Abstract

Zak phase, that is, the Berry phase acquired during an adiabatic motion of a Bloch particle across the Brillouin zone, provides a measure of the topological invariant of Bloch bands in one-dimensional crystalline potentials. Here a photonic structure, based on engineered lattices of evanescently coupled optical waveguides, is proposed to detect Zak phase difference of photons undergoing Bloch oscillations in topologically distinct Bloch bands of dimerized superlattices.

© 2013 Optical Society of America

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  1. R. Resta, J. Phys. Condens. Matter 12, R107 (2000).
    [CrossRef]
  2. D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010).
    [CrossRef]
  3. J. C. Gutiérrez-Vega, Opt. Lett. 36, 1143 (2011).
    [CrossRef]
  4. J. Zak, Phys. Rev. Lett. 62, 2747 (1989).
    [CrossRef]
  5. R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47, 1651 (1993).
    [CrossRef]
  6. J. E. Avron, D. Osadchy, and R. Seiler, Phys. Today 56(8), 38 (2003).
    [CrossRef]
  7. S. Ryu and Y. Hatsugai, Phys. Rev. Lett. 89, 077002 (2002).
    [CrossRef]
  8. P. Delplace, D. Ullmo, and G. Montambaux, Phys. Rev. B 84, 195452 (2011).
    [CrossRef]
  9. W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979).
    [CrossRef]
  10. S. Ryu and Y. Hatsugai, Phys. Rev. B 73, 245115 (2006).
    [CrossRef]
  11. H. T. Cui and J. Yi, Phys. Rev. A 78, 022101 (2008).
    [CrossRef]
  12. T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Nat. Commun. 3, 882 (2012).
    [CrossRef]
  13. M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silberberg, Phys. Rev. Lett. 110, 076403 (2013).
    [CrossRef]
  14. M. C. Rechtsman, Y. Plotnik, J. M. Zeuner, A. Szameit, and M. Segev, “Topological creation and destruction of edge states in photonic grapheme,” arXiv:1211.5683 (2012).
  15. N. Goldman, J. Dalibard, A. Dauphin, F. Gerbier, M. Lewenstein, P. Zoller, and I. B. Spielman, Proc. Natl. Acad. Sci. U.S.A. 110, 6736 (2013).
    [CrossRef]
  16. D. A. Abanin, T. Kitagawa, I. Bloch, and E. Demler, Phys. Rev. Lett. 110, 16530 (2013).
    [CrossRef]
  17. M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” arXiv:1212.0572v1 (2012).
  18. U. Peschel, T. Pertsch, and F. Lederer, Opt. Lett. 23, 1701 (1998).
    [CrossRef]
  19. D. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
    [CrossRef]
  20. H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, Phys. Rev. Lett. 96, 023901 (2006).
    [CrossRef]
  21. A. Szameit and S. Nolte, J. Phys. B 43, 163001 (2010).
    [CrossRef]
  22. I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rep. 518, 1 (2012).
    [CrossRef]
  23. S. Longhi, Phys. Rev. B 79, 245108 (2009).
    [CrossRef]
  24. F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 102, 076802 (2009).
    [CrossRef]
  25. G. Corrielli, A. Crespi, G. Della Valle, S. Longhi, and O. Osellame, Nat. Commun. 4, 1555 (2013).
    [CrossRef]
  26. G. D. Marshall, A. Politi, J. C. F. Matthews, P. Dekker, M. Ams, M. J. Withford, and J. L. O’Brien, Opt. Express 17, 12546 (2009).
    [CrossRef]
  27. M. S. Rudner and L. S. Levitov, Phys. Rev. Lett. 102, 065703 (2009).
    [CrossRef]

2013

M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silberberg, Phys. Rev. Lett. 110, 076403 (2013).
[CrossRef]

N. Goldman, J. Dalibard, A. Dauphin, F. Gerbier, M. Lewenstein, P. Zoller, and I. B. Spielman, Proc. Natl. Acad. Sci. U.S.A. 110, 6736 (2013).
[CrossRef]

D. A. Abanin, T. Kitagawa, I. Bloch, and E. Demler, Phys. Rev. Lett. 110, 16530 (2013).
[CrossRef]

G. Corrielli, A. Crespi, G. Della Valle, S. Longhi, and O. Osellame, Nat. Commun. 4, 1555 (2013).
[CrossRef]

2012

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rep. 518, 1 (2012).
[CrossRef]

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Nat. Commun. 3, 882 (2012).
[CrossRef]

2011

J. C. Gutiérrez-Vega, Opt. Lett. 36, 1143 (2011).
[CrossRef]

P. Delplace, D. Ullmo, and G. Montambaux, Phys. Rev. B 84, 195452 (2011).
[CrossRef]

2010

D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010).
[CrossRef]

A. Szameit and S. Nolte, J. Phys. B 43, 163001 (2010).
[CrossRef]

2009

G. D. Marshall, A. Politi, J. C. F. Matthews, P. Dekker, M. Ams, M. J. Withford, and J. L. O’Brien, Opt. Express 17, 12546 (2009).
[CrossRef]

M. S. Rudner and L. S. Levitov, Phys. Rev. Lett. 102, 065703 (2009).
[CrossRef]

S. Longhi, Phys. Rev. B 79, 245108 (2009).
[CrossRef]

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 102, 076802 (2009).
[CrossRef]

2008

H. T. Cui and J. Yi, Phys. Rev. A 78, 022101 (2008).
[CrossRef]

2006

S. Ryu and Y. Hatsugai, Phys. Rev. B 73, 245115 (2006).
[CrossRef]

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef]

2003

D. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef]

J. E. Avron, D. Osadchy, and R. Seiler, Phys. Today 56(8), 38 (2003).
[CrossRef]

2002

S. Ryu and Y. Hatsugai, Phys. Rev. Lett. 89, 077002 (2002).
[CrossRef]

2000

R. Resta, J. Phys. Condens. Matter 12, R107 (2000).
[CrossRef]

1998

1993

R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47, 1651 (1993).
[CrossRef]

1989

J. Zak, Phys. Rev. Lett. 62, 2747 (1989).
[CrossRef]

1979

W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979).
[CrossRef]

Abanin, D.

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” arXiv:1212.0572v1 (2012).

Abanin, D. A.

D. A. Abanin, T. Kitagawa, I. Bloch, and E. Demler, Phys. Rev. Lett. 110, 16530 (2013).
[CrossRef]

Aidelsburger, M.

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” arXiv:1212.0572v1 (2012).

Ams, M.

Aspuru-Guzik, A.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Nat. Commun. 3, 882 (2012).
[CrossRef]

Atala, M.

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” arXiv:1212.0572v1 (2012).

Avron, J. E.

J. E. Avron, D. Osadchy, and R. Seiler, Phys. Today 56(8), 38 (2003).
[CrossRef]

Barreiro, J. T.

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” arXiv:1212.0572v1 (2012).

Berg, E.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Nat. Commun. 3, 882 (2012).
[CrossRef]

Bloch, I.

D. A. Abanin, T. Kitagawa, I. Bloch, and E. Demler, Phys. Rev. Lett. 110, 16530 (2013).
[CrossRef]

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” arXiv:1212.0572v1 (2012).

Bräuer, A.

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef]

Broome, M. A.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Nat. Commun. 3, 882 (2012).
[CrossRef]

Chang, M.-C.

D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010).
[CrossRef]

Christodoulides, D.

D. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef]

Corrielli, G.

G. Corrielli, A. Crespi, G. Della Valle, S. Longhi, and O. Osellame, Nat. Commun. 4, 1555 (2013).
[CrossRef]

Crespi, A.

G. Corrielli, A. Crespi, G. Della Valle, S. Longhi, and O. Osellame, Nat. Commun. 4, 1555 (2013).
[CrossRef]

Cui, H. T.

H. T. Cui and J. Yi, Phys. Rev. A 78, 022101 (2008).
[CrossRef]

Dalibard, J.

N. Goldman, J. Dalibard, A. Dauphin, F. Gerbier, M. Lewenstein, P. Zoller, and I. B. Spielman, Proc. Natl. Acad. Sci. U.S.A. 110, 6736 (2013).
[CrossRef]

Dauphin, A.

N. Goldman, J. Dalibard, A. Dauphin, F. Gerbier, M. Lewenstein, P. Zoller, and I. B. Spielman, Proc. Natl. Acad. Sci. U.S.A. 110, 6736 (2013).
[CrossRef]

Dekker, P.

Della Valle, G.

G. Corrielli, A. Crespi, G. Della Valle, S. Longhi, and O. Osellame, Nat. Commun. 4, 1555 (2013).
[CrossRef]

Delplace, P.

P. Delplace, D. Ullmo, and G. Montambaux, Phys. Rev. B 84, 195452 (2011).
[CrossRef]

Demler, E.

D. A. Abanin, T. Kitagawa, I. Bloch, and E. Demler, Phys. Rev. Lett. 110, 16530 (2013).
[CrossRef]

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Nat. Commun. 3, 882 (2012).
[CrossRef]

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” arXiv:1212.0572v1 (2012).

Dreisow, F.

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 102, 076802 (2009).
[CrossRef]

Fedrizzi, A.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Nat. Commun. 3, 882 (2012).
[CrossRef]

Garanovich, I. L.

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rep. 518, 1 (2012).
[CrossRef]

Gerbier, F.

N. Goldman, J. Dalibard, A. Dauphin, F. Gerbier, M. Lewenstein, P. Zoller, and I. B. Spielman, Proc. Natl. Acad. Sci. U.S.A. 110, 6736 (2013).
[CrossRef]

Goldman, N.

N. Goldman, J. Dalibard, A. Dauphin, F. Gerbier, M. Lewenstein, P. Zoller, and I. B. Spielman, Proc. Natl. Acad. Sci. U.S.A. 110, 6736 (2013).
[CrossRef]

Gutiérrez-Vega, J. C.

Hatsugai, Y.

S. Ryu and Y. Hatsugai, Phys. Rev. B 73, 245115 (2006).
[CrossRef]

S. Ryu and Y. Hatsugai, Phys. Rev. Lett. 89, 077002 (2002).
[CrossRef]

Heeger, A. J.

W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979).
[CrossRef]

Heinrich, M.

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 102, 076802 (2009).
[CrossRef]

Kassal, I.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Nat. Commun. 3, 882 (2012).
[CrossRef]

King-Smith, R. D.

R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47, 1651 (1993).
[CrossRef]

Kitagawa, T.

D. A. Abanin, T. Kitagawa, I. Bloch, and E. Demler, Phys. Rev. Lett. 110, 16530 (2013).
[CrossRef]

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Nat. Commun. 3, 882 (2012).
[CrossRef]

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” arXiv:1212.0572v1 (2012).

Kivshar, Y. S.

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rep. 518, 1 (2012).
[CrossRef]

Kraus, Y. E.

M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silberberg, Phys. Rev. Lett. 110, 076403 (2013).
[CrossRef]

Lahini, Y.

M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silberberg, Phys. Rev. Lett. 110, 076403 (2013).
[CrossRef]

Lederer, F.

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef]

D. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef]

U. Peschel, T. Pertsch, and F. Lederer, Opt. Lett. 23, 1701 (1998).
[CrossRef]

Levitov, L. S.

M. S. Rudner and L. S. Levitov, Phys. Rev. Lett. 102, 065703 (2009).
[CrossRef]

Lewenstein, M.

N. Goldman, J. Dalibard, A. Dauphin, F. Gerbier, M. Lewenstein, P. Zoller, and I. B. Spielman, Proc. Natl. Acad. Sci. U.S.A. 110, 6736 (2013).
[CrossRef]

Longhi, S.

G. Corrielli, A. Crespi, G. Della Valle, S. Longhi, and O. Osellame, Nat. Commun. 4, 1555 (2013).
[CrossRef]

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rep. 518, 1 (2012).
[CrossRef]

S. Longhi, Phys. Rev. B 79, 245108 (2009).
[CrossRef]

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 102, 076802 (2009).
[CrossRef]

Marshall, G. D.

Matthews, J. C. F.

Michaelis, D.

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef]

Montambaux, G.

P. Delplace, D. Ullmo, and G. Montambaux, Phys. Rev. B 84, 195452 (2011).
[CrossRef]

Niu, Q.

D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010).
[CrossRef]

Nolte, S.

A. Szameit and S. Nolte, J. Phys. B 43, 163001 (2010).
[CrossRef]

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 102, 076802 (2009).
[CrossRef]

O’Brien, J. L.

Osadchy, D.

J. E. Avron, D. Osadchy, and R. Seiler, Phys. Today 56(8), 38 (2003).
[CrossRef]

Osellame, O.

G. Corrielli, A. Crespi, G. Della Valle, S. Longhi, and O. Osellame, Nat. Commun. 4, 1555 (2013).
[CrossRef]

Pertsch, T.

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 102, 076802 (2009).
[CrossRef]

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef]

U. Peschel, T. Pertsch, and F. Lederer, Opt. Lett. 23, 1701 (1998).
[CrossRef]

Peschel, U.

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef]

U. Peschel, T. Pertsch, and F. Lederer, Opt. Lett. 23, 1701 (1998).
[CrossRef]

Plotnik, Y.

M. C. Rechtsman, Y. Plotnik, J. M. Zeuner, A. Szameit, and M. Segev, “Topological creation and destruction of edge states in photonic grapheme,” arXiv:1211.5683 (2012).

Politi, A.

Rechtsman, M. C.

M. C. Rechtsman, Y. Plotnik, J. M. Zeuner, A. Szameit, and M. Segev, “Topological creation and destruction of edge states in photonic grapheme,” arXiv:1211.5683 (2012).

Resta, R.

R. Resta, J. Phys. Condens. Matter 12, R107 (2000).
[CrossRef]

Rudner, M. S.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Nat. Commun. 3, 882 (2012).
[CrossRef]

M. S. Rudner and L. S. Levitov, Phys. Rev. Lett. 102, 065703 (2009).
[CrossRef]

Ryu, S.

S. Ryu and Y. Hatsugai, Phys. Rev. B 73, 245115 (2006).
[CrossRef]

S. Ryu and Y. Hatsugai, Phys. Rev. Lett. 89, 077002 (2002).
[CrossRef]

Schrieffer, J. R.

W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979).
[CrossRef]

Segev, M.

M. C. Rechtsman, Y. Plotnik, J. M. Zeuner, A. Szameit, and M. Segev, “Topological creation and destruction of edge states in photonic grapheme,” arXiv:1211.5683 (2012).

Seiler, R.

J. E. Avron, D. Osadchy, and R. Seiler, Phys. Today 56(8), 38 (2003).
[CrossRef]

Silberberg, Y.

M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silberberg, Phys. Rev. Lett. 110, 076403 (2013).
[CrossRef]

D. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef]

Spielman, I. B.

N. Goldman, J. Dalibard, A. Dauphin, F. Gerbier, M. Lewenstein, P. Zoller, and I. B. Spielman, Proc. Natl. Acad. Sci. U.S.A. 110, 6736 (2013).
[CrossRef]

Streppel, U.

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef]

Su, W. P.

W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979).
[CrossRef]

Sukhorukov, A. A.

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rep. 518, 1 (2012).
[CrossRef]

Szameit, A.

A. Szameit and S. Nolte, J. Phys. B 43, 163001 (2010).
[CrossRef]

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 102, 076802 (2009).
[CrossRef]

M. C. Rechtsman, Y. Plotnik, J. M. Zeuner, A. Szameit, and M. Segev, “Topological creation and destruction of edge states in photonic grapheme,” arXiv:1211.5683 (2012).

Trompeter, H.

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef]

Tünnermann, A.

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 102, 076802 (2009).
[CrossRef]

Ullmo, D.

P. Delplace, D. Ullmo, and G. Montambaux, Phys. Rev. B 84, 195452 (2011).
[CrossRef]

Vanderbilt, D.

R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47, 1651 (1993).
[CrossRef]

Verbin, M.

M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silberberg, Phys. Rev. Lett. 110, 076403 (2013).
[CrossRef]

White, A. G.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Nat. Commun. 3, 882 (2012).
[CrossRef]

Withford, M. J.

Xiao, D.

D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010).
[CrossRef]

Yi, J.

H. T. Cui and J. Yi, Phys. Rev. A 78, 022101 (2008).
[CrossRef]

Zak, J.

J. Zak, Phys. Rev. Lett. 62, 2747 (1989).
[CrossRef]

Zeuner, J. M.

M. C. Rechtsman, Y. Plotnik, J. M. Zeuner, A. Szameit, and M. Segev, “Topological creation and destruction of edge states in photonic grapheme,” arXiv:1211.5683 (2012).

Zilberberg, O.

M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silberberg, Phys. Rev. Lett. 110, 076403 (2013).
[CrossRef]

Zoller, P.

N. Goldman, J. Dalibard, A. Dauphin, F. Gerbier, M. Lewenstein, P. Zoller, and I. B. Spielman, Proc. Natl. Acad. Sci. U.S.A. 110, 6736 (2013).
[CrossRef]

J. Phys. B

A. Szameit and S. Nolte, J. Phys. B 43, 163001 (2010).
[CrossRef]

J. Phys. Condens. Matter

R. Resta, J. Phys. Condens. Matter 12, R107 (2000).
[CrossRef]

Nat. Commun.

G. Corrielli, A. Crespi, G. Della Valle, S. Longhi, and O. Osellame, Nat. Commun. 4, 1555 (2013).
[CrossRef]

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Nat. Commun. 3, 882 (2012).
[CrossRef]

Nature

D. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rep.

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rep. 518, 1 (2012).
[CrossRef]

Phys. Rev. A

H. T. Cui and J. Yi, Phys. Rev. A 78, 022101 (2008).
[CrossRef]

Phys. Rev. B

S. Ryu and Y. Hatsugai, Phys. Rev. B 73, 245115 (2006).
[CrossRef]

P. Delplace, D. Ullmo, and G. Montambaux, Phys. Rev. B 84, 195452 (2011).
[CrossRef]

R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47, 1651 (1993).
[CrossRef]

S. Longhi, Phys. Rev. B 79, 245108 (2009).
[CrossRef]

Phys. Rev. Lett.

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 102, 076802 (2009).
[CrossRef]

M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silberberg, Phys. Rev. Lett. 110, 076403 (2013).
[CrossRef]

D. A. Abanin, T. Kitagawa, I. Bloch, and E. Demler, Phys. Rev. Lett. 110, 16530 (2013).
[CrossRef]

J. Zak, Phys. Rev. Lett. 62, 2747 (1989).
[CrossRef]

S. Ryu and Y. Hatsugai, Phys. Rev. Lett. 89, 077002 (2002).
[CrossRef]

W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979).
[CrossRef]

M. S. Rudner and L. S. Levitov, Phys. Rev. Lett. 102, 065703 (2009).
[CrossRef]

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef]

Phys. Today

J. E. Avron, D. Osadchy, and R. Seiler, Phys. Today 56(8), 38 (2003).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A.

N. Goldman, J. Dalibard, A. Dauphin, F. Gerbier, M. Lewenstein, P. Zoller, and I. B. Spielman, Proc. Natl. Acad. Sci. U.S.A. 110, 6736 (2013).
[CrossRef]

Rev. Mod. Phys.

D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010).
[CrossRef]

Other

M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” arXiv:1212.0572v1 (2012).

M. C. Rechtsman, Y. Plotnik, J. M. Zeuner, A. Szameit, and M. Segev, “Topological creation and destruction of edge states in photonic grapheme,” arXiv:1211.5683 (2012).

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Figures (4)

Fig. 1.
Fig. 1.

(a) Schematic of a 1D superlattice with period d that realizes the SSH model [Eq. (3)] of polyacetylene. The lattice unit cell contains two sites, an and bn. The two realizations D1 and D2, with either κ1>κ2 or κ1<κ2, correspond to two distinct topological phases. The Zak phases in the topological states D1 and D2 differ by π. (b) Photonic realization of the SSH model based on an array of coupled optical waveguides with equal spacing d/2. The auxiliary waveguides at sites cn are used to control the hopping rates.

Fig. 2.
Fig. 2.

(a) Setup to detect the π phase shift of Zak phases in the topological states D1 and D2 of the SSH model. The two vertically displaced circularly bent waveguide arrays realize the D1 and D2 topological phases. The arrays are illuminated at the input plane by a broad Gaussian beam at normal incidence (the circular spot in the figure). (b) Schematic of the waveguide tritter used to detect the π phase shift after one BO cycle.

Fig. 3.
Fig. 3.

(a) Evolution of the site occupation amplitudes dn(z) (real and imaginary parts) over one BO cycle (dn=a(n+1)/2 for n odd, dn=bn/2 for n even) in the two lattices D1 and D2 of Fig. 2(a) for a Gaussian beam excitation. Parameter values are given in the text. (b) Phase of dn at z=zB, i.e., after one BO cycle, for the dimerizations D1 and D2. The left panels in the figure refer to the model Eq. (4), whereas the right panels are the predictions of the SSH model Eq. (3).

Fig. 4.
Fig. 4.

Evolution of the site occupation probabilities |dn(z)|2 in the straight arrays D1 and D2 corresponding to the single-site excitation of the left-edge site a1. The light trapped at the boundary of the array D2 is the clear signature of the existence of an edge state.

Equations (6)

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izψ(x,z)=H^0ψ(x,z)Fxψ(x,z),
γα(Z)=π/dπ/dXα,α(k),
H^0=n(κ1a^nb^n+κ2a^nb^n1+H.c.),
H^0=nκ2(a^nb^n+a^nb^n1+H.c.)nρ(a^nc^n+b^nc^n+H.c.)+nUc^nc^n.
κ1=κ2+ρ2/U,
H^e=n(κ1a^nb^n+κ2a^nb^n1+H.c.)ρ2Un(a^na^n+b^nb^n),

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