J. Carolan, Y. N. Joglekar, T. Hashimoto, N. Matsuda, J. L. O’Brien, D. P. Tew, and A. Laing, “Simulating the vibrational quantum dynamics of molecules using photonics,” Nature 557, 660–667 (2018).
[Crossref]

Meschede, D.

M. Genske, W. Alt, A. Steffen, A. H. Werner, R. F. Werner, D. Meschede, and A. Alberti, “Electric quantum walks with individual atoms,” Phys. Rev. Lett. 110, 190601 (2013).
[Crossref]

Migdall, A.

S. Mittal, J. Fan, S. Faez, A. Migdall, J. M. Taylor, and M. Hafezi, “Topologically robust transport of photons in a synthetic gauge field,” Phys. Rev. Lett. 113, 087403 (2014).
[Crossref]

Miller, D. A. B.

A. Dutt, M. Minkov, Q. Lin, L. Yuan, D. A. B. Miller, and S. Fan, “Experimental band structure spectroscopy along a synthetic dimension,” Nat. Commun. 10, 3122 (2019).
[Crossref]

Minkov, M.

A. Dutt, M. Minkov, Q. Lin, L. Yuan, D. A. B. Miller, and S. Fan, “Experimental band structure spectroscopy along a synthetic dimension,” Nat. Commun. 10, 3122 (2019).
[Crossref]

Miri, M.-A.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref]

Mittal, S.

H. Chalabi, S. Barik, S. Mittal, T. E. Murphy, M. Hafezi, and E. Waks, “Synthetic gauge field for two-dimensional time-multiplexed quantum random walks,” Phys. Rev. Lett. 123, 150503 (2019).
[Crossref]

V. V. Orre, E. A. Goldschmidt, A. Deshpande, A. V. Gorshkov, V. Tamma, M. Hafezi, and S. Mittal, “Interference of temporally distinguishable photons using frequency-resolved detection,” Phys. Rev. Lett. 123, 123603 (2019).
[Crossref]

S. Mittal, S. Ganeshan, J. Fan, A. Vaezi, and M. Hafezi, “Measurement of topological invariants in a 2D photonic system,” Nat. Photonics 10, 180–183 (2016).
[Crossref]

S. Mittal, J. Fan, S. Faez, A. Migdall, J. M. Taylor, and M. Hafezi, “Topologically robust transport of photons in a synthetic gauge field,” Phys. Rev. Lett. 113, 087403 (2014).
[Crossref]

Mohseni, M.

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nat. Chem. 2, 106–111 (2010).
[Crossref]

Morandotti, R.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[Crossref]

Mosley, P. J.

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref]

Mossmann, S.

D. Witthaut, F. Keck, H. J. Korsch, and S. Mossmann, “Bloch oscillations in two-dimensional lattices,” New J. Phys. 6, 41 (2004).
[Crossref]

Murphy, T. E.

H. Chalabi, S. Barik, S. Mittal, T. E. Murphy, M. Hafezi, and E. Waks, “Synthetic gauge field for two-dimensional time-multiplexed quantum random walks,” Phys. Rev. Lett. 123, 150503 (2019).
[Crossref]

Nägerl, H.-C.

E. Haller, R. Hart, M. J. Mark, J. G. Danzl, L. Reichsöllner, and H.-C. Nägerl, “Inducing transport in a dissipation-free lattice with super Bloch oscillations,” Phys. Rev. Lett. 104, 200403 (2010).
[Crossref]

Navarrete-Benlloch, C.

C. Navarrete-Benlloch, A. Pérez, and E. Roldán, “Nonlinear optical Galton board,” Phys. Rev. A 75, 062333 (2007).
[Crossref]

Naylor, W.

X.-S. Ma, B. Dakic, W. Naylor, A. Zeilinger, and P. Walther, “Quantum simulation of the wavefunction to probe frustrated Heisenberg spin systems,” Nat. Phys. 7, 399–405 (2011).
[Crossref]

Neshev, D. N.

B. A. Bell, K. Wang, A. S. Solntsev, D. N. Neshev, A. A. Sukhorukov, and B. J. Eggleton, “Spectral photonic lattices with complex long-range coupling,” Optica 4, 1433–1436 (2017).
[Crossref]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[Crossref]

Neville, A.

C. Sparrow, E. Martín-López, N. Maraviglia, A. Neville, C. Harrold, J. Carolan, Y. N. Joglekar, T. Hashimoto, N. Matsuda, J. L. O’Brien, D. P. Tew, and A. Laing, “Simulating the vibrational quantum dynamics of molecules using photonics,” Nature 557, 660–667 (2018).
[Crossref]

Nitsche, T.

S. Barkhofen, T. Nitsche, F. Elster, L. Lorz, A. Gábris, I. Jex, and C. Silberhorn, “Measuring topological invariants in disordered discrete-time quantum walks,” Phys. Rev. A 96, 033846 (2017).
[Crossref]

T. Nitsche, F. Elster, J. Novotný, A. Gábris, I. Jex, S. Barkhofen, and C. Silberhorn, “Quantum walks with dynamical control: graph engineering, initial state preparation and state transfer,” New J. Phys. 18, 063017 (2016).
[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]

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, “Bloch-Zener oscillations in binary superlattices,” Phys. Rev. Lett. 102, 076802 (2009).
[Crossref]

Novotný, J.

T. Nitsche, F. Elster, J. Novotný, A. Gábris, I. Jex, S. Barkhofen, and C. Silberhorn, “Quantum walks with dynamical control: graph engineering, initial state preparation and state transfer,” New J. Phys. 18, 063017 (2016).
[Crossref]

O’Brien, J. L.

C. Sparrow, E. Martín-López, N. Maraviglia, A. Neville, C. Harrold, J. Carolan, Y. N. Joglekar, T. Hashimoto, N. Matsuda, J. L. O’Brien, D. P. Tew, and A. Laing, “Simulating the vibrational quantum dynamics of molecules using photonics,” Nature 557, 660–667 (2018).
[Crossref]

Onishchukov, G.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref]

C. Bersch, G. Onishchukov, and U. Peschel, “Experimental observation of spectral Bloch oscillations,” Opt. Lett. 34, 2372–2374 (2009).
[Crossref]

U. Peschel, C. Bersch, and G. Onishchukov, “Discreteness in time,” Open Phys. 6, 619–627 (2008).
[Crossref]

Orre, V. V.

V. V. Orre, E. A. Goldschmidt, A. Deshpande, A. V. Gorshkov, V. Tamma, M. Hafezi, and S. Mittal, “Interference of temporally distinguishable photons using frequency-resolved detection,” Phys. Rev. Lett. 123, 123603 (2019).
[Crossref]

Osellame, R.

G. Corrielli, A. Crespi, G. Della Valle, S. Longhi, and R. Osellame, “Fractional Bloch oscillations in photonic lattices,” Nat. Commun. 4, 1555 (2013).
[Crossref]

Ozawa, T.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91, 015006 (2019).
[Crossref]

Pan, J.-W.

C. Chen, X. Ding, J. Qin, Y. He, Y.-H. Luo, M.-C. Chen, C. Liu, X.-L. Wang, W.-J. Zhang, H. Li, L.-X. You, Z. Wang, D.-W. Wang, B. C. Sanders, C.-Y. Lu, and J.-W. Pan, “Observation of topologically protected edge states in a photonic two-dimensional quantum walk,” Phys. Rev. Lett. 121, 100502 (2018).
[Crossref]

Paparo, D.

F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, “Quantum walks and wavepacket dynamics on a lattice with twisted photons,” Sci. Adv. 1, e1500087 (2015).
[Crossref]

Parker, R.

G. Lenz, R. Parker, M. Wanke, and C. de Sterke, “Dynamical localization and AC Bloch oscillations in periodic optical waveguide arrays,” Opt. Commun. 218, 87–92 (2003).
[Crossref]

Pasek, M.

W. Hu, J. C. Pillay, K. Wu, M. Pasek, P. P. Shum, and Y. D. Chong, “Measurement of a topological edge invariant in a microwave network,” Phys. Rev. X 5, 011012 (2015).
[Crossref]

Peik, E.

M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, “Bloch oscillations of atoms in an optical potential,” Phys. Rev. Lett. 76, 4508–4511 (1996).
[Crossref]

Peng, Y.

C. Qin, F. Zhou, Y. Peng, D. Sounas, X. Zhu, B. Wang, J. Dong, X. Zhang, A. Alù, and P. Lu, “Spectrum control through discrete frequency diffraction in the presence of photonic gauge potentials,” Phys. Rev. Lett. 120, 133901 (2018).
[Crossref]

Pérez, A.

C. Navarrete-Benlloch, A. Pérez, and E. Roldán, “Nonlinear optical Galton board,” Phys. Rev. A 75, 062333 (2007).
[Crossref]

Pertsch, T.

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, “Bloch-Zener oscillations in binary superlattices,” Phys. Rev. Lett. 102, 076802 (2009).
[Crossref]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[Crossref]

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[Crossref]

Peschel, U.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref]

C. Bersch, G. Onishchukov, and U. Peschel, “Experimental observation of spectral Bloch oscillations,” Opt. Lett. 34, 2372–2374 (2009).
[Crossref]

U. Peschel, C. Bersch, and G. Onishchukov, “Discreteness in time,” Open Phys. 6, 619–627 (2008).
[Crossref]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[Crossref]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[Crossref]

Piccirillo, B.

A. D’Errico, F. Cardano, M. Maffei, A. Dauphin, R. Barboza, C. Esposito, B. Piccirillo, M. Lewenstein, P. Massignan, and L. Marrucci, “Two-dimensional topological quantum walks in the momentum space of structured light,” Optica 7, 108–114 (2018).
[Crossref]

F. Cardano, A. D’Errico, A. Dauphin, M. Maffei, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, L. Marrucci, M. Lewenstein, and P. Massignan, “Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons,” Nat. Commun. 8, 15516 (2017).
[Crossref]

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7, 11439 (2016).
[Crossref]

Pichler, H.

H. Pichler, S. Choi, P. Zoller, and M. D. Lukin, “Universal photonic quantum computation via time-delayed feedback,” Proc. Natl. Acad. Sci. USA 114, 11362–11367 (2017).
[Crossref]

Pillay, J. C.

W. Hu, J. C. Pillay, K. Wu, M. Pasek, P. P. Shum, and Y. D. Chong, “Measurement of a topological edge invariant in a microwave network,” Phys. Rev. X 5, 011012 (2015).
[Crossref]

Plotnik, Y.

E. Lustig, S. Weimann, Y. Plotnik, Y. Lumer, M. A. Bandres, A. Szameit, and M. Segev, “Photonic topological insulator in synthetic dimensions,” Nature 567, 356–360 (2019).
[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]

Poon, J. K. S.

A. Joushaghani, R. Iyer, J. K. S. Poon, J. S. Aitchison, C. M. de Sterke, J. Wan, and M. M. Dignam, “Generalized exact dynamic localization in curved coupled optical waveguide arrays,” Phys. Rev. Lett. 109, 103901 (2012).
[Crossref]

A. Joushaghani, R. Iyer, J. K. S. Poon, J. S. Aitchison, C. M. de Sterke, J. Wan, and M. M. Dignam, “Quasi-Bloch oscillations in curved coupled optical waveguides,” Phys. Rev. Lett. 103, 143903 (2009).
[Crossref]

Potocek, V.

A. Schreiber, A. Gabris, P. P. Rohde, K. Laiho, M. Stefanak, V. Potocek, C. Hamilton, I. Jex, and C. Silberhorn, “A 2D quantum walk simulation of two-particle dynamics,” Science 336, 55–58 (2012).
[Crossref]

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, I. Jex, and C. Silberhorn, “Decoherence and disorder in quantum walks: from ballistic spread to localization,” Phys. Rev. Lett. 106, 180403 (2011).
[Crossref]

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref]

Powell, B. J.

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nat. Chem. 2, 106–111 (2010).
[Crossref]

Price, H. M.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91, 015006 (2019).
[Crossref]

Qassim, H.

F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, “Quantum walks and wavepacket dynamics on a lattice with twisted photons,” Sci. Adv. 1, e1500087 (2015).
[Crossref]

Qin, C.

C. Qin, F. Zhou, Y. Peng, D. Sounas, X. Zhu, B. Wang, J. Dong, X. Zhang, A. Alù, and P. Lu, “Spectrum control through discrete frequency diffraction in the presence of photonic gauge potentials,” Phys. Rev. Lett. 120, 133901 (2018).
[Crossref]

Qin, H.

P. Xue, R. Zhang, H. Qin, X. Zhan, Z. H. Bian, J. Li, and B. C. Sanders, “Experimental quantum-walk revival with a time-dependent coin,” Phys. Rev. Lett. 114, 140502 (2015).
[Crossref]

Qin, J.

C. Chen, X. Ding, J. Qin, Y. He, Y.-H. Luo, M.-C. Chen, C. Liu, X.-L. Wang, W.-J. Zhang, H. Li, L.-X. You, Z. Wang, D.-W. Wang, B. C. Sanders, C.-Y. Lu, and J.-W. Pan, “Observation of topologically protected edge states in a photonic two-dimensional quantum walk,” Phys. Rev. Lett. 121, 100502 (2018).
[Crossref]

Ramponi, R.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[Crossref]

Rechtsman, M. C.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91, 015006 (2019).
[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]

Regensburger, A.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref]

Reichel, J.

M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, “Bloch oscillations of atoms in an optical potential,” Phys. Rev. Lett. 76, 4508–4511 (1996).
[Crossref]

Reichsöllner, L.

E. Haller, R. Hart, M. J. Mark, J. G. Danzl, L. Reichsöllner, and H.-C. Nägerl, “Inducing transport in a dissipation-free lattice with super Bloch oscillations,” Phys. Rev. Lett. 104, 200403 (2010).
[Crossref]

Rohde, P. P.

A. Schreiber, A. Gabris, P. P. Rohde, K. Laiho, M. Stefanak, V. Potocek, C. Hamilton, I. Jex, and C. Silberhorn, “A 2D quantum walk simulation of two-particle dynamics,” Science 336, 55–58 (2012).
[Crossref]

Roldán, E.

L. A. Bru, M. Hinarejos, F. Silva, G. J. de Valcárcel, and E. Roldán, “Electric quantum walks in two dimensions,” Phys. Rev. A 93, 032333 (2016).
[Crossref]

C. Navarrete-Benlloch, A. Pérez, and E. Roldán, “Nonlinear optical Galton board,” Phys. Rev. A 75, 062333 (2007).
[Crossref]

Roux, F. S.

S. K. Goyal, F. S. Roux, A. Forbes, and T. Konrad, “Implementing quantum walks using orbital angular momentum of classical light,” Phys. Rev. Lett. 110, 263602 (2013).
[Crossref]

Rybár, T.

C. Cedzich, T. Rybár, A. H. Werner, A. Alberti, M. Genske, and R. F. Werner, “Propagation of quantum walks in electric fields,” Phys. Rev. Lett. 111, 160601 (2013).
[Crossref]

Salomon, C.

M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, “Bloch oscillations of atoms in an optical potential,” Phys. Rev. Lett. 76, 4508–4511 (1996).
[Crossref]

Sanders, B. C.

C. Chen, X. Ding, J. Qin, Y. He, Y.-H. Luo, M.-C. Chen, C. Liu, X.-L. Wang, W.-J. Zhang, H. Li, L.-X. You, Z. Wang, D.-W. Wang, B. C. Sanders, C.-Y. Lu, and J.-W. Pan, “Observation of topologically protected edge states in a photonic two-dimensional quantum walk,” Phys. Rev. Lett. 121, 100502 (2018).
[Crossref]

P. Xue, R. Zhang, H. Qin, X. Zhan, Z. H. Bian, J. Li, and B. C. Sanders, “Experimental quantum-walk revival with a time-dependent coin,” Phys. Rev. Lett. 114, 140502 (2015).
[Crossref]

Santamato, E.

F. Cardano, A. D’Errico, A. Dauphin, M. Maffei, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, L. Marrucci, M. Lewenstein, and P. Massignan, “Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons,” Nat. Commun. 8, 15516 (2017).
[Crossref]

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7, 11439 (2016).
[Crossref]

F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, “Quantum walks and wavepacket dynamics on a lattice with twisted photons,” Sci. Adv. 1, e1500087 (2015).
[Crossref]

Schleich, W.

D. Bouwmeester, I. Marzoli, G. P. Karman, W. Schleich, and J. P. Woerdman, “Optical Galton board,” Phys. Rev. A 61, 013410 (1999).
[Crossref]

Schreiber, A.

A. Schreiber, A. Gabris, P. P. Rohde, K. Laiho, M. Stefanak, V. Potocek, C. Hamilton, I. Jex, and C. Silberhorn, “A 2D quantum walk simulation of two-particle dynamics,” Science 336, 55–58 (2012).
[Crossref]

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, I. Jex, and C. Silberhorn, “Decoherence and disorder in quantum walks: from ballistic spread to localization,” Phys. Rev. Lett. 106, 180403 (2011).
[Crossref]

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref]

Schuster, D.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91, 015006 (2019).
[Crossref]

Sciarrino, F.

F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, “Quantum walks and wavepacket dynamics on a lattice with twisted photons,” Sci. Adv. 1, e1500087 (2015).
[Crossref]

Segev, M.

E. Lustig, S. Weimann, Y. Plotnik, Y. Lumer, M. A. Bandres, A. Szameit, and M. Segev, “Photonic topological insulator in synthetic dimensions,” Nature 567, 356–360 (2019).
[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]

Shum, P. P.

W. Hu, J. C. Pillay, K. Wu, M. Pasek, P. P. Shum, and Y. D. Chong, “Measurement of a topological edge invariant in a microwave network,” Phys. Rev. X 5, 011012 (2015).
[Crossref]

Silberberg, Y.

Y. Bromberg, Y. Lahini, and Y. Silberberg, “Bloch oscillations of path-entangled photons,” Phys. Rev. Lett. 105, 263604 (2010).
[Crossref]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[Crossref]

Silberhorn, C.

S. Barkhofen, T. Nitsche, F. Elster, L. Lorz, A. Gábris, I. Jex, and C. Silberhorn, “Measuring topological invariants in disordered discrete-time quantum walks,” Phys. Rev. A 96, 033846 (2017).
[Crossref]

T. Nitsche, F. Elster, J. Novotný, A. Gábris, I. Jex, S. Barkhofen, and C. Silberhorn, “Quantum walks with dynamical control: graph engineering, initial state preparation and state transfer,” New J. Phys. 18, 063017 (2016).
[Crossref]

A. Schreiber, A. Gabris, P. P. Rohde, K. Laiho, M. Stefanak, V. Potocek, C. Hamilton, I. Jex, and C. Silberhorn, “A 2D quantum walk simulation of two-particle dynamics,” Science 336, 55–58 (2012).
[Crossref]

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, I. Jex, and C. Silberhorn, “Decoherence and disorder in quantum walks: from ballistic spread to localization,” Phys. Rev. Lett. 106, 180403 (2011).
[Crossref]

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref]

Silva, F.

L. A. Bru, M. Hinarejos, F. Silva, G. J. de Valcárcel, and E. Roldán, “Electric quantum walks in two dimensions,” Phys. Rev. A 93, 032333 (2016).
[Crossref]

Simon, J.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91, 015006 (2019).
[Crossref]

Slussarenko, S.

F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, “Quantum walks and wavepacket dynamics on a lattice with twisted photons,” Sci. Adv. 1, e1500087 (2015).
[Crossref]

Solntsev, A. S.

Sounas, D.

C. Qin, F. Zhou, Y. Peng, D. Sounas, X. Zhu, B. Wang, J. Dong, X. Zhang, A. Alù, and P. Lu, “Spectrum control through discrete frequency diffraction in the presence of photonic gauge potentials,” Phys. Rev. Lett. 120, 133901 (2018).
[Crossref]

Sparrow, C.

C. Sparrow, E. Martín-López, N. Maraviglia, A. Neville, C. Harrold, J. Carolan, Y. N. Joglekar, T. Hashimoto, N. Matsuda, J. L. O’Brien, D. P. Tew, and A. Laing, “Simulating the vibrational quantum dynamics of molecules using photonics,” Nature 557, 660–667 (2018).
[Crossref]

Stefanak, M.

A. Schreiber, A. Gabris, P. P. Rohde, K. Laiho, M. Stefanak, V. Potocek, C. Hamilton, I. Jex, and C. Silberhorn, “A 2D quantum walk simulation of two-particle dynamics,” Science 336, 55–58 (2012).
[Crossref]

Steffen, A.

M. Genske, W. Alt, A. Steffen, A. H. Werner, R. F. Werner, D. Meschede, and A. Alberti, “Electric quantum walks with individual atoms,” Phys. Rev. Lett. 110, 190601 (2013).
[Crossref]

Sukhorukov, A. A.

B. A. Bell, K. Wang, A. S. Solntsev, D. N. Neshev, A. A. Sukhorukov, and B. J. Eggleton, “Spectral photonic lattices with complex long-range coupling,” Optica 4, 1433–1436 (2017).
[Crossref]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[Crossref]

Sun, X.-Q.

Q. Lin, X.-Q. Sun, M. Xiao, S.-C. Zhang, and S. Fan, “A three-dimensional photonic topological insulator using a two-dimensional ring resonator lattice with a synthetic frequency dimension,” Sci. Adv. 4, eaat2774 (2018).
[Crossref]

Szameit, A.

E. Lustig, S. Weimann, Y. Plotnik, Y. Lumer, M. A. Bandres, A. Szameit, and M. Segev, “Photonic topological insulator in synthetic dimensions,” Nature 567, 356–360 (2019).
[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]

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, “Bloch-Zener oscillations in binary superlattices,” Phys. Rev. Lett. 102, 076802 (2009).
[Crossref]

Talanina, I.

G. Lenz, I. Talanina, and C. M. de Sterke, “Bloch Oscillations in an array of curved optical waveguides,” Phys. Rev. Lett. 83, 963–966 (1999).
[Crossref]

Tamma, V.

V. V. Orre, E. A. Goldschmidt, A. Deshpande, A. V. Gorshkov, V. Tamma, M. Hafezi, and S. Mittal, “Interference of temporally distinguishable photons using frequency-resolved detection,” Phys. Rev. Lett. 123, 123603 (2019).
[Crossref]

Taylor, J. M.

S. Mittal, J. Fan, S. Faez, A. Migdall, J. M. Taylor, and M. Hafezi, “Topologically robust transport of photons in a synthetic gauge field,” Phys. Rev. Lett. 113, 087403 (2014).
[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]

Tew, D. P.

C. Sparrow, E. Martín-López, N. Maraviglia, A. Neville, C. Harrold, J. Carolan, Y. N. Joglekar, T. Hashimoto, N. Matsuda, J. L. O’Brien, D. P. Tew, and A. Laing, “Simulating the vibrational quantum dynamics of molecules using photonics,” Nature 557, 660–667 (2018).
[Crossref]

Trompeter, H.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[Crossref]

Tünnermann, A.

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, “Bloch-Zener oscillations in binary superlattices,” Phys. Rev. Lett. 102, 076802 (2009).
[Crossref]

Vaezi, A.

S. Mittal, S. Ganeshan, J. Fan, A. Vaezi, and M. Hafezi, “Measurement of topological invariants in a 2D photonic system,” Nat. Photonics 10, 180–183 (2016).
[Crossref]

Waks, E.

H. Chalabi, S. Barik, S. Mittal, T. E. Murphy, M. Hafezi, and E. Waks, “Synthetic gauge field for two-dimensional time-multiplexed quantum random walks,” Phys. Rev. Lett. 123, 150503 (2019).
[Crossref]

H. Chalabi and E. Waks, “Interaction of photons with a coupled atom-cavity system through a bidirectional time-delayed feedback,” Phys. Rev. A 98, 063832 (2018).
[Crossref]

Walther, P.

A. Aspuru-Guzik and P. Walther, “Photonic quantum simulators,” Nat. Phys. 8, 285–291 (2012).
[Crossref]

X.-S. Ma, B. Dakic, W. Naylor, A. Zeilinger, and P. Walther, “Quantum simulation of the wavefunction to probe frustrated Heisenberg spin systems,” Nat. Phys. 7, 399–405 (2011).
[Crossref]

Wan, J.

A. Joushaghani, R. Iyer, J. K. S. Poon, J. S. Aitchison, C. M. de Sterke, J. Wan, and M. M. Dignam, “Generalized exact dynamic localization in curved coupled optical waveguide arrays,” Phys. Rev. Lett. 109, 103901 (2012).
[Crossref]

A. Joushaghani, R. Iyer, J. K. S. Poon, J. S. Aitchison, C. M. de Sterke, J. Wan, and M. M. Dignam, “Quasi-Bloch oscillations in curved coupled optical waveguides,” Phys. Rev. Lett. 103, 143903 (2009).
[Crossref]

R. Iyer, J. S. Aitchison, J. Wan, M. M. Dignam, and C. M. de Sterke, “Exact dynamic localization in curved AlGaAs optical waveguide arrays,” Opt. Express 15, 3212–3223 (2007).
[Crossref]

Wang, B.

C. Qin, F. Zhou, Y. Peng, D. Sounas, X. Zhu, B. Wang, J. Dong, X. Zhang, A. Alù, and P. Lu, “Spectrum control through discrete frequency diffraction in the presence of photonic gauge potentials,” Phys. Rev. Lett. 120, 133901 (2018).
[Crossref]

Wang, D.-W.

C. Chen, X. Ding, J. Qin, Y. He, Y.-H. Luo, M.-C. Chen, C. Liu, X.-L. Wang, W.-J. Zhang, H. Li, L.-X. You, Z. Wang, D.-W. Wang, B. C. Sanders, C.-Y. Lu, and J.-W. Pan, “Observation of topologically protected edge states in a photonic two-dimensional quantum walk,” Phys. Rev. Lett. 121, 100502 (2018).
[Crossref]

Wang, K.

Wang, X.-L.

C. Chen, X. Ding, J. Qin, Y. He, Y.-H. Luo, M.-C. Chen, C. Liu, X.-L. Wang, W.-J. Zhang, H. Li, L.-X. You, Z. Wang, D.-W. Wang, B. C. Sanders, C.-Y. Lu, and J.-W. Pan, “Observation of topologically protected edge states in a photonic two-dimensional quantum walk,” Phys. Rev. Lett. 121, 100502 (2018).
[Crossref]

Wang, Z.

C. Chen, X. Ding, J. Qin, Y. He, Y.-H. Luo, M.-C. Chen, C. Liu, X.-L. Wang, W.-J. Zhang, H. Li, L.-X. You, Z. Wang, D.-W. Wang, B. C. Sanders, C.-Y. Lu, and J.-W. Pan, “Observation of topologically protected edge states in a photonic two-dimensional quantum walk,” Phys. Rev. Lett. 121, 100502 (2018).
[Crossref]

Wanke, M.

G. Lenz, R. Parker, M. Wanke, and C. de Sterke, “Dynamical localization and AC Bloch oscillations in periodic optical waveguide arrays,” Opt. Commun. 218, 87–92 (2003).
[Crossref]

Weimann, S.

E. Lustig, S. Weimann, Y. Plotnik, Y. Lumer, M. A. Bandres, A. Szameit, and M. Segev, “Photonic topological insulator in synthetic dimensions,” Nature 567, 356–360 (2019).
[Crossref]

Werner, A. H.

C. Cedzich, T. Geib, A. H. Werner, and R. F. Werner, “Quantum walks in external gauge fields,” J. Math. Phys. 60, 012107 (2019).
[Crossref]

M. Genske, W. Alt, A. Steffen, A. H. Werner, R. F. Werner, D. Meschede, and A. Alberti, “Electric quantum walks with individual atoms,” Phys. Rev. Lett. 110, 190601 (2013).
[Crossref]

C. Cedzich, T. Rybár, A. H. Werner, A. Alberti, M. Genske, and R. F. Werner, “Propagation of quantum walks in electric fields,” Phys. Rev. Lett. 111, 160601 (2013).
[Crossref]

Werner, R. F.

C. Cedzich, T. Geib, A. H. Werner, and R. F. Werner, “Quantum walks in external gauge fields,” J. Math. Phys. 60, 012107 (2019).
[Crossref]

C. Cedzich and R. F. Werner, “Revivals in quantum walks with a quasiperiodically-time-dependent coin,” Phys. Rev. A 93, 032329 (2016).
[Crossref]

M. Genske, W. Alt, A. Steffen, A. H. Werner, R. F. Werner, D. Meschede, and A. Alberti, “Electric quantum walks with individual atoms,” Phys. Rev. Lett. 110, 190601 (2013).
[Crossref]

C. Cedzich, T. Rybár, A. H. Werner, A. Alberti, M. Genske, and R. F. Werner, “Propagation of quantum walks in electric fields,” Phys. Rev. Lett. 111, 160601 (2013).
[Crossref]

White, A. G.

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nat. Chem. 2, 106–111 (2010).
[Crossref]

Whitfield, J. D.

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nat. Chem. 2, 106–111 (2010).
[Crossref]

Witthaut, D.

D. Witthaut, F. Keck, H. J. Korsch, and S. Mossmann, “Bloch oscillations in two-dimensional lattices,” New J. Phys. 6, 41 (2004).
[Crossref]

Woerdman, J. P.

D. Bouwmeester, I. Marzoli, G. P. Karman, W. Schleich, and J. P. Woerdman, “Optical Galton board,” Phys. Rev. A 61, 013410 (1999).
[Crossref]

Wu, K.

W. Hu, J. C. Pillay, K. Wu, M. Pasek, P. P. Shum, and Y. D. Chong, “Measurement of a topological edge invariant in a microwave network,” Phys. Rev. X 5, 011012 (2015).
[Crossref]

Xiao, M.

Q. Lin, X.-Q. Sun, M. Xiao, S.-C. Zhang, and S. Fan, “A three-dimensional photonic topological insulator using a two-dimensional ring resonator lattice with a synthetic frequency dimension,” Sci. Adv. 4, eaat2774 (2018).
[Crossref]

L. Yuan, M. Xiao, Q. Lin, and S. Fan, “Synthetic space with arbitrary dimensions in a few rings undergoing dynamic modulation,” Phys. Rev. B 97, 104105 (2018).
[Crossref]

Q. Lin, M. Xiao, L. Yuan, and S. Fan, “Photonic Weyl point in a two-dimensional resonator lattice with a synthetic frequency dimension,” Nat. Commun. 7, 13731 (2016).
[Crossref]

Xue, P.

P. Xue, R. Zhang, H. Qin, X. Zhan, Z. H. Bian, J. Li, and B. C. Sanders, “Experimental quantum-walk revival with a time-dependent coin,” Phys. Rev. Lett. 114, 140502 (2015).
[Crossref]

You, L.-X.

C. Chen, X. Ding, J. Qin, Y. He, Y.-H. Luo, M.-C. Chen, C. Liu, X.-L. Wang, W.-J. Zhang, H. Li, L.-X. You, Z. Wang, D.-W. Wang, B. C. Sanders, C.-Y. Lu, and J.-W. Pan, “Observation of topologically protected edge states in a photonic two-dimensional quantum walk,” Phys. Rev. Lett. 121, 100502 (2018).
[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]

Yuan, L.

A. Dutt, M. Minkov, Q. Lin, L. Yuan, D. A. B. Miller, and S. Fan, “Experimental band structure spectroscopy along a synthetic dimension,” Nat. Commun. 10, 3122 (2019).
[Crossref]

L. Yuan, M. Xiao, Q. Lin, and S. Fan, “Synthetic space with arbitrary dimensions in a few rings undergoing dynamic modulation,” Phys. Rev. B 97, 104105 (2018).
[Crossref]

Q. Lin, M. Xiao, L. Yuan, and S. Fan, “Photonic Weyl point in a two-dimensional resonator lattice with a synthetic frequency dimension,” Nat. Commun. 7, 13731 (2016).
[Crossref]

L. Yuan and S. Fan, “Bloch oscillation and unidirectional translation of frequency in a dynamically modulated ring resonator,” Optica 3, 1014–1018 (2016).
[Crossref]

L. Yuan and S. Fan, “Three-dimensional dynamic localization of light from a time-dependent effective gauge field for photons,” Phys. Rev. Lett. 114, 243901 (2015).
[Crossref]

Zeilinger, A.

X.-S. Ma, B. Dakic, W. Naylor, A. Zeilinger, and P. Walther, “Quantum simulation of the wavefunction to probe frustrated Heisenberg spin systems,” Nat. Phys. 7, 399–405 (2011).
[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]

Zhan, X.

P. Xue, R. Zhang, H. Qin, X. Zhan, Z. H. Bian, J. Li, and B. C. Sanders, “Experimental quantum-walk revival with a time-dependent coin,” Phys. Rev. Lett. 114, 140502 (2015).
[Crossref]

Zhang, R.

P. Xue, R. Zhang, H. Qin, X. Zhan, Z. H. Bian, J. Li, and B. C. Sanders, “Experimental quantum-walk revival with a time-dependent coin,” Phys. Rev. Lett. 114, 140502 (2015).
[Crossref]

Zhang, S.-C.

Q. Lin, X.-Q. Sun, M. Xiao, S.-C. Zhang, and S. Fan, “A three-dimensional photonic topological insulator using a two-dimensional ring resonator lattice with a synthetic frequency dimension,” Sci. Adv. 4, eaat2774 (2018).
[Crossref]

Zhang, W.-J.

C. Chen, X. Ding, J. Qin, Y. He, Y.-H. Luo, M.-C. Chen, C. Liu, X.-L. Wang, W.-J. Zhang, H. Li, L.-X. You, Z. Wang, D.-W. Wang, B. C. Sanders, C.-Y. Lu, and J.-W. Pan, “Observation of topologically protected edge states in a photonic two-dimensional quantum walk,” Phys. Rev. Lett. 121, 100502 (2018).
[Crossref]

Zhang, X.

C. Qin, F. Zhou, Y. Peng, D. Sounas, X. Zhu, B. Wang, J. Dong, X. Zhang, A. Alù, and P. Lu, “Spectrum control through discrete frequency diffraction in the presence of photonic gauge potentials,” Phys. Rev. Lett. 120, 133901 (2018).
[Crossref]

Zhou, F.

C. Qin, F. Zhou, Y. Peng, D. Sounas, X. Zhu, B. Wang, J. Dong, X. Zhang, A. Alù, and P. Lu, “Spectrum control through discrete frequency diffraction in the presence of photonic gauge potentials,” Phys. Rev. Lett. 120, 133901 (2018).
[Crossref]

Zhu, X.

C. Qin, F. Zhou, Y. Peng, D. Sounas, X. Zhu, B. Wang, J. Dong, X. Zhang, A. Alù, and P. Lu, “Spectrum control through discrete frequency diffraction in the presence of photonic gauge potentials,” Phys. Rev. Lett. 120, 133901 (2018).
[Crossref]

Zilberberg, O.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91, 015006 (2019).
[Crossref]

Zoller, P.

H. Pichler, S. Choi, P. Zoller, and M. D. Lukin, “Universal photonic quantum computation via time-delayed feedback,” Proc. Natl. Acad. Sci. USA 114, 11362–11367 (2017).
[Crossref]

Europhys. Lett. (1)

S. Longhi, “Optical Zener-Bloch oscillations in binary waveguide arrays,” Europhys. Lett. 76, 416–421 (2006).
[Crossref]

J. Math. Phys. (1)

C. Cedzich, T. Geib, A. H. Werner, and R. F. Werner, “Quantum walks in external gauge fields,” J. Math. Phys. 60, 012107 (2019).
[Crossref]

Nat. Chem. (1)

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nat. Chem. 2, 106–111 (2010).
[Crossref]

Nat. Commun. (5)

A. Dutt, M. Minkov, Q. Lin, L. Yuan, D. A. B. Miller, and S. Fan, “Experimental band structure spectroscopy along a synthetic dimension,” Nat. Commun. 10, 3122 (2019).
[Crossref]

Q. Lin, M. Xiao, L. Yuan, and S. Fan, “Photonic Weyl point in a two-dimensional resonator lattice with a synthetic frequency dimension,” Nat. Commun. 7, 13731 (2016).
[Crossref]

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7, 11439 (2016).
[Crossref]

F. Cardano, A. D’Errico, A. Dauphin, M. Maffei, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, L. Marrucci, M. Lewenstein, and P. Massignan, “Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons,” Nat. Commun. 8, 15516 (2017).
[Crossref]

G. Corrielli, A. Crespi, G. Della Valle, S. Longhi, and R. Osellame, “Fractional Bloch oscillations in photonic lattices,” Nat. Commun. 4, 1555 (2013).
[Crossref]

Nat. Photonics (2)

S. Mittal, S. Ganeshan, J. Fan, A. Vaezi, and M. Hafezi, “Measurement of topological invariants in a 2D photonic system,” Nat. Photonics 10, 180–183 (2016).
[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. (3)

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]

X.-S. Ma, B. Dakic, W. Naylor, A. Zeilinger, and P. Walther, “Quantum simulation of the wavefunction to probe frustrated Heisenberg spin systems,” Nat. Phys. 7, 399–405 (2011).
[Crossref]

A. Aspuru-Guzik and P. Walther, “Photonic quantum simulators,” Nat. Phys. 8, 285–291 (2012).
[Crossref]

Nature (4)

C. Sparrow, E. Martín-López, N. Maraviglia, A. Neville, C. Harrold, J. Carolan, Y. N. Joglekar, T. Hashimoto, N. Matsuda, J. L. O’Brien, D. P. Tew, and A. Laing, “Simulating the vibrational quantum dynamics of molecules using photonics,” Nature 557, 660–667 (2018).
[Crossref]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref]

E. Lustig, S. Weimann, Y. Plotnik, Y. Lumer, M. A. Bandres, A. Szameit, and M. Segev, “Photonic topological insulator in synthetic dimensions,” Nature 567, 356–360 (2019).
[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]

New J. Phys. (2)

T. Nitsche, F. Elster, J. Novotný, A. Gábris, I. Jex, S. Barkhofen, and C. Silberhorn, “Quantum walks with dynamical control: graph engineering, initial state preparation and state transfer,” New J. Phys. 18, 063017 (2016).
[Crossref]

D. Witthaut, F. Keck, H. J. Korsch, and S. Mossmann, “Bloch oscillations in two-dimensional lattices,” New J. Phys. 6, 41 (2004).
[Crossref]

Open Phys. (1)

U. Peschel, C. Bersch, and G. Onishchukov, “Discreteness in time,” Open Phys. 6, 619–627 (2008).
[Crossref]

Opt. Commun. (1)

G. Lenz, R. Parker, M. Wanke, and C. de Sterke, “Dynamical localization and AC Bloch oscillations in periodic optical waveguide arrays,” Opt. Commun. 218, 87–92 (2003).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Optica (3)

Phys. Rev. A (6)

H. Chalabi and E. Waks, “Interaction of photons with a coupled atom-cavity system through a bidirectional time-delayed feedback,” Phys. Rev. A 98, 063832 (2018).
[Crossref]

C. Cedzich and R. F. Werner, “Revivals in quantum walks with a quasiperiodically-time-dependent coin,” Phys. Rev. A 93, 032329 (2016).
[Crossref]

L. A. Bru, M. Hinarejos, F. Silva, G. J. de Valcárcel, and E. Roldán, “Electric quantum walks in two dimensions,” Phys. Rev. A 93, 032333 (2016).
[Crossref]

S. Barkhofen, T. Nitsche, F. Elster, L. Lorz, A. Gábris, I. Jex, and C. Silberhorn, “Measuring topological invariants in disordered discrete-time quantum walks,” Phys. Rev. A 96, 033846 (2017).
[Crossref]

C. Navarrete-Benlloch, A. Pérez, and E. Roldán, “Nonlinear optical Galton board,” Phys. Rev. A 75, 062333 (2007).
[Crossref]

D. Bouwmeester, I. Marzoli, G. P. Karman, W. Schleich, and J. P. Woerdman, “Optical Galton board,” Phys. Rev. A 61, 013410 (1999).
[Crossref]

Phys. Rev. B (2)

L. Yuan, M. Xiao, Q. Lin, and S. Fan, “Synthetic space with arbitrary dimensions in a few rings undergoing dynamic modulation,” Phys. Rev. B 97, 104105 (2018).
[Crossref]

D. H. Dunlap and V. M. Kenkre, “Dynamic localization of a charged particle moving under the influence of an electric field,” Phys. Rev. B 34, 3625–3633 (1986).
[Crossref]

Phys. Rev. Lett. (23)

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[Crossref]

A. Joushaghani, R. Iyer, J. K. S. Poon, J. S. Aitchison, C. M. de Sterke, J. Wan, and M. M. Dignam, “Quasi-Bloch oscillations in curved coupled optical waveguides,” Phys. Rev. Lett. 103, 143903 (2009).
[Crossref]

A. Joushaghani, R. Iyer, J. K. S. Poon, J. S. Aitchison, C. M. de Sterke, J. Wan, and M. M. Dignam, “Generalized exact dynamic localization in curved coupled optical waveguide arrays,” Phys. Rev. Lett. 109, 103901 (2012).
[Crossref]

L. Yuan and S. Fan, “Three-dimensional dynamic localization of light from a time-dependent effective gauge field for photons,” Phys. Rev. Lett. 114, 243901 (2015).
[Crossref]

V. V. Orre, E. A. Goldschmidt, A. Deshpande, A. V. Gorshkov, V. Tamma, M. Hafezi, and S. Mittal, “Interference of temporally distinguishable photons using frequency-resolved detection,” Phys. Rev. Lett. 123, 123603 (2019).
[Crossref]

H. Chalabi, S. Barik, S. Mittal, T. E. Murphy, M. Hafezi, and E. Waks, “Synthetic gauge field for two-dimensional time-multiplexed quantum random walks,” Phys. Rev. Lett. 123, 150503 (2019).
[Crossref]

C. Cedzich, T. Rybár, A. H. Werner, A. Alberti, M. Genske, and R. F. Werner, “Propagation of quantum walks in electric fields,” Phys. Rev. Lett. 111, 160601 (2013).
[Crossref]

M. Genske, W. Alt, A. Steffen, A. H. Werner, R. F. Werner, D. Meschede, and A. Alberti, “Electric quantum walks with individual atoms,” Phys. Rev. Lett. 110, 190601 (2013).
[Crossref]

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, “Bloch-Zener oscillations in binary superlattices,” Phys. Rev. Lett. 102, 076802 (2009).
[Crossref]

P. Xue, R. Zhang, H. Qin, X. Zhan, Z. H. Bian, J. Li, and B. C. Sanders, “Experimental quantum-walk revival with a time-dependent coin,” Phys. Rev. Lett. 114, 140502 (2015).
[Crossref]

S. K. Goyal, F. S. Roux, A. Forbes, and T. Konrad, “Implementing quantum walks using orbital angular momentum of classical light,” Phys. Rev. Lett. 110, 263602 (2013).
[Crossref]

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref]

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, I. Jex, and C. Silberhorn, “Decoherence and disorder in quantum walks: from ballistic spread to localization,” Phys. Rev. Lett. 106, 180403 (2011).
[Crossref]

C. Qin, F. Zhou, Y. Peng, D. Sounas, X. Zhu, B. Wang, J. Dong, X. Zhang, A. Alù, and P. Lu, “Spectrum control through discrete frequency diffraction in the presence of photonic gauge potentials,” Phys. Rev. Lett. 120, 133901 (2018).
[Crossref]

C. Chen, X. Ding, J. Qin, Y. He, Y.-H. Luo, M.-C. Chen, C. Liu, X.-L. Wang, W.-J. Zhang, H. Li, L.-X. You, Z. Wang, D.-W. Wang, B. C. Sanders, C.-Y. Lu, and J.-W. Pan, “Observation of topologically protected edge states in a photonic two-dimensional quantum walk,” Phys. Rev. Lett. 121, 100502 (2018).
[Crossref]

S. Mittal, J. Fan, S. Faez, A. Migdall, J. M. Taylor, and M. Hafezi, “Topologically robust transport of photons in a synthetic gauge field,” Phys. Rev. Lett. 113, 087403 (2014).
[Crossref]

Y. Bromberg, Y. Lahini, and Y. Silberberg, “Bloch oscillations of path-entangled photons,” Phys. Rev. Lett. 105, 263604 (2010).
[Crossref]

E. Haller, R. Hart, M. J. Mark, J. G. Danzl, L. Reichsöllner, and H.-C. Nägerl, “Inducing transport in a dissipation-free lattice with super Bloch oscillations,” Phys. Rev. Lett. 104, 200403 (2010).
[Crossref]

M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, “Bloch oscillations of atoms in an optical potential,” Phys. Rev. Lett. 76, 4508–4511 (1996).
[Crossref]

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[Crossref]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[Crossref]

G. Lenz, I. Talanina, and C. M. de Sterke, “Bloch Oscillations in an array of curved optical waveguides,” Phys. Rev. Lett. 83, 963–966 (1999).
[Crossref]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[Crossref]

Phys. Rev. X (1)

W. Hu, J. C. Pillay, K. Wu, M. Pasek, P. P. Shum, and Y. D. Chong, “Measurement of a topological edge invariant in a microwave network,” Phys. Rev. X 5, 011012 (2015).
[Crossref]

Proc. Natl. Acad. Sci. USA (1)

H. Pichler, S. Choi, P. Zoller, and M. D. Lukin, “Universal photonic quantum computation via time-delayed feedback,” Proc. Natl. Acad. Sci. USA 114, 11362–11367 (2017).
[Crossref]

Rev. Mod. Phys. (1)

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91, 015006 (2019).
[Crossref]

Sci. Adv. (2)

F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, “Quantum walks and wavepacket dynamics on a lattice with twisted photons,” Sci. Adv. 1, e1500087 (2015).
[Crossref]

Q. Lin, X.-Q. Sun, M. Xiao, S.-C. Zhang, and S. Fan, “A three-dimensional photonic topological insulator using a two-dimensional ring resonator lattice with a synthetic frequency dimension,” Sci. Adv. 4, eaat2774 (2018).
[Crossref]

Science (1)

A. Schreiber, A. Gabris, P. P. Rohde, K. Laiho, M. Stefanak, V. Potocek, C. Hamilton, I. Jex, and C. Silberhorn, “A 2D quantum walk simulation of two-particle dynamics,” Science 336, 55–58 (2012).
[Crossref]

Z. Phys. (1)

F. Bloch, “Über die Quantenmechanik der Elektronen in Kristallgittern,” Z. Phys. 52, 555–600 (1929).
[Crossref]

Supplementary Material (1)

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» Supplement 1       Experimental details and theoretical analysis

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Figures (8)

Fig. 1.
Fig. 1. Applied phase modulation scheme for the $n$th step of the quantum walk.
Fig. 2.
Fig. 2. Pseudo energy band diagrams of the two-dimensional quantum walk for different gauge field strengths: (a) $\phi = \pi /2$, (b) $\phi = \pi /3$, and (c) $\phi = \pi /4$. In each row the left figure shows the full band diagram, while the right one shows the zoomed in part of the band diagram.
Fig. 3.
Fig. 3. (a) Experimental observations and (b) theoretical predictions of the evolution of the quantum walk distribution under the application of the time-dependent phase modulation with $\phi = \pi /4$. The columns from left to right show the distributions at time steps of 0, 2, 4, 6, and 8, respectively. In these plots, all the distributions are normalized to their maximum.
Fig. 4.
Fig. 4. Theoretical values for the quadratic means of (a) $y$ and (c) $x$ as a function of the time step for different gauge field strengths. Experimental values for the quadratic means of (b) $y$ and (d) $x$ as a function of the time step for different gauge field strengths.
Fig. 5.
Fig. 5. Theoretical values for the norm ones of (a) $y$ and (c) $x$ as a function of the time step for different gauge field strengths. Experimental values for the norm ones of (b) $y$ and (d) $x$ as a function of the time step for different gauge field strengths.
Fig. 6.
Fig. 6. Probability of revival: experimental (blue solid squares) and theoretical (red solid circles) probabilities of the walker returning to the origin with respect to the required number of steps for the revival to happen. The error bars in the measurements are smaller than the size of the plotted data points.
Fig. 7.
Fig. 7. Confinement of the quantum walker through the application of a discontinuous electric field. (a) Schematic describing the phase modulation pattern in the synthetic space that leads to a zero electric field for ${-}2 \lt y \lt 2$ and a nonzero electric field outside this range. (b) Experimental observations and (c) theoretical predictions of the evolution of the quantum walk distribution under the discontinuous electric field. The columns from left to right show the distributions at time steps of 1, 5, and 9, respectively. All the distributions are normalized to their maximum. (d) Experimentally measured and theoretically predicted quadratic means of $x$ and $y$ as a function of the time step. The error bars in the measurements are smaller than the size of the plotted data points.
Fig. 8.
Fig. 8. Theoretical predictions of the evolution of the quantum walk distribution under the discontinuous electric field for the thickness of (a) $t = 2$, (b) $t = 6$, and (c) $t = 8$. The columns from left to right show the distributions at time steps of 0, 4, 8, and 12, respectively. All the distributions are normalized to their maximum. Quadratic means of (d) $x$ and (e) $y$ as a function of the time step for different thicknesses of the discontinuity.

Equations (1)

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E n , ± , k x , k y = 2 n π q ± 1 q arccos ( cos ( π q 2 ) ( sin q k y 1 ) cos ( q k x + π q 2 ) sin q k y ) ,