Abstract

We demonstrate that a flat-band state in a quasi-one-dimensional rhombic lattice is robust in the presence of external drivings along the lattice axis. The lattice was formed by periodic arrays of evanescently coupled optical waveguides, and the external drivings were realized by modulating the paths of the waveguides. We excited a superposition of flat-band eigenmodes at the input and observed that this state does not diffract in the presence of static, as well as high-frequency sinusoidal drivings. This robust localization is due to destructive interference of the analogous wavefunction and is associated with the symmetry in the lattice geometry. We then excited the dispersive bands and observed Bloch oscillations and coherent destruction of tunneling.

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2016 (9)

S. Xia, Y. Hu, D. Song, Y. Zong, L. Tang, and Z. Chen, Opt. Lett. 41, 1435 (2016).
[Crossref]

A. Nandy and A. Chakrabarti, Phys. Rev. A 93, 013807 (2016).
[Crossref]

F. Diebel, D. Leykam, S. Kroesen, C. Denz, and A. S. Desyatnikov, Phys. Rev. Lett. 116, 183902 (2016).
[Crossref]

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[Crossref]

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[Crossref]

R. Khomeriki and S. Flach, Phys. Rev. Lett. 116, 245301 (2016).
[Crossref]

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[Crossref]

F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Lematre, L. Le Gratiet, I. Sagnes, S. Schmidt, H. Türeci, A. Amo, and J. Bloch, Phys. Rev. Lett. 116, 066402 (2016).
[Crossref]

S. Mukherjee, M. Valiente, N. Goldman, A. Spracklen, E. Andersson, P. Öhberg, and R. R. Thomson, Phys. Rev. A 94, 053853 (2016).
[Crossref]

2015 (6)

S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman, P. Öhberg, E. Andersson, and R. R. Thomson, New J. Phys. 17, 115002 (2015).
[Crossref]

S. Mukherjee and R. R. Thomson, Opt. Lett. 40, 5443 (2015).
[Crossref]

M. I. Molina, Phys. Rev. A 92, 063813 (2015).
[Crossref]

S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman, P. Öhberg, E. Andersson, and R. R. Thomson, Phys. Rev. Lett. 114, 245504 (2015).
[Crossref]

R. A. Vicencio, C. Cantillano, L. Morales-Inostroza, B. Real, C. Mejía-Cortés, S. Weimann, A. Szameit, and M. I. Molina, Phys. Rev. Lett. 114, 245503 (2015).
[Crossref]

S. Taie, H. Ozawa, T. Ichinose, T. Nishio, S. Nakajima, and Y. Takahashi, Sci. Adv. 1, e1500854 (2015).

2014 (2)

D. Guzmán-Silva, C. Meja-Cortés, M. Bandres, M. Rechtsman, S. Weimann, S. Nolte, M. Segev, A. Szameit, and R. Vicencio, New J. Phys. 16, 063061 (2014).
[Crossref]

S. Longhi, Opt. Lett. 39, 5892 (2014).
[Crossref]

2013 (4)

D. Leykam, S. Flach, O. Bahat-Treidel, and A. S. Desyatnikov, Phys. Rev. B 88, 224203 (2013).
[Crossref]

A. Yulin and V. Konotop, Opt. Lett. 38, 4880 (2013).
[Crossref]

X. Cai, S. Chen, and Y. Wang, Phys. Rev. A 87, 013607 (2013).
[Crossref]

M. Niţă, B. Ostahie, and A. Aldea, Phys. Rev. B 87, 125428 (2013).
[Crossref]

2012 (2)

I. Bloch, J. Dalibard, and S. Nascimbene, Nat. Phys. 8, 267 (2012).
[Crossref]

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

2011 (2)

2010 (2)

C. E. Creffield and G. Platero, Phys. Rev. Lett. 105, 086804 (2010).
[Crossref]

S. D. Huber and E. Altman, Phys. Rev. B 82, 184502 (2010).
[Crossref]

2009 (2)

S. Longhi, Laser Photon. Rev. 3, 243 (2009).
[Crossref]

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and Y. S. Kivshar, Nat. Phys. 5, 271 (2009).
[Crossref]

2008 (2)

2007 (2)

G. Della Valle, M. Ornigotti, E. Cianci, V. Foglietti, P. Laporta, and S. Longhi, Phys. Rev. Lett. 98, 263601 (2007).
[Crossref]

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[Crossref]

2006 (3)

2000 (1)

J. Vidal, B. Douçot, R. Mosseri, and P. Butaud, Phys. Rev. Lett. 85, 3906 (2000).
[Crossref]

1999 (2)

R. Morandotti, U. Peschel, J. Aitchison, H. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[Crossref]

G. Lenz, I. Talanina, and C. M. De Sterke, Phys. Rev. Lett. 83, 963 (1999).
[Crossref]

1998 (2)

1996 (2)

M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, Phys. Rev. Lett. 76, 4508 (1996).
[Crossref]

K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, Opt. Lett. 21, 1729 (1996).
[Crossref]

1993 (1)

M. Holthaus and D. Hone, Phys. Rev. B 47, 6499 (1993).
[Crossref]

1986 (1)

D. Dunlap and V. Kenkre, Phys. Rev. B 34, 3625 (1986).
[Crossref]

1975 (1)

M. Heiblum and J. H. Harris, IEEE J. Quantum Electron. 11, 75 (1975).
[Crossref]

1973 (1)

H. Fukuyama, R. A. Bari, and H. C. Fogedby, Phys. Rev. B 8, 5579 (1973).
[Crossref]

1929 (1)

F. Bloch, Z. Phys. 52, 555 (1929).
[Crossref]

Abouraddy, A. F.

Aitchison, J.

R. Morandotti, U. Peschel, J. Aitchison, H. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[Crossref]

Aldea, A.

M. Niţă, B. Ostahie, and A. Aldea, Phys. Rev. B 87, 125428 (2013).
[Crossref]

Altman, E.

S. D. Huber and E. Altman, Phys. Rev. B 82, 184502 (2010).
[Crossref]

Amo, A.

F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Lematre, L. Le Gratiet, I. Sagnes, S. Schmidt, H. Türeci, A. Amo, and J. Bloch, Phys. Rev. Lett. 116, 066402 (2016).
[Crossref]

Andersson, E.

S. Mukherjee, M. Valiente, N. Goldman, A. Spracklen, E. Andersson, P. Öhberg, and R. R. Thomson, Phys. Rev. A 94, 053853 (2016).
[Crossref]

S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman, P. Öhberg, E. Andersson, and R. R. Thomson, New J. Phys. 17, 115002 (2015).
[Crossref]

S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman, P. Öhberg, E. Andersson, and R. R. Thomson, Phys. Rev. Lett. 114, 245504 (2015).
[Crossref]

Baboux, F.

F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Lematre, L. Le Gratiet, I. Sagnes, S. Schmidt, H. Türeci, A. Amo, and J. Bloch, Phys. Rev. Lett. 116, 066402 (2016).
[Crossref]

Bahat-Treidel, O.

D. Leykam, S. Flach, O. Bahat-Treidel, and A. S. Desyatnikov, Phys. Rev. B 88, 224203 (2013).
[Crossref]

Bandres, M.

D. Guzmán-Silva, C. Meja-Cortés, M. Bandres, M. Rechtsman, S. Weimann, S. Nolte, M. Segev, A. Szameit, and R. Vicencio, New J. Phys. 16, 063061 (2014).
[Crossref]

Bari, R. A.

H. Fukuyama, R. A. Bari, and H. C. Fogedby, Phys. Rev. B 8, 5579 (1973).
[Crossref]

Bartal, G.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[Crossref]

Biondi, M.

F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Lematre, L. Le Gratiet, I. Sagnes, S. Schmidt, H. Türeci, A. Amo, and J. Bloch, Phys. Rev. Lett. 116, 066402 (2016).
[Crossref]

Bloch, F.

F. Bloch, Z. Phys. 52, 555 (1929).
[Crossref]

Bloch, I.

I. Bloch, J. Dalibard, and S. Nascimbene, Nat. Phys. 8, 267 (2012).
[Crossref]

Bloch, J.

F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Lematre, L. Le Gratiet, I. Sagnes, S. Schmidt, H. Türeci, A. Amo, and J. Bloch, Phys. Rev. Lett. 116, 066402 (2016).
[Crossref]

Brown, O. T.

E. T. Owen, O. T. Brown, and M. J. Hartmann, “Dissipation-induced mobility in frustrated lattices,” arXiv:1612.07243 (2016).

Burghoff, J.

Butaud, P.

J. Vidal, B. Douçot, R. Mosseri, and P. Butaud, Phys. Rev. Lett. 85, 3906 (2000).
[Crossref]

Cai, X.

X. Cai, S. Chen, and Y. Wang, Phys. Rev. A 87, 013607 (2013).
[Crossref]

Cantillano, C.

S. Weimann, L. Morales-Inostroza, B. Real, C. Cantillano, A. Szameit, and R. A. Vicencio, Opt. Lett. 41, 2414 (2016).
[Crossref]

R. A. Vicencio, C. Cantillano, L. Morales-Inostroza, B. Real, C. Mejía-Cortés, S. Weimann, A. Szameit, and M. I. Molina, Phys. Rev. Lett. 114, 245503 (2015).
[Crossref]

Castin, Y.

M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, Phys. Rev. Lett. 76, 4508 (1996).
[Crossref]

Cerullo, G.

Chakrabarti, A.

A. Nandy and A. Chakrabarti, Phys. Rev. A 93, 013807 (2016).
[Crossref]

Chen, S.

X. Cai, S. Chen, and Y. Wang, Phys. Rev. A 87, 013607 (2013).
[Crossref]

Chen, Z.

Chiodo, N.

Choudhury, D.

S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman, P. Öhberg, E. Andersson, and R. R. Thomson, Phys. Rev. Lett. 114, 245504 (2015).
[Crossref]

S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman, P. Öhberg, E. Andersson, and R. R. Thomson, New J. Phys. 17, 115002 (2015).
[Crossref]

Christodoulides, D. N.

Cianci, E.

G. Della Valle, M. Ornigotti, E. Cianci, V. Foglietti, P. Laporta, and S. Longhi, Phys. Rev. Lett. 98, 263601 (2007).
[Crossref]

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, Phys. Rev. Lett. 96, 243901 (2006).
[Crossref]

Creffield, C. E.

C. E. Creffield and G. Platero, Phys. Rev. Lett. 105, 086804 (2010).
[Crossref]

Dahan, M. B.

M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, Phys. Rev. Lett. 76, 4508 (1996).
[Crossref]

Dalibard, J.

I. Bloch, J. Dalibard, and S. Nascimbene, Nat. Phys. 8, 267 (2012).
[Crossref]

Davis, K. M.

De Sterke, C. M.

G. Lenz, I. Talanina, and C. M. De Sterke, Phys. Rev. Lett. 83, 963 (1999).
[Crossref]

Della Valle, G.

Denz, C.

F. Diebel, D. Leykam, S. Kroesen, C. Denz, and A. S. Desyatnikov, Phys. Rev. Lett. 116, 183902 (2016).
[Crossref]

Desyatnikov, A. S.

F. Diebel, D. Leykam, S. Kroesen, C. Denz, and A. S. Desyatnikov, Phys. Rev. Lett. 116, 183902 (2016).
[Crossref]

D. Leykam, S. Flach, O. Bahat-Treidel, and A. S. Desyatnikov, Phys. Rev. B 88, 224203 (2013).
[Crossref]

Di Giuseppe, G.

Diebel, F.

F. Diebel, D. Leykam, S. Kroesen, C. Denz, and A. S. Desyatnikov, Phys. Rev. Lett. 116, 183902 (2016).
[Crossref]

Doering, S.

Douçot, B.

J. Vidal, B. Douçot, R. Mosseri, and P. Butaud, Phys. Rev. Lett. 85, 3906 (2000).
[Crossref]

Dreisow, F.

Dunlap, D.

D. Dunlap and V. Kenkre, Phys. Rev. B 34, 3625 (1986).
[Crossref]

Eisenberg, H.

R. Morandotti, U. Peschel, J. Aitchison, H. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[Crossref]

Fahr, S.

Fishman, S.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[Crossref]

Flach, S.

R. Khomeriki and S. Flach, Phys. Rev. Lett. 116, 245301 (2016).
[Crossref]

D. Leykam, S. Flach, O. Bahat-Treidel, and A. S. Desyatnikov, Phys. Rev. B 88, 224203 (2013).
[Crossref]

Fogedby, H. C.

H. Fukuyama, R. A. Bari, and H. C. Fogedby, Phys. Rev. B 8, 5579 (1973).
[Crossref]

Foglietti, V.

G. Della Valle, M. Ornigotti, E. Cianci, V. Foglietti, P. Laporta, and S. Longhi, Phys. Rev. Lett. 98, 263601 (2007).
[Crossref]

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, Phys. Rev. Lett. 96, 243901 (2006).
[Crossref]

Fukuyama, H.

H. Fukuyama, R. A. Bari, and H. C. Fogedby, Phys. Rev. B 8, 5579 (1973).
[Crossref]

Galopin, E.

F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Lematre, L. Le Gratiet, I. Sagnes, S. Schmidt, H. Türeci, A. Amo, and J. Bloch, Phys. Rev. Lett. 116, 066402 (2016).
[Crossref]

Gao, J.

Z. Yang, Y. Wang, Z. Xue, W. Yang, Y. Hu, J. Gao, and Y. Wu, Phys. Rev. A 93, 062319 (2016).
[Crossref]

Garanovich, I. L.

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

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and Y. S. Kivshar, Nat. Phys. 5, 271 (2009).
[Crossref]

Ge, L.

F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Lematre, L. Le Gratiet, I. Sagnes, S. Schmidt, H. Türeci, A. Amo, and J. Bloch, Phys. Rev. Lett. 116, 066402 (2016).
[Crossref]

Goldman, N.

S. Mukherjee, M. Valiente, N. Goldman, A. Spracklen, E. Andersson, P. Öhberg, and R. R. Thomson, Phys. Rev. A 94, 053853 (2016).
[Crossref]

S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman, P. Öhberg, E. Andersson, and R. R. Thomson, New J. Phys. 17, 115002 (2015).
[Crossref]

S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman, P. Öhberg, E. Andersson, and R. R. Thomson, Phys. Rev. Lett. 114, 245504 (2015).
[Crossref]

Grifoni, M.

M. Grifoni and P. Hänggi, Phys. Rep. 304, 229 (1998).
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Figures (7)

Fig. 1.
Fig. 1. (a) Quasi-one-dimensional rhombic lattice driven by an external force, F . The unit cell contains three sites: a , b , and c . (b) In the photonic setup, the static and sinusoidal drivings are implemented by modulating the paths of the waveguides; see the text. (c) Band structure calculated by diagonalizing the Fourier transformed Hamiltonian with F = 0 . In this situation, the non-dispersive (flat) band can be excited by initially exciting the { E s b , E s c } = { 1 / 2 , 1 / 2 } state; see [17].
Fig. 2.
Fig. 2. Robust flat-band localization in the presence of static driving. (a)–(c) Measured output intensity distributions after a propagation of 10, 30, and 70 mm, respectively. The flat-band was excited by launching the { E 6 b , E 6 c } = { 1 / 2 , 1 / 2 } state at the input of circularly curved photonic rhombic lattices. Each image is normalized such that the total output power is 1. The circles indicate the initially (at z = 0 ) excited lattice sites.
Fig. 3.
Fig. 3. Observation of Bloch oscillations when the dispersive bands are excited by launching the equal-phase state, { E 6 b , E 6 c } = { 1 / 2 , 1 / 2 } ; see Fig. 4. The propagation distances were 10, 30, and 70 mm for (a), (b), and (c), respectively.
Fig. 4.
Fig. 4. Numerically calculated variation of light intensity along the propagation direction of a circularly curved photonic rhombic lattice for the initial condition, { E 6 b , E 6 c } = { 1 / 2 , 1 / 2 } . Here κ = 0.034    mm 1 and R = 1.8    m . Note that the period of the breathing motion of the intensity distribution is z B = 2 R λ / ( n 0 d ) = 77.87    mm . The red circles and the blue squares are the measured values of the average intensities at the waveguides that were excited at the input [i.e., ( | E 6 b | 2 + | E 6 c | 2 ) / 2 ] and their nearest neighbor sites [i.e., ( | E 6 a | 2 + | E 7 a | 2 ) / 2 ], respectively.
Fig. 5.
Fig. 5. (a) Numerically calculated quasienergy spectrum for a sinusoidally driven rhombic lattice with 12 unit cells (open squares). Here κ = 0.034    mm 1 , ω / κ = 46.2 , and K / ω = 0.912 ( A = 4    μm ). Note that the driving does not destroy the flat-band, shown by the open blue squares. The blue (red) circles show the calculated values of the overlap of the initial flat-band state (equal-phase state) with the Floquet eigenstates. (b) Floquet spectrum as a function of K / ω . When the { E 6 b , E 6 c } = { 1 / 2 , 1 / 2 } state is excited, only the eigenstates at zero quasienergy (shown by the blue circles) are excited, regardless of the value of K / ω . The dispersive bands collapse at K / ω = 2.405 , causing coherent destruction of tunneling; see Fig. 6(g).
Fig. 6.
Fig. 6. Experimentally measured intensity distributions at the output of the 70 mm long modulated lattices for four different values of amplitude of modulations ( A = 6 , 8, 10, and 12 μm, respectively). For the left column, (a)–(d), the flat-band state was launched at the input. For the right column, (e)–(h), the dispersive bands were excited by launching the equal-phase state at the input. The flat-band state is robust in the presence of sinusoidal drivings. For the equal-phase input state, the output intensity distribution is determined by K / ω , as the effective coupling constant is renormalized as | κ eff / κ | = | J 0 ( K / ω ) | . Note that for (g)  K / ω is close to the first root of J 0 ( K / ω ) , causing coherent destruction of tunneling.
Fig. 7.
Fig. 7. (a) High-frequency sinusoidal driving renormalizes the effective coupling constant, as shown by the solid line. The filled circles indicate the values of | κ eff / κ | , estimated from the measured intensity distributions. (b) Analogous site energy of the lattice sites at a given z , in the presence of an external field along x -axis. The linear ramp of the site energy is determined by the strength of the external field and its direction. An initial flat-band state remains localized as the b s and c s sites have the same site energy which is shifted by ± β , compared to the a s and the a s + 1 sites, respectively.

Equations (2)

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i ƛ z E ( x , z ) = [ ƛ 2 2 n 0 2 x 2 Δ n ( x ) + F ( z ) x ] E ( x , z ) ,
[ i z + 2 s β ] E s a = κ ( E s b + E s 1 b + E s c + E s 1 c ) , [ i z + ( 2 s + 1 ) β ] E s b = κ ( E s a + E s + 1 a ) , [ i z + ( 2 s + 1 ) β ] E s c = κ ( E s a + E s + 1 a ) ,

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