Abstract

Topological insulators have attracted abundant attention for a variety of reasons—notably, the possibility for lossless energy transport through edge states “protected” against disorder. Topological effects such as the quantum Hall state can be induced through a gauge field, which is, however, hard to create in practice, especially for charge-neutral particles. One way to induce an effective gauge potential is through a dynamic, time-periodic modulation of the lattice confining such particles. In this way, the Haldane quantum Hall effect was recently observed in a cold-atom system. Here, we show how this same effect can be induced for light confined to a lattice of identical optical resonators, using an on-site modulation of the resonant frequencies. In this system, coupled-mode analysis shows the presence of one-direction edge states immune to backscattering losses. We also discuss possible realizations of the model, which could enable slow-light devices of unprecedented quality.

© 2016 Optical Society of America

Full Article  |  PDF Article
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References

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2015 (2)

V. Peano, C. Brendel, M. Schmidt, and F. Marquardt, “Topological phases of sound and light,” Phys. Rev. X 031011, 1–18 (2015).

M. Schmidt, S. Kessler, V. Peano, O. Painter, and F. Marquardt, “Optomechanical creation of magnetic fields for photons on a lattice,” Optica 2, 635–641 (2015).
[Crossref]

2014 (10)

H. Sekoguchi, Y. Takahashi, T. Asano, and S. Noda, “Photonic crystal nanocavity with a q-factor of 9 million,” Opt. Express 22, 916–924 (2014).
[Crossref]

A. Y. Piggott, K. G. Lagoudakis, T. Sarmiento, M. Bajcsy, G. Shambat, and J. Vučković, “Photo-oxidative tuning of individual and coupled gas photonic crystal cavities,” Opt. Express 22, 15017–15023 (2014).
[Crossref]

Y. Lai, S. Pirotta, G. Urbinati, D. Gerace, M. Minkov, V. Savona, A. Badolato, and M. Galli, “Genetically designed L3 photonic crystal nanocavities with measured quality factor exceeding one million,” Appl. Phys. Lett. 104, 241101 (2014).
[Crossref]

A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, J. Chan, S. Gröblacher, and O. Painter, “Two-dimensional phononic-photonic band gap optomechanical crystal cavity,” Phys. Rev. Lett. 112, 153603 (2014).
[Crossref]

H. Takesue, N. Matsuda, E. Kuramochi, and M. Notomi, “Entangled photons from on-chip slow light,” Sci. Rep. 4, 3913 (2014).
[Crossref]

L. D. Tzuang, K. Fang, P. Nussenzveig, S. Fan, and M. Lipson, “Non-reciprocal phase shift induced by an effective magnetic flux for light,” Nat. Photonics 8, 701–705 (2014).
[Crossref]

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

N. Goldman and J. Dalibard, “Periodically driven quantum systems: effective Hamiltonians and engineered gauge fields,” Phys. Rev. X 4, 031027 (2014).

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8, 821–829 (2014).

2013 (6)

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7, 1001–1006 (2013).

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref]

M. Minkov and V. Savona, “Radiative coupling of quantum dots in photonic crystal structures,” Phys. Rev. B 87, 125306 (2013).
[Crossref]

E. Yüce, G. Ctistis, J. Claudon, E. Dupuy, R. D. Buijs, B. de Ronde, A. P. Mosk, J. Gérard, and W. L. Vos, “All-optical switching of a microcavity repeated at terahertz rates,” Opt. Lett. 38, 374–376 (2013).
[Crossref]

S. Longhi, “Effective magnetic fields for photons in waveguide and coupled resonator lattices,” Opt. Lett. 38, 3570–3573 (2013).
[Crossref]

M. Minkov, U. P. Dharanipathy, R. Houdré, and V. Savona, “Statistics of the disorder-induced losses of high-q photonic crystal cavities,” Opt. Express 21, 28233–28245 (2013).
[Crossref]

2012 (9)

M. Hafezi and P. Rabl, “Optomechanically induced non-reciprocity in microring resonators,” Opt. Express 20, 7672–7684 (2012).
[Crossref]

S. Azzini, D. Grassani, M. J. Strain, M. Sorel, L. G. Helt, J. E. Sipe, M. Liscidini, M. Galli, and D. Bajoni, “Ultra-low power generation of twin photons in a compact silicon ring resonator,” Opt. Express 20, 23100–23107 (2012).
[Crossref]

F. Intonti, N. Caselli, S. Vignolini, F. Riboli, S. Kumar, A. Rastelli, O. G. Schmidt, M. Francardi, A. Gerardino, L. Balet, L. H. Li, A. Fiore, and M. Gurioli, “Mode tuning of photonic crystal nanocavities by photoinduced non-thermal oxidation,” Appl. Phys. Lett. 100, 033116 (2012).
[Crossref]

H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip,” Phys. Rev. Lett. 109, 1–5 (2012).
[Crossref]

R. O. Umucalılar and I. Carusotto, “Fractional quantum Hall states of photons in an array of dissipative coupled cavities,” Phys. Rev. Lett. 108, 206809 (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]

P. Hauke, O. Tieleman, A. Celi, C. Ölschläger, J. Simonet, J. Struck, M. Weinberg, P. Windpassinger, K. Sengstock, M. Lewenstein, and A. Eckardt, “Non-Abelian gauge fields and topological insulators in shaken optical lattices,” Phys. Rev. Lett. 109, 1–6 (2012).
[Crossref]

J. Struck, C. Olschläger, M. Weinberg, P. Hauke, J. Simonet, A. Eckardt, M. Lewenstein, K. Sengstock, and P. Windpassinger, “Tunable gauge potential for neutral and spinless particles in driven optical lattices,” Phys. Rev. Lett. 108, 1–5 (2012).
[Crossref]

K. Fang, Z. Yu, and S. Fan, “Photonic Aharonov-Bohm effect based on dynamic modulation,” Phys. Rev. Lett. 108, 1–5 (2012).

2011 (2)

R. O. Umucalılar and I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A 84, 043804 (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]

2010 (4)

J. Koch, A. A. Houck, K. L. Hur, and S. M. Girvin, “Time-reversal-symmetry breaking in circuit-QED-based photon lattices,” Phys. Rev. A 82, 1–18 (2010).

M. Z. Hasan and C. L. Kane, “Colloquium: topological insulators,” Rev. Mod. Phys. 82, 3045–3067 (2010).
[Crossref]

A. Eckardt, P. Hauke, P. Soltan-Panahi, C. Becker, K. Sengstock, and M. Lewenstein, “Frustrated quantum antiferromagnetism with ultracold bosons in a triangular lattice,” Europhys. Lett. 89, 10010 (2010).
[Crossref]

C. Monat, M. Ebnali-Heidari, C. Grillet, B. Corcoran, B. J. Eggleton, T. P. White, L. O’Faolain, J. Li, and T. F. Krauss, “Four-wave mixing in slow light engineered silicon photonic crystal waveguides,” Opt. Express 18, 22915–22927 (2010).
[Crossref]

2009 (6)

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
[Crossref]

H. Hagino, Y. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Effects of fluctuation in air hole radii and positions on optical characteristics in photonic crystal heterostructure nanocavities,” Phys. Rev. B 79, 085112 (2009).
[Crossref]

H. M. Guo and M. Franz, “Topological insulator on the kagome lattice,” Phys. Rev. B 80, 113102 (2009).

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

M. Patterson, S. Hughes, S. Combrié, N. V. Q. Tran, A. De Rossi, R. Gabet, and Y. Jaouën, “Disorder-induced coherent scattering in slow-light photonic crystal waveguides,” Phys. Rev. Lett. 102, 1–4 (2009).
[Crossref]

S. Mazoyer, J. P. Hugonin, and P. Lalanne, “Disorder-induced multiple scattering in photonic-crystal waveguides,” Phys. Rev. Lett. 103, 063903 (2009).
[Crossref]

2008 (5)

F. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[Crossref]

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 033834 (2008).
[Crossref]

T. F. Krauss, “Why do we need slow light?” Nat. Photonics 2, 448–450 (2008).
[Crossref]

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008).
[Crossref]

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

2007 (1)

S. Hughes, “Coupled-cavity QED using planar photonic crystals,” Phys. Rev. Lett. 98, 083603 (2007).
[Crossref]

2005 (4)

C. L. Kane and E. J. Mele, “Quantum spin Hall effect in graphene,” Phys. Rev. Lett. 95, 226801 (2005).
[Crossref]

A. Eckardt, C. Weiss, and M. Holthaus, “Superfluid-insulator transition in a periodically driven optical lattice,” Phys. Rev. Lett. 95, 1–4 (2005).

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325–327 (2005).
[Crossref]

Y.-H. Kuo, Y. K. Lee, Y. Ge, S. Ren, J. E. Roth, T. I. Kamins, D. A. B. Miller, and J. S. Harris, “Strong quantum-confined Stark effect in germanium quantum-well structures on silicon,” Nature 437, 1334–1336 (2005).
[Crossref]

2000 (2)

K. Ohgushi, S. Murakami, and N. Nagaosa, “Spin anisotropy and quantum Hall effect in the kagomé lattice: chiral spin state based on a ferromagnet,” Phys. Rev. B 62, R6065–R6068 (2000).
[Crossref]

R. Resta, “Manifestations of Berry’s phase in molecules and condensed matter,” J. Phys. 12, R107–R143 (2000).

1989 (1)

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

1988 (1)

F. D. M. Haldane, “Model for a quantum Hall effect without Landau levels: condensed-matter realization of the ‘parity anomaly’,” Phys. Rev. Lett. 61, 2015–2018 (1988).
[Crossref]

1987 (1)

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[Crossref]

1984 (1)

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. London Ser. A 392, 45–57 (1984).
[Crossref]

1973 (1)

H. Sambe, “Steady states and quasienergies of a quantum-mechanical system in an oscillating field,” Phys. Rev. A 7, 2203–2213 (1973).
[Crossref]

1965 (1)

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

1933 (1)

R. Peierls, “On the theory of the diamagnetism of conduction electrons,” Z. Phys. 80, 763–791 (1933).
[Crossref]

Asano, T.

H. Sekoguchi, Y. Takahashi, T. Asano, and S. Noda, “Photonic crystal nanocavity with a q-factor of 9 million,” Opt. Express 22, 916–924 (2014).
[Crossref]

H. Hagino, Y. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Effects of fluctuation in air hole radii and positions on optical characteristics in photonic crystal heterostructure nanocavities,” Phys. Rev. B 79, 085112 (2009).
[Crossref]

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

Azzini, S.

Baba, T.

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008).
[Crossref]

Badolato, A.

Y. Lai, S. Pirotta, G. Urbinati, D. Gerace, M. Minkov, V. Savona, A. Badolato, and M. Galli, “Genetically designed L3 photonic crystal nanocavities with measured quality factor exceeding one million,” Appl. Phys. Lett. 104, 241101 (2014).
[Crossref]

Bajcsy, M.

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Appl. Phys. Lett. (2)

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Supplementary Material (1)

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Figures (6)

Fig. 1.
Fig. 1. (a) Honeycomb lattice with two sites A and B in the primitive cell (highlighted in orange). The Haldane model involves a complex second-neighbor hopping, i.e., along the red dashed lines. The Brillouin zone in reciprocal space is also shown. (b) Band structure of the lattice with first-neighbor coupling J and zero second-neighbor coupling. Six Dirac cones at the K-points are present.
Fig. 2.
Fig. 2. (a) Kagomé lattice with three sites in the primitive cell, and the corresponding Brillouin zone. (b) Corresponding band structure with first-neighbor coupling J. There are six Dirac cones, and in addition a flat band. (c) The dynamic modulation results in effective imaginary couplings along the dashed lines. These are both first- and second-neighbor (brown and red lines, respectively). The three first-neighbor vectors a1, a2, and a3 appearing in Eq. (11) are also defined.
Fig. 3.
Fig. 3. (a) Quasi-energy bands computed through diagonalization on the Floquet basis, for J=0.1Ω, A0=0.9Ω, and φ=2.1. The bands are repeated in orders of mΩ, with m an integer. (b) Zoom-in on the m=0 region of (a). (c) Bands computed through a perturbative expansion of the effective time-independent Hamiltonian. In (b) and (c), the Chern number for each band is indicated.
Fig. 4.
Fig. 4. (Largest) width of the opened band gap due to the dynamic modulation of frequency Ω versus the amplitude A0 and the phase angle φ for the Kagomé lattice with first-neighbor coupling J=0.1Ω. (a) Floquet perturbation theory. (b) Expansion on the Floquet basis. (c),(d) Same as (a) and (b), but for J=0.5Ω. The color scheme is the same in panels (a) and (b), as well as in panels (c) and (d).
Fig. 5.
Fig. 5. Bands structure with the largest possible band gap for various values of J/Ω. (a) J=0.3Ω, A0=0.5Ω, and φ=2.1. (b) J=0.5Ω, A0=1.6Ω, and φ=2.1. (c) J=0.7Ω, A0=3.05Ω, and φ=2.67. The Chern number for each band is indicated.
Fig. 6.
Fig. 6. (a) Floquet bands for the ribbon geometry shown in (b), with a finite number of sites in one direction (the system is truncated at the solid black lines), and periodic boundary conditions in the other (along the dashed black lines). The parameters are as in Fig. 4(b): J=0.5Ω, A0=1.6Ω, and ϕ=2.1. (c) Spatial dependence of the eigenstates marked in blue and red, respectively, in panel (a). The y axis is aligned with the y axis of panel (b). (d)–(f) Same as (a)–(c), but for a different truncation [compare (b) and (e)].

Equations (11)

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H=i(ωi+Aicos(Ωt+ϕi))aiaiijJijaiaj,
|{ni},m=|{ni}exp(iΩiAisin(Ωt+ϕi)ni+imΩt),
Jij=JijJ0(ρij),
Jij=2ijm=1(1)mΩmpJm(ρip)Jm(ρpj)JipJpj×sin(m(ϕipϕpj))aiaj,
ρijeiϕij=(AjeiϕjAieiϕi)/Ω.
Aicos(Ωt+ϕi)=A0xircos(Ωt)+A0yirsin(Ωt),  i.
un(k,t)=i,mvi,m(k,n)eikRieimΩt,
H˜=kAk(ω0+H(k))Ak
Ak=(aA,k,aB,k,aC,k),
H(k)=2J(0tAB(k)tAC(k)tAB*(k)0tBC(k)tAC*(k)tBC*(k)0),
tAB(k)=(tAB,0+tAB,1)cos(ka1)+tAB,1cos(k(a2+a3)),tAC(k)=(tAC,0+tAC,1)cos(ka2)+tAC,1cos(k(a1a3)),tBC(k)=(tBC,0+tBC,1)cos(ka2)+tBC,1cos(k(a1+a2)),

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