Abstract

Based on the recently proposed concept of effective gauge potential and magnetic field for photons, we numerically demonstrate a photonic de Haas-van Alphen effect. We show that in a dynamically modulated photonic resonator lattice exhibiting an effect magnetic field, the trajectories of the light beam at a given frequency have the same shape as the constant energy contour for the photonic band structure of the lattice in the absence of the effective magnetic field.

© 2013 OSA

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  1. U. Peschel, T. Pertsch, and F. Lederer, “Optical Bloch oscillations in waveguide arrays,” Opt. Lett.23, 1701–1703 (1998).
    [CrossRef]
  2. P. St. J. Russell and T. A. Birks, “Hamiltonian optics of nonuniform photonic crystals,” J. of Lightw. Tech17, 1982–1988 (1999).
    [CrossRef]
  3. 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]
  4. 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]
  5. R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, “Optical analogue of electronic Bloch oscillations,” Phys. Rev. Lett.91, 263902 (2003).
    [CrossRef]
  6. R. Khomeriki and S. Ruffo, “Nonadiabatic Landau-Zener tunneling in waveguide arrays with a step in the refractive index,” Phys. Rev. Lett.94, 113904 (2005).
    [CrossRef] [PubMed]
  7. 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] [PubMed]
  8. L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett.103, 033902 (2009).
    [CrossRef] [PubMed]
  9. K. Fang, Z. Yu, and S. Fan, “Photonic Aharonov-Bohm effect based on dynamic modulation,” Phys. Rev. Lett.108, 153901 (2012).
    [CrossRef] [PubMed]
  10. K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nature Photons.6, 782–787 (2012).
    [CrossRef]
  11. K. Fang, Z. Yu, and S. Fan, “Experimental demonstration of a photonic Aharonov-Bohm effect at radio frequencies,” Phys. Rev. B87, 060301(R) (2013).
    [CrossRef]
  12. Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nature Photon.3, 91–94 (2009).
    [CrossRef]
  13. M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nature Phys.7, 907–912 (2011).
    [CrossRef]
  14. R. O. Umucallar and I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A84, 043804 (2011).
    [CrossRef]
  15. A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater.12, 233–239 (2013).
    [CrossRef]
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    [CrossRef] [PubMed]
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  22. H. Sambe, “Steady states and quasienergies of a quantum-mechanical system in an oscillating field,” Phys. Rev. A7, 2203–2213 (1973).
    [CrossRef]
  23. J. M. Luttinger, “The effect of a magnetic field on electrons in a periodic potential,” Phys. Rev.84, 814–817 (1951).
    [CrossRef]
  24. R. Kosloff, “Time-dependent quantum-mechanical methods for molecular dynamics,” J. Phys. Chem92, 2087–2100 (1988).
    [CrossRef]
  25. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Brooks Cole, 1976).
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    [CrossRef]
  28. S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Science394, 251–253 (1998).

2013

K. Fang, Z. Yu, and S. Fan, “Experimental demonstration of a photonic Aharonov-Bohm effect at radio frequencies,” Phys. Rev. B87, 060301(R) (2013).
[CrossRef]

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater.12, 233–239 (2013).
[CrossRef]

M. C. Rechtsman, J. M. Zeuner, A. Tnnermann, S. Nolte, M. Segev, and A. Szameit, “Strain-induced pseudo-magnetic field and photonic Landau levels in dielectric structures,” Nature Photon.7, 153–158 (2013).
[CrossRef]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature496, 196 (2013).
[CrossRef] [PubMed]

2012

K. Fang, Z. Yu, and S. Fan, “Photonic Aharonov-Bohm effect based on dynamic modulation,” Phys. Rev. Lett.108, 153901 (2012).
[CrossRef] [PubMed]

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

2011

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

R. O. Umucallar and I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A84, 043804 (2011).
[CrossRef]

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

2010

J. Koch, A. A. Houck, K. Le Hur, and S. M. Girvin, “Time-reversal-symmetry breaking in circuit-QED-based photon lattices,” Phys. Rev. A82, 043811 (2010).
[CrossRef]

T. Kitagawa, E. Berg, M. Rudner, and E. Demler, “Topological characterization of periodically driven quantum systems, ” Phys. Rev. B82, 235114 (2010).
[CrossRef]

2009

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

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett.103, 033902 (2009).
[CrossRef] [PubMed]

2008

M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nature Photons.2, 741–747 (2008).
[CrossRef]

2007

2006

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] [PubMed]

2005

R. Khomeriki and S. Ruffo, “Nonadiabatic Landau-Zener tunneling in waveguide arrays with a step in the refractive index,” Phys. Rev. Lett.94, 113904 (2005).
[CrossRef] [PubMed]

2003

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, “Optical analogue of electronic Bloch oscillations,” Phys. Rev. Lett.91, 263902 (2003).
[CrossRef]

1999

P. St. J. Russell and T. A. Birks, “Hamiltonian optics of nonuniform photonic crystals,” J. of Lightw. Tech17, 1982–1988 (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]

1998

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Science394, 251–253 (1998).

U. Peschel, T. Pertsch, and F. Lederer, “Optical Bloch oscillations in waveguide arrays,” Opt. Lett.23, 1701–1703 (1998).
[CrossRef]

1988

R. Kosloff, “Time-dependent quantum-mechanical methods for molecular dynamics,” J. Phys. Chem92, 2087–2100 (1988).
[CrossRef]

1973

H. Sambe, “Steady states and quasienergies of a quantum-mechanical system in an oscillating field,” Phys. Rev. A7, 2203–2213 (1973).
[CrossRef]

1965

J. H. Shirley, “Solution of the Schrdinger Equation with a Hamiltonian periodic in time,” Phys. Rev.138, B979–B987 (1965).
[CrossRef]

1951

J. M. Luttinger, “The effect of a magnetic field on electrons in a periodic potential,” Phys. Rev.84, 814–817 (1951).
[CrossRef]

Aitchison, J. 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]

Asano, T.

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Brooks Cole, 1976).

Berg, E.

T. Kitagawa, E. Berg, M. Rudner, and E. Demler, “Topological characterization of periodically driven quantum systems, ” Phys. Rev. B82, 235114 (2010).
[CrossRef]

Birks, T. A.

P. St. J. Russell and T. A. Birks, “Hamiltonian optics of nonuniform photonic crystals,” J. of Lightw. Tech17, 1982–1988 (1999).
[CrossRef]

Biswas, R.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Science394, 251–253 (1998).

Bur, J.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Science394, 251–253 (1998).

Carusotto, I.

R. O. Umucallar and I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A84, 043804 (2011).
[CrossRef]

Catrysse, P. B.

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett.103, 033902 (2009).
[CrossRef] [PubMed]

Costantino, P.

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, “Optical analogue of electronic Bloch oscillations,” Phys. Rev. Lett.91, 263902 (2003).
[CrossRef]

de Sterke, C.M.

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]

Demler, E.

T. Kitagawa, E. Berg, M. Rudner, and E. Demler, “Topological characterization of periodically driven quantum systems, ” Phys. Rev. B82, 235114 (2010).
[CrossRef]

Demler, E. A.

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

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,” Nature496, 196 (2013).
[CrossRef] [PubMed]

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]

Fan, S.

K. Fang, Z. Yu, and S. Fan, “Experimental demonstration of a photonic Aharonov-Bohm effect at radio frequencies,” Phys. Rev. B87, 060301(R) (2013).
[CrossRef]

K. Fang, Z. Yu, and S. Fan, “Photonic Aharonov-Bohm effect based on dynamic modulation,” Phys. Rev. Lett.108, 153901 (2012).
[CrossRef] [PubMed]

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

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett.103, 033902 (2009).
[CrossRef] [PubMed]

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

Fang, K.

K. Fang, Z. Yu, and S. Fan, “Experimental demonstration of a photonic Aharonov-Bohm effect at radio frequencies,” Phys. Rev. B87, 060301(R) (2013).
[CrossRef]

K. Fang, Z. Yu, and S. Fan, “Photonic Aharonov-Bohm effect based on dynamic modulation,” Phys. Rev. Lett.108, 153901 (2012).
[CrossRef] [PubMed]

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

Fleming, J. G.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Science394, 251–253 (1998).

Galitski, V.

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

Ghulinyan, M.

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, “Optical analogue of electronic Bloch oscillations,” Phys. Rev. Lett.91, 263902 (2003).
[CrossRef]

Girvin, S. M.

J. Koch, A. A. Houck, K. Le Hur, and S. M. Girvin, “Time-reversal-symmetry breaking in circuit-QED-based photon lattices,” Phys. Rev. A82, 043811 (2010).
[CrossRef]

Hafezi, M.

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

Hagino, H.

Hetherington, D. L.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Science394, 251–253 (1998).

Ho, K. M.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Science394, 251–253 (1998).

Houck, A. A.

J. Koch, A. A. Houck, K. Le Hur, and S. M. Girvin, “Time-reversal-symmetry breaking in circuit-QED-based photon lattices,” Phys. Rev. A82, 043811 (2010).
[CrossRef]

Kargarian, M.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater.12, 233–239 (2013).
[CrossRef]

Khanikaev, A. B.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater.12, 233–239 (2013).
[CrossRef]

Khomeriki, R.

R. Khomeriki and S. Ruffo, “Nonadiabatic Landau-Zener tunneling in waveguide arrays with a step in the refractive index,” Phys. Rev. Lett.94, 113904 (2005).
[CrossRef] [PubMed]

Kitagawa, T.

T. Kitagawa, E. Berg, M. Rudner, and E. Demler, “Topological characterization of periodically driven quantum systems, ” Phys. Rev. B82, 235114 (2010).
[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] [PubMed]

Koch, J.

J. Koch, A. A. Houck, K. Le Hur, and S. M. Girvin, “Time-reversal-symmetry breaking in circuit-QED-based photon lattices,” Phys. Rev. A82, 043811 (2010).
[CrossRef]

Kosloff, R.

R. Kosloff, “Time-dependent quantum-mechanical methods for molecular dynamics,” J. Phys. Chem92, 2087–2100 (1988).
[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] [PubMed]

Kuramochi, E.

M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nature Photons.2, 741–747 (2008).
[CrossRef]

Kurtz, S. R.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Science394, 251–253 (1998).

Le Hur, K.

J. Koch, A. A. Houck, K. Le Hur, and S. M. Girvin, “Time-reversal-symmetry breaking in circuit-QED-based photon lattices,” Phys. Rev. A82, 043811 (2010).
[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] [PubMed]

U. Peschel, T. Pertsch, and F. Lederer, “Optical Bloch oscillations in waveguide arrays,” Opt. Lett.23, 1701–1703 (1998).
[CrossRef]

Lenz, G.

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]

Lin, S. Y.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Science394, 251–253 (1998).

Lindner, N. H.

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

Lukin, M. D.

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

Lumer, Y.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature496, 196 (2013).
[CrossRef] [PubMed]

Luttinger, J. M.

J. M. Luttinger, “The effect of a magnetic field on electrons in a periodic potential,” Phys. Rev.84, 814–817 (1951).
[CrossRef]

MacDonald, A. H.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater.12, 233–239 (2013).
[CrossRef]

Mermin, N. D.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Brooks Cole, 1976).

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]

Mousavi, S. H.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater.12, 233–239 (2013).
[CrossRef]

Neshev, D. N.

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] [PubMed]

Noda, S.

Nolte, S.

M. C. Rechtsman, J. M. Zeuner, A. Tnnermann, S. Nolte, M. Segev, and A. Szameit, “Strain-induced pseudo-magnetic field and photonic Landau levels in dielectric structures,” Nature Photon.7, 153–158 (2013).
[CrossRef]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature496, 196 (2013).
[CrossRef] [PubMed]

Notomi, M.

M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nature Photons.2, 741–747 (2008).
[CrossRef]

Oton, C. J.

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, “Optical analogue of electronic Bloch oscillations,” Phys. Rev. Lett.91, 263902 (2003).
[CrossRef]

Pavesi, L.

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, “Optical analogue of electronic Bloch oscillations,” Phys. Rev. Lett.91, 263902 (2003).
[CrossRef]

Pertsch, T.

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] [PubMed]

U. Peschel, T. Pertsch, and F. Lederer, “Optical Bloch oscillations in waveguide arrays,” Opt. Lett.23, 1701–1703 (1998).
[CrossRef]

Peschel, U.

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] [PubMed]

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]

U. Peschel, T. Pertsch, and F. Lederer, “Optical Bloch oscillations in waveguide arrays,” Opt. Lett.23, 1701–1703 (1998).
[CrossRef]

Plotnik, Y.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature496, 196 (2013).
[CrossRef] [PubMed]

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,” Nature496, 196 (2013).
[CrossRef] [PubMed]

Rechtsman, M. C.

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,” Nature496, 196 (2013).
[CrossRef] [PubMed]

M. C. Rechtsman, J. M. Zeuner, A. Tnnermann, S. Nolte, M. Segev, and A. Szameit, “Strain-induced pseudo-magnetic field and photonic Landau levels in dielectric structures,” Nature Photon.7, 153–158 (2013).
[CrossRef]

Refael, G.

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

Rudner, M.

T. Kitagawa, E. Berg, M. Rudner, and E. Demler, “Topological characterization of periodically driven quantum systems, ” Phys. Rev. B82, 235114 (2010).
[CrossRef]

Ruffo, S.

R. Khomeriki and S. Ruffo, “Nonadiabatic Landau-Zener tunneling in waveguide arrays with a step in the refractive index,” Phys. Rev. Lett.94, 113904 (2005).
[CrossRef] [PubMed]

Russell, P. St. J.

P. St. J. Russell and T. A. Birks, “Hamiltonian optics of nonuniform photonic crystals,” J. of Lightw. Tech17, 1982–1988 (1999).
[CrossRef]

Sambe, H.

H. Sambe, “Steady states and quasienergies of a quantum-mechanical system in an oscillating field,” Phys. Rev. A7, 2203–2213 (1973).
[CrossRef]

Sapienza, R.

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, “Optical analogue of electronic Bloch oscillations,” Phys. Rev. Lett.91, 263902 (2003).
[CrossRef]

Segev, M.

M. C. Rechtsman, J. M. Zeuner, A. Tnnermann, S. Nolte, M. Segev, and A. Szameit, “Strain-induced pseudo-magnetic field and photonic Landau levels in dielectric structures,” Nature Photon.7, 153–158 (2013).
[CrossRef]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature496, 196 (2013).
[CrossRef] [PubMed]

Shirley, J. H.

J. H. Shirley, “Solution of the Schrdinger Equation with a Hamiltonian periodic in time,” Phys. Rev.138, B979–B987 (1965).
[CrossRef]

Shvets, G.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater.12, 233–239 (2013).
[CrossRef]

Sigalas, M. M.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Science394, 251–253 (1998).

Silberberg, Y.

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]

Smith, B. K.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Science394, 251–253 (1998).

Song, B. S.

Sukhorukov, A. A.

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] [PubMed]

Szameit, A.

M. C. Rechtsman, J. M. Zeuner, A. Tnnermann, S. Nolte, M. Segev, and A. Szameit, “Strain-induced pseudo-magnetic field and photonic Landau levels in dielectric structures,” Nature Photon.7, 153–158 (2013).
[CrossRef]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature496, 196 (2013).
[CrossRef] [PubMed]

Takahashi, Y.

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]

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M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nature Photons.2, 741–747 (2008).
[CrossRef]

Tanaka, Y.

Taylor, J. M.

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

Tnnermann, A.

M. C. Rechtsman, J. M. Zeuner, A. Tnnermann, S. Nolte, M. Segev, and A. Szameit, “Strain-induced pseudo-magnetic field and photonic Landau levels in dielectric structures,” Nature Photon.7, 153–158 (2013).
[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] [PubMed]

Tse, W.-K.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater.12, 233–239 (2013).
[CrossRef]

Umucallar, R. O.

R. O. Umucallar and I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A84, 043804 (2011).
[CrossRef]

Verslegers, L.

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett.103, 033902 (2009).
[CrossRef] [PubMed]

Wiersma, D.

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, “Optical analogue of electronic Bloch oscillations,” Phys. Rev. Lett.91, 263902 (2003).
[CrossRef]

Yu, Z.

K. Fang, Z. Yu, and S. Fan, “Experimental demonstration of a photonic Aharonov-Bohm effect at radio frequencies,” Phys. Rev. B87, 060301(R) (2013).
[CrossRef]

K. Fang, Z. Yu, and S. Fan, “Photonic Aharonov-Bohm effect based on dynamic modulation,” Phys. Rev. Lett.108, 153901 (2012).
[CrossRef] [PubMed]

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

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett.103, 033902 (2009).
[CrossRef] [PubMed]

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nature Photon.3, 91–94 (2009).
[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,” Nature496, 196 (2013).
[CrossRef] [PubMed]

M. C. Rechtsman, J. M. Zeuner, A. Tnnermann, S. Nolte, M. Segev, and A. Szameit, “Strain-induced pseudo-magnetic field and photonic Landau levels in dielectric structures,” Nature Photon.7, 153–158 (2013).
[CrossRef]

Zubrzycki, W.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Science394, 251–253 (1998).

J. of Lightw. Tech

P. St. J. Russell and T. A. Birks, “Hamiltonian optics of nonuniform photonic crystals,” J. of Lightw. Tech17, 1982–1988 (1999).
[CrossRef]

J. Phys. Chem

R. Kosloff, “Time-dependent quantum-mechanical methods for molecular dynamics,” J. Phys. Chem92, 2087–2100 (1988).
[CrossRef]

Nat. Mater.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater.12, 233–239 (2013).
[CrossRef]

Nature

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,” Nature496, 196 (2013).
[CrossRef] [PubMed]

Nature Photon.

M. C. Rechtsman, J. M. Zeuner, A. Tnnermann, S. Nolte, M. Segev, and A. Szameit, “Strain-induced pseudo-magnetic field and photonic Landau levels in dielectric structures,” Nature Photon.7, 153–158 (2013).
[CrossRef]

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

Nature Photons.

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

M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nature Photons.2, 741–747 (2008).
[CrossRef]

Nature Phys.

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

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

Opt. express

Opt. Lett.

Phys. Rev.

J. M. Luttinger, “The effect of a magnetic field on electrons in a periodic potential,” Phys. Rev.84, 814–817 (1951).
[CrossRef]

J. H. Shirley, “Solution of the Schrdinger Equation with a Hamiltonian periodic in time,” Phys. Rev.138, B979–B987 (1965).
[CrossRef]

Phys. Rev. A

H. Sambe, “Steady states and quasienergies of a quantum-mechanical system in an oscillating field,” Phys. Rev. A7, 2203–2213 (1973).
[CrossRef]

R. O. Umucallar and I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A84, 043804 (2011).
[CrossRef]

J. Koch, A. A. Houck, K. Le Hur, and S. M. Girvin, “Time-reversal-symmetry breaking in circuit-QED-based photon lattices,” Phys. Rev. A82, 043811 (2010).
[CrossRef]

Phys. Rev. B

T. Kitagawa, E. Berg, M. Rudner, and E. Demler, “Topological characterization of periodically driven quantum systems, ” Phys. Rev. B82, 235114 (2010).
[CrossRef]

K. Fang, Z. Yu, and S. Fan, “Experimental demonstration of a photonic Aharonov-Bohm effect at radio frequencies,” Phys. Rev. B87, 060301(R) (2013).
[CrossRef]

Phys. Rev. Lett.

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]

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, “Optical analogue of electronic Bloch oscillations,” Phys. Rev. Lett.91, 263902 (2003).
[CrossRef]

R. Khomeriki and S. Ruffo, “Nonadiabatic Landau-Zener tunneling in waveguide arrays with a step in the refractive index,” Phys. Rev. Lett.94, 113904 (2005).
[CrossRef] [PubMed]

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] [PubMed]

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett.103, 033902 (2009).
[CrossRef] [PubMed]

K. Fang, Z. Yu, and S. Fan, “Photonic Aharonov-Bohm effect based on dynamic modulation,” Phys. Rev. Lett.108, 153901 (2012).
[CrossRef] [PubMed]

Science

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Science394, 251–253 (1998).

Other

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Brooks Cole, 1976).

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Figures (2)

Fig. 1
Fig. 1

a Schematic of a photonic crystal resonator lattice with dynamically modulated nearest-neighbor coupling. The modulation phase is zero along the x direcition, and varies in space along the y direction as indicated in the figure. b Constant energy contours in the first Brillouin zone of a square lattice with lattice constant a and nearest-neighbor coupling strength V in the absence of effective magnetic field, which corresponds to the case with ϕ = 0 everywhere in a.

Fig. 2
Fig. 2

Beam trajectories for different initial momenta. The unit of axes is a. The initial momenta are: (a) k 1 = 0.41 ( π / a ) x ^, (b) k 2 = 0.48 ( π / a ) x ^ + 0.48 ( π / a ) y ^, (c) k 3 = 0.59 ( π / a ) x ^ + ( π / a ) y ^, (d) k 4 = 0.5 ( π / a ) x ^ + 0.5 ( π / a ) y ^ as labeled in Fig. 1(b). The width w of the source is 50 a. Red arrows indicate the initial propagation direction. For (d), the beam will eventually trace out a large square grid. Here we show only part of such a grid that has been traced out in the duration of a finite-time simulation.

Equations (24)

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

i d d t | ψ = H ( t ) | ψ ,
H ( t ) = ω A i a i a i + ω B j b j b j + i j V cos ( Ω t + ϕ i j ) ( a i b j + b j a i ) ,
( i t H ( t ) ) | χ ( t ) = ε | χ ( t ) ,
| χ ( t ) = n = | χ n e i n Ω t ,
H ( t ) = H 0 + H 1 e i Ω t + H 1 e i Ω t .
( H 0 ε + n Ω ) | χ n + H 1 | χ n 1 + H 1 | χ n + 1 = 0 .
H ( t ) ω A i a i a i + ω B j b j b j + i j [ V 2 e i ( Ω t + ϕ i j ) a i b j + V 2 e i ( Ω t + ϕ i j ) b j a i ] .
H rwa = U H U 1 + i d U d t U 1 = i j V 2 ( e i ϕ i j c i c j + e i ϕ i j c j c i ) .
i j A eff d l = ϕ i j .
B eff = 1 a 2 A eff d l ,
( H 0 , A ε + n Ω ) | χ n , A + H 1 | χ n + 1 , B = 0 ,
( H 0 , B ε + n Ω ) | χ n , B + H 1 | χ n 1 , A = 0 ,
( H 0 , A + n Ω H 1 H 1 H 0 , B + ( n + 1 ) Ω ) ( | χ n , A | χ n + 1 , B ) = ε ( | χ n , A | χ n + 1 , B ) .
ε rwa ( k x , k y ) = V ( cos ( a k x ) + cos ( a k y ) ) .
i d | ψ d t = H ( t ) | ψ + | s ,
| s = θ ( t t 0 ) x , y e ( ( x x 0 ) 2 + ( y y 0 ) 2 ) / w 2 e i ( k x 0 x + k y 0 y ) i ( ω x , y + ε 0 ) ( t t 0 ) a ( b ) { x , y } | 0 ,
| ψ ( t n + 1 ) = | ψ ( t n 1 ) 2 i H ( t n ) | ψ ( t n ) δ t 2 i | s ( t n ) δ t .
d r d t = v g k ε ,
d k d t = v g × q B z ^
k x ( t ) k x ( t = 0 ) = q B [ y ( t ) y ( t = 0 ) ] ,
k y ( t ) k y ( t = 0 ) = q B [ x ( t ) x ( t = 0 ) ] .
ε ( q B [ y ( t ) y ( t = 0 ) ] + k x ( t = 0 ) , q B [ x ( t ) x ( t = 0 ) ] + k y ( t = 0 ) ) = ε 0 .
ε ( B eff [ y ( t ) y ( t = 0 ) ] + k x ( t = 0 ) , B eff [ x ( t ) x ( t = 0 ) ] + k y ( t = 0 ) ) = ε 0 .
d 2 r d t 2 = B eff m d r d t × z ^ .

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