Abstract
Topological insulators are a new class of materials that exhibit robust and scatterfree transport along their edges — independently of the fine details of the system and of the edge — due to topological protection. To classify the topological character of twodimensional systems without additional symmetries, one commonly uses Chern numbers, as their sum computed from all bands below a specific bandgap is equal to the net number of chiral edge modes traversing this gap. However, this is strictly valid only in settings with static Hamiltonians. The Chern numbers do not give a full characterization of the topological properties of periodically driven systems. In our work, we implement a system where chiral edge modes exist although the Chern numbers of all bands are zero. We employ periodically driven photonic waveguide lattices and demonstrate topologically protected scatterfree edge transport in such anomalous Floquet topological insulators.
Introduction
The discovery of the quantized Hall effect^{1} revealed the existence of a new class of extremely robust transport phenomena, which are largely independent of sample size, shape and composition. The scatterfree nature of these phenomena can be linked to the existence of nontrivial topological invariants associated with the systems’ bulk bands^{2}. Shortly after the discovery of topological insulators^{2,3,4,5,6,7}, the concept of topology was transferred to the photonic domain of electromagnetic waves^{8} with the first realization in the microwave regime implementing the photonic analogue of the quantum Hall effect^{9}. The search for an optical realization of topological insulators has prompted a number of proposals^{10,11,12,13,14}, and culminated in various experimental realizations^{5,6}. Photonic topological insulators may enable novel and more robust photonic devices such as waveguides, interconnects, delay lines, isolators and couplers (or anything susceptible to parasitic scattering by fabrication disorder). The field of topological photonics^{15} evolved well afterwards and resulted in various further studies, such as nonlinear waves in topological insulators and the prediction of topological gap solitons^{16}, topological states in passive PTsymmetric media^{17}, topological subwavelength settings^{18} and even threedimensional systems exhibiting Weyl points^{19}.
It is commonly accepted that for twodimensional spindecoupled topological systems a complete topological characterization is provided by the Chern numbers of each band, which represent a set of integer topological invariants^{20,21}. The number of chiral edge modes residing in a bandgap is given by the sum of the Chern numbers of all bands below this gap. Hence, the Chern number is equal to the difference between the chiral edge modes entering the band from below and exiting it above^{15}. However, this is strictly true only for systems that are static, that is, where the Hamiltonian is constant in time. In periodically driven (Floquet) systems, the Chern numbers employed in the static case do not give a full characterization of the topological properties^{22}. The reason is that in these systems, the fixed energy in the band structure is replaced by a periodic quasienergy. As a consequence, the Chern numbers of all bands lying below a certain gap cannot be summed up since there exists no lowest band in the (periodic) band structure. Moreover, in such systems chiral edge modes are possible^{10,23}, although the Chern numbers of all bands may be zero (see Fig. 1 for an illustrative sketch). These materials are called anomalous Floquet topological insulators (AFTI)^{22,24}. Recently, it was shown that the appropriate topological invariants for characterizing these new phenomena are winding numbers^{22}, which utilize the information in the Hamiltonian for all times within a single driving period. This is in contrast to the Chern numbers of the individual bands, which only depend on the Hamiltonian evaluated stroboscopically once per driving cycle. Recently, anomalous edge states were shown in static network systems that are described by a scattering matrix and can be mapped onto a Floquet lattice^{25,26}. However, to date the experimental demonstration of an AFTI in an explicitly driven system is still elusive.
Results
Tightbinding and Floquet description of the lattice
In our work, we experimentally demonstrate an AFTI in a twodimensional driven system being not only periodical in the lattice directions x and y but also along the evolution coordinate. To this end, we work in the photonic regime and employ arrays of evanescently coupled waveguides. In such structures, the light evolution is governed by the paraxial Helmholtz equation, which is mathematically equivalent to the Schrödinger equation (see ref. 27 for details). Therefore, evanescently coupled waveguide lattices are an excellent platform for testing Schrödinger physics.
We consider a bipartite square lattice with two site species A and B (with same onsite potential), as it was suggested in ref. 22. Along the propagation direction, the structure consists of four sections with each having length T/4 and the entire period is T. In the first section, a particular Asite couples to neighbouring Bsite on its right, in the second section to its neighbouring Bsite above, and in the third and fourth section to its left and below, respectively (as sketched in Fig. 2a). If a 100%coupling per section is present, this lattice structure exhibits no transport in the bulk, as an excitation is trapped by moving only in loops, whereas at the edge transport occurs (see Fig. 2b). Figure 2c shows a sketch of how we realized this lattice in our experiments. The intersite coupling in the individual sections n is achieved by appropriately engineering directional couplers^{28}. This system is described by the Bloch Hamiltonian
where the vectors {b_{j}} are given by b_{1}=−b_{3}=(a,0) and b_{2}=−b_{4}=(0,a), with a being the distance between adjacent lattice sites. In addition, for each partial step n, the coupling coefficients {c_{j}(z)} are defined as c_{j}=δ_{jn}c. We start our analysis by choosing the coupling coefficient , such that during each step complete coupling into the respective neighbouring waveguide occurs. Obviously, the Hamiltonian is zdependent, which for waveguide lattices is analogue to timedependence in quantum mechanics^{27} and, hence, no eigenstates exist. However, due to the periodicity in z, Floquet theory can be applied to derive a band structure of socalled quasienergies ɛ (ref. 22). A solution of such a timedependent Schrödinger equation are the Floquet states with φ(t+T)=φ(t). Consequently, the Floquet spectrum is periodic in its quasienergies, in full correspondence to the periodicity in the transverse momentum caused by Bloch’s theorem. The temporal evolution of the system is described by , such that . Note that P is the timeordering operator. The time evolution operator includes the effective stroboscopic dynamics after multiples of the period T and the micro motion within a single period. This represents the full Floquet regime, in contrast to the adiabatic system used in our recent work^{5}, in which the highfrequency driving allows for the description with an effective timeindependent Hamiltonian H_{eff} for all times t as .
Topological characterization by winding number
Our lattice structure exhibits two flat degenerate bands (that appear as a single band), as the bipartite character of the lattice arises only from the sequential coupling steps with four equal coupling coefficients c_{j} and not from a sublattice potential. Since the sum of the Chern numbers of all bands has to be zero, we find that the Chern number of the flat band in our system is zero. Although, when considering a finite system, we observe the formation of chiral edge states (see Fig. 2d). In this vein, the Chern number is not the appropriate topological invariant that characterizes the existence and the amount of chiral edge states in our system. This is the very nature of an AFTI. As it was shown earlier^{22}, in periodically driven systems, the topological invariant characterizing the number of chiral edge modes is the winding number W_{ɛ} , which is equal to the number of chiral edge modes n_{edge} in a bandgap at a certain quasienergy ɛ:
The winding number is directly related to the Chern number^{22}:
where is the sum of the Chern numbers of all bands residing between ɛ_{1} and ɛ_{2}. Therefore, the difference of the number of chiral edge modes entering a band from below and exiting it above is equal to the Chern number of the respective band. We describe the approach for calculating the winding number in Methods section.
Experimental realization of flat band structure
For our experiments, we fabricate the lattice sketched in Fig. 2c using the laser directwriting technology^{27}. For details regarding the fabrication, the lattice parameters and the characterization setup we refer to Methods section. We start by launching light into single sites of the lattice and observe light dynamics that is summarized in Fig. 3. As clearly shown, the excited edge state travels dispersionless and without any scattering along edges, around corners and various defects (Fig. 3a–c). This highly robust, unidirectional state is a clear signature of topological protection. However, as opposed to a common Floquet topological insulator, in our system we find a flat band of bulk modes. This is shown by exciting the sites in the bulk of the lattice and observing that light is trapped in a loop, indicating the excitation of only localized modes (see Fig. 3d for one example). As we observe the same dynamics for any bulk site, we can conclude that there is indeed only one band, which consists of localized degenerate states: a single flat band, which has to have a Chern number of zero. This is the unequivocal proof of having implemented an AFTI, as clearly the Chern number does not predict the existence of the chiral edge states.
Examination of the winding number transition
In the next step, we will analyse the impact of the intersite coupling on the topological nature of the system. So far we considered perfect hopping , that is, in each section n the light completely couples to the neighbouring site, which results in an AFTI phase. However, when decreasing the hopping rate (which results in only partial coupling), one will eventually leave the topologically nontrivial phase^{22} and enter the trivial phase exactly at . This is clearly visible in the edge band structures: one example of the topological nontrivial regime is shown in Fig. 4a and an example of the trivial regime in Fig. 4b . Whereas for chiral edge states exist (topological phase, Fig. 4a), at a phase transition occurs and the edge states disappear, such that for the system is in a trivial phase (Fig. 4b). Note, that in both phases the Chern number of the band is zero, and only the value of the winding number changes. To study this phase transition, we perform various measurements in systems with decreasing coupling constant (see Methods section for the experimental approach). We launch light into a single site at the edge of the structure, as this populates the entire band structure, and analyse the diffraction pattern. If there is an edge state present, it is partially excited by the singlesite excitation and some of the evolving light will remain at the edge during propagation. However, if there is no edge state present, after a certain propagation length all of the light will have diffracted into the bulk of the system. Our experimental results are summarized in Fig. 4c, where we plot the intensity ratio I_{edge}/I_{tot} as a function of the coupling constant c. The error bars are due to slightly fluctuating power of the writing laser and the signal to noise ratio of the recorded chargecoupled device (CCD) images. For indeed almost all of the light remains at the edge, as suggested by the edge band structure shown in Fig. 2d. For a decreasing coupling constant the fraction of light that remains at the edge monotonously decreases until at no edge states are present, as the trivial phase is reached.
Edge state observation for specific momentum excitation
Importantly, the region in reciprocal space where chiral edge states being separated from the bulk bands are found reduces for decreasing coupling strength: whereas in the centre of the edge band structure around k_{x}=0 such states are always found until the phase transition occurs, separated edge modes close to the edge of the band structure (around ) continuously cease to exist for decreasing coupling. This is illustrated when exciting the band structure only at the specific momentum by using an appropriately tilted broad beam^{29}. In Fig. 5a–c, it is shown that the state completely remains at the edge of the lattice for (Fig. 5a), and partially spreads into the bulk for while a significant fraction is still trapped at the edge (Fig. 5b). However, for the light almost completely diffracts away from the edge as no separated chiral edge modes remain at for this low coupling strength (Fig. 5c). Note that the chiral edge states reside on every second waveguide solely, such that we excited only those with the broad beam. Our results are summarized in Fig. 5d, where the fraction of the light trapped at the lattice edge is plotted as a function of the coupling strength. One clearly sees the drop in light intensity at the edge, proving the disappearance of the edge states for decreasing coupling strength. In addition, the edge band structures for and are shown as insets to see the region in kspace in which the topological edge states exist. For , the respective edge band structure is equal to Fig. 2d.
Discussion
Summarizing our work, the results presented here clearly demonstrate the significance of the winding number as the appropriate topological invariant characterizing periodically driven systems. Moreover, the chiral edge states in AFTIs are highly robust to distortions in the lattice structure (including defects and imperfect hopping). Hence, our experimental observation of an AFTI opens a new chapter in the field of topological physics. Only recently, a novel topological phase was predicted in disordered AFTI: the anomalous FloquetAnderson insulator^{30}. But there are many more puzzles to solve: What is the impact of nonlinearity on the formation of these chiral edge states? Does the dimensionality play a significant role? What are the possibilities to obtain different phases than reported here? The answer to these and other intriguing questions are now in reach. The authors of this work would like to point out that a related work with similar results is published in ref. 32.
Methods
Winding number
To calculate the number of chiral edge modes in a periodically driven system, the behaviour of the system during a full driving period has to be taken into account, by employing the time evolution operator , with k as the momentum and P as the timeordering operator. In a system exhibiting a flat band at quasienergy ɛ=0, the winding number W can be calculated as^{22}:
In the case of curved (dispersive) bands, the winding number in a gap is W_{ɛ}=, with U_{ɛ} being constructed as follows^{22}:
Here, V_{ɛ}(k,t)=exp(−iH_{eff} (k)t) with . The branch cut of the logarithm is chosen such that:
Fabrication of the structures
The singlemode waveguides were written^{27} inside a highpurity 15cmlong fused silica wafer (Corning 7980) using a RegA 9000 seeded by a Mira Ti:Al_{2}O_{3} femtosecond laser. Pulses centred at 800 nm with duration of 150 fs were used at a repetition rate of 100 kHz and energy of 450 nJ. The pulses were focused 671 to 883 μm under the sample surface using an objective with a numerical aperture (NA) of 0.35, while the sample was translated at constant speed of 100 mm min^{−1} by highprecision positioning stages (ALS130, Aerotech Inc.). The refractive index increase of each guide is ∼8 × 10^{−4}, the mode field diameters of the guided mode were 10.4 μm × 8.0 μm at 633 nm. Propagation losses and birefringence were estimated at 0.2 dB cm^{−1} and in the order of 10^{−7}, respectively. The site spacing a=40 μm ensures that there is no unwanted coupling between adjacent waveguides. In the individual sections of the lattice (shown in Fig. 2c) the waveguides that couple converge to a spacing of 9.7 μm to ensure significant intersite hopping. For perfect coupling , the length of a full period is T=40 mm. Each bending is 4.17 mm long, such that the additional losses caused by the bending are as low as 4%. The coupling strength between the guides was determined in preliminary experiments as a function of intersite spacing and interaction length; experimental errors arising due to uncertainties in the fabrication and the measurement are ∼5%. For obtaining the different coupling strengths between the individual sites the length of the coupling region was appropriately designed, taking into account the weak coupling that occurs already in the bends^{28}. All samples contain 3 full periods, whereas the remaining 3 cm were used for preparation of the injection distribution required in each case.
Characterization of the structures
For the observation of the light evolution, light from a tunable Helium Neon laser (Thorlabs HTPSEC1) was launched into the system using a NA=0.35 objective. Whereas this is sufficient for singlesite excitation, for the broad excitation the beam was expanded with a slit and a biconvex lens (f=35 mm) perpendicular to the orientation of the slit. Together with every other waveguide starting 2 cm later in propagation direction and an appropriate tilt of the sample we excite the correct transverse momentum^{29}. We fabricated several structures with different coupling strengths as described above. However, in order to achieve more data points, we used different excitation wavelengths (633, 604, 594 and 543 nm) that allowed us to further manipulate the coupling strength^{31}. We calibrated the wavelengthdependent coupling strength for the different interaction lengths of the individual sections in independent directional couplers.
Data availability
The data that support the findings of this study are available from the corresponding author (A.S.) upon reasonable request.
Additional information
How to cite this article: Maczewsky, L. J. et al. Observation of photonic anomalous Floquet topological insulators. Nat. Commun. 8, 13756 doi: 10.1038/ncomms13756 (2017).
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
 1
von Klitzing, K., Dorda, G. & Pepper, M. New method for highaccuracy determination of the finestructure constant based on quantized Hall Resistance. Phys. Rev. Lett. 45, 494–497 (1980).
 2
König, M. et al. Quantum spin Hall insulator state in HgTe quantum wells. Science 318, 766–770 (2007).
 3
Kane, C. L. & Mele, E. J. Z2 topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005).
 4
Hsieh, D. et al. A topological Dirac insulator in a 3D quantum spin Hall phase. Nature 452, 970–974 (2008).
 5
Rechtsman, M. C. et al. Photonic Floquet topological insulators. Nature 496, 196–200 (2013).
 6
Hafezi, M., Mittal, S., Fan, J., Migdall, A. & Taylor, J. M. Imaging topological edge states in silicon photonics. Nat. Photon. 7, 1001–1005 (2013).
 7
Bernevig, B. A., Hughes, T. L. & Zhang, S.C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757–1761 (2006).
 8
Haldane, F. D. M. & Raghu, S. Possible realization of directional optical waveguides in photonic crystals with broken timereversal symmetry. Phys. Rev. Lett. 100, 013904 (2008).
 9
Wang, Z., Chong, Y., Joannopoulos, J. D. & Soljačić, M. Observation of unidirectional backscatteringimmune topological electromagnetic states. Nature 461, 772–775 (2009).
 10
Kitagawa, T., Berg, E., Rudner, M. & Demler, E. Topological characterization of periodically driven quantum systems. Phys. Rev. B 82, 235114 (2010).
 11
Umucalılar, R. O. & Carusotto, I. Artificial gauge field for photons in coupled cavity arrays. Phys. Rev. A 84, 043804 (2011).
 12
Hafezi, M., Demler, E. A., Lukin, M. D. & Taylor, J. M. Robust optical delay lines with topological protection. Nat. Phys. 7, 907–912 (2011).
 13
Fang, K., Yu, Z. & Fan, S. Realizing effective magnetic field for photons by controlling the phase of dynamic modulation. Nat. Photon. 6, 782–787 (2012).
 14
Khanikaev, A. B. et al. Photonic topological insulators. Nat. Mater. 12, 233–239 (2013).
 15
Lu, L., Joannopoulos, J. D. & Soljačić, M. Topological photonics. Nat. Photon. 8, 821–829 (2014).
 16
Lumer, Y., Plotnik, Y., Rechtsman, M. C. & Segev, M. Selflocalized states in photonic topological insulators. Phys. Rev. Lett. 111, 243905 (2013).
 17
Zeuner, J. M. et al. Observation of a topological transition in the bulk of a nonHermitian system. Phys. Rev. Lett. 115, 040402 (2015).
 18
Slobozhanyuk, A. P., Poddubny, A. N., Miroshnichenko, A. E., Belov, P. A. & Kivshar, Y. S. Subwavelength topological edge states in optically resonant dielectric structures. Phys. Rev. Lett. 114, 123901 (2015).
 19
Lu, L. et al. Experimental observation of Weyl points. Science 349, 622–624 (2015).
 20
Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized Hall conductance in a twodimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982).
 21
Schnyder, A. P., Ryu, S., Furusaki, A. & Ludwig, A. W. W. Classification of topological insulators and superconductors in three spatial dimensions. Phys. Rev. B 78, 195125 (2008).
 22
Rudner, M. S., Lindner, N. H., Berg, E. & Levin, M. Anomalous edge states and the bulkedge correspondence for periodically driven twodimensional systems. Phys. Rev. X 3, 031005 (2013).
 23
Kitagawa, T. et al. Observation of topologically protected bound states in photonic quantum walks. Nat. Commun. 3, 882 (2012).
 24
Lindner, N. H., Refael, G. & Galitski, V. Floquet topological insulator in semiconductor quantum wells. Nat. Phys. 7, 490–495 (2011).
 25
Hu, W. et al. Measurement of a topological edge invariant in a microwave network. Phys. Rev. X 5, 011012 (2015).
 26
Gao, F. et al. Probing topological protection using a designer surface plasmon structure. Nat. Commun. 7, 11619 (2016).
 27
Szameit, A. & Nolte, S. Discrete optics in femtosecondlaserwritten photonic structures. J. Phys. B 43, 163001 (2010).
 28
Gräfe, M. et al. Onchip generation of highorder singlephoton Wstates. Nat. Photon. 8, 791–795 (2014).
 29
Eisenberg, H. S., Silberberg, Y., Morandotti, R. & Aitchison, J. S. Diffraction management. Phys. Rev. Lett. 85, 1863 (2000).
 30
Titum, P., Berg, E., Rudner, M. S., Refael, G. & Lindner, N. H. Anomalous FloquetAnderson insulator as a nonadiabatic quantized charge pump. Phys. Rev. X 6, 021013 (2016).
 31
Szameit, A., Dreisow, F., Pertsch, T., Nolte, S. & Tünnermann, A. Control of directional evanescent coupling in fs laser written waveguides. Opt. Express 15, 1579 (2007).
 32
Mukherjee, S. et al. Experimental observation of anomalous topological edge modes in a slowly driven photonic lattice. Nat. Commun 8, 13918 (2017).
Acknowledgements
We gratefully acknowledge financial support from the Deutsche Forschungsgemeinschaft (grants SZ 276/71, SZ 276/91, BL 574/131, GRK 2101/1) and the German Ministry for Science and Education (grant 03Z1HN31).
Author information
Affiliations
Contributions
L.J.M. performed the measurements, J.M.Z. elaborated on the theory and A.S. supervised the project. All authors discussed the results and cowrote the paper.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Supplementary information
Rights and permissions
This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
About this article
Cite this article
Maczewsky, L., Zeuner, J., Nolte, S. et al. Observation of photonic anomalous Floquet topological insulators. Nat Commun 8, 13756 (2017). https://doi.org/10.1038/ncomms13756
Received:
Accepted:
Published:
Further reading

Topological holographic quench dynamics in a synthetic frequency dimension
Light: Science & Applications (2021)

Electronic Floquet gyroliquid crystal
Nature Communications (2021)

Floquet Spectrum and Dynamics for NonHermitian Floquet OneDimension Lattice Model
International Journal of Theoretical Physics (2021)

Realization of an anomalous Floquet topological system with ultracold atoms
Nature Physics (2020)

Driving toward hot new phases
Nature Physics (2020)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.