Observation of photonic anomalous Floquet topological insulators

Abstract

Topological insulators are a new class of materials that exhibit robust and scatter-free 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 two-dimensional 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 scatter-free edge transport in such anomalous Floquet topological insulators.

ARTICLE

Received 5 Aug 2016 | Accepted 31 Oct 2016 | Published 4 Jan 2017

Topological insulators are a new class of materials that exhibit robust and scatter-free transport along their edges independently of the ne details of the system and of the edge due to topological protection. To classify the topological character of two-dimensional systems without additional symmetries, one commonly uses Chern numbers, as their sum computed from all bands below a specic 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 scatter-free edge transport in such anomalous Floquet topological insulators.

DOI: 10.1038/ncomms13756 OPEN

Observation of photonic anomalous Floquet topological insulators

Lukas J. Maczewsky1,*, Julia M. Zeuner1,*, Stefan Nolte1 & Alexander Szameit1

1 Institute of Applied Physics, Abbe Center of Photonics, Friedrich-Schiller-Universitat Jena, Max-Wien-Platz 1, 07743 Jena, Germany. * These authors contributed equally to this work. Correspondence and requests for materials should be addressed to A.S. (email: mailto:[email protected]

Web End [email protected] ).

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The discovery of the quantized Hall effect1 revealed the existence of a new class of extremely robust transport phenomena, which are largely independent of sample size,

shape and composition. The scatter-free nature of these phenomena can be linked to the existence of non-trivial topological invariants associated with the systems bulk bands2. Shortly after the discovery of topological insulators27, the concept of topology was transferred to the photonic domain of electromagnetic waves8 with the rst realization in the microwave regime implementing the photonic analogue of the quantum Hall effect9. The search for an optical realization of topological insulators has prompted a number of proposals1014, and culminated in various experimental realizations5,6. 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 eld of topological photonics15 evolved well afterwards and resulted in various further studies, such as nonlinear waves in topological insulators and the prediction of topological gap solitons16, topological states in passive PT-symmetric media17, topological sub-wavelength

c1 c1

c1 c1

T

Quasienergy

T

Edge mode

Chern number = 0

Edge mode

k

Figure 1 | Floquet band structure in a driven system. Conceptual sketch of the band structure in a driven system, which is periodic in momentum k and quasi-energy e. Essentially, the band structure is analogue to a torus (see inset). This allows chiral edge modes to exist even if the Chern numbers of all bands are equal to zero.

a b

c1

c1

3 1 1

2

a

STEP 1

STEP 2

STEP 3

STEP 4

c1

4

c1

y

c1

c1

31 3

x

c2

c2

c

STEP 4

STEP 1 STEP 3

STEP 2

Quasienergy

c2

c2

c2

c2 c2

c2

c2

c3

c3 c3c3

c3

c3

c3

d

T

c3 c3 c3

c4 c4 c4

c4

c4 c4c4 c4 c4

0

T

2

0

kxa 2

Figure 2 | Bipartite lattice structure with periodic driving. (a) The coupling to the neighbouring waveguides occurs in four steps of equal length; in each step, hopping takes place solely along the highlighted bonds with a coupling strength cj; all other couplings are zero. (b) If the coupling during each step is 100% cj 2pT

,

after a full driving period T, one observes the formation of localized bulk modes without dispersion and chiral edge modes travelling along the lattice boundaries. (c) A schematic sketch of four lattice sites of the fabricated sample, in which the waveguides are drawn pairwise together to enable evanescent coupling. The initial waveguide spacing is a 40 mm ensuring negligible coupling between adjacent guides. (d) The edge band structure of

periodic quasi-energies in the case cj 2pT, exhibiting a at bulk band (brown line) and dispersionless chiral edge modes. Dotted and solid orange lines

describe the dispersion of opposite edges, respectively.

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NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13756 ARTICLE

settings18 and even three-dimensional systems exhibiting Weyl points19.

It is commonly accepted that for two-dimensional spin-decoupled topological systems a complete topological characterization is provided by the Chern numbers of each band, which represent a set of integer topological invariants20,21. 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 above15. 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 properties22. The reason is that in these systems, the xed energy in the band structure is replaced by a periodic quasi-energy. 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 possible10,23, 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 (A-FTI)22,24. Recently, it was shown that the appropriate topological invariants for characterizing these new phenomena are winding numbers22, 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 lattice25,26. However, to date the experimental demonstration of an A-FTI in an explicitly driven system is still elusive.

ResultsTight-binding and Floquet description of the lattice. In our work, we experimentally demonstrate an A-FTI in a two-

dimensional 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 Schrdinger equation (see ref. 27 for details). Therefore, evanescently coupled waveguide lattices are an excellent platform for testing Schrdinger physics.

We consider a bipartite square lattice with two site species A and B (with same on-site 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 rst section, a particular A-site couples to neighbouring B-site on its right, in the second section to its neighbouring B-site 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 cj 2pT

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 inter-site coupling in the individual sections n is achieved by appropriately engineering directional couplers28. This system is described by the Bloch Hamiltonian

HB k; z X

4 ;

where the vectors {bj} are given by b1 b3 (a,0) and

b2 b4 (0,a), with a being the distance between adjacent

lattice sites. In addition, for each partial step n, the coupling coefcients {cj(z)} are dened as cj djnc. We start our analysis by

choosing the coupling coefcient c 2pT, such that during each

step complete coupling into the respective neighbouring waveguide occurs. Obviously, the Hamiltonian is z-dependent, which for waveguide lattices is analogue to time-dependence in

10.80.60.40.2 0

0 cj z eibjk

cj z e ibjk 0

a b

c d

Figure 3 | Experimental demonstration of at band structure. Light distribution in the lattice after single-waveguide excitation for perfect coupling with strength cj 2pT 2p40mm 1 and after 3 full periods. The waveguide positions of the lattice are marked by white ellipses, the excited site is marked by a red

ellipse, and the trajectory is visualized with a white arrow. Evolution of the excited chiral edge state (a) along the edge, (b) around a corner and (c) along articial defects in the lattice structure. (d) If a bulk waveguide is excited, light follows a loop trajectory, as only localized at band modes are excited. The intensity in each gure is normalized to its maximum in the gure. One can clearly see that no scattering and no dispersion occurs, supporting the claim that a dispersion-free chiral edge state was excited.

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quantum mechanics27 and, hence, no eigenstates exist. However, due to the periodicity in z, Floquet theory can be applied to derive a band structure of so-called quasi-energies e (ref. 22). A solution of such a time-dependent Schrdinger equation are the Floquet states c t

f t

e iet with f(t T) f(t). Consequently, the

Floquet spectrum is periodic in its quasi-energies, in full correspondence to the periodicity in the transverse momentum caused by Blochs theorem. The temporal evolution of the system

is described by c t

Pe iR

t0 Htdtc 0 , such that

c T

e ieTc 0

. Note that P is the time-ordering operator. The time evolution operator U t

Pe iR

t0 Htdt 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 work5, in which the high-frequency driving allows for the description with an effective time-independent Hamiltonian Heff for all times t as c t

e itHeff c 0

.

Topological characterization by winding number. Our lattice structure exhibits two at 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 coefcients cj and not from a sublattice potential. Since the sum of the Chern numbers of all bands has to be zero, we nd that the Chern number of the at band in our system is zero. Although, when considering a nite 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 A-FTI. As it was shown earlier22, in periodically driven systems, the topological invariant characterizing the number of chiral edge modes is the winding number We , which is equal to the number of chiral edge modes nedge in a bandgap at a certain quasi-energy e:

nedgee We:

a b

1

0.5

0

0.5

11 0.5 0 0.5 1

kxa kxa

]2

1

0.5

0

0.5

1

1

c = 1.5 T

c = 0.85T

]

]

T

T

]

]

]

0.5 0 0.5 1]2

]

c

1.0

0.8

0.6

0.4

0.2

0.0

= 0W /T = 0 W /T = 1

= 0

I edge/ I tot

0.8 1.0 1.2 1.4 1.6 1.8 2.0 c / T

]

]

Figure 4 | Winding number transition. (a) If the coupling coefcients are reduced to c 1:5pT, the bulk band is not at anymore. However, chiral edge

modes still exist in the center of the Brillouin zone, and the winding number remains Wp/T 1. (b) For c 0:85pT, the chiral edge states have disappeared,

and the winding number is Wp/T 0. (c) Visualization of the phase transition using a single-waveguide excitation. As this populates all modes, the fraction

of the trapped intensity at the surface Iedge with respect to the total intensity Itot as a function of the coupling strength indicates the amount of existing chiral edge modes. When the coupling is copT, these modes disappear and essentially all light diffracts into the bulk. The different topological phases are marked with dark orange (Wp/T 1) and light orange (Wp/T 0), in both regimes, however, the Chern number C is zero. The error bars result due to

uncertainties of the fabrication process and are estimated via linear propagation of errors.

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The winding number is directly related to the Chern number22:

We2 We1 Ce1e2;where Ce1e2 is the sum of the Chern numbers of all bands residing

between e1 and e2. 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 at band structure. For our experiments, we fabricate the lattice sketched in Fig. 2c using the laser direct-writing technology27. 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. 3ac). 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 nd a at 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 at band, which has to have a Chern number of zero. This is the unequivocal proof of having implemented an A-FTI, 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 inter-site coupling on the topological nature of the system. So far we considered perfect hopping c 2pT

,

that is, in each section n the light completely couples to the neighbouring site, which results in an A-FTI phase. However, when decreasing the hopping rate (which results in only partial coupling), one will eventually leave the topologically non-trivial phase22 and enter the trivial phase exactly at c pT.

This is clearly visible in the edge band structures: one example of the topological non-trivial regime is shown in Fig. 4a c 1:5pT

and an example of the trivial regime in Fig. 4b c 0:85pT .

Whereas for c4pT chiral edge states exist (topological phase, Fig. 4a), at c pT a phase transition occurs and the edge states

disappear, such that for copT 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 single-site 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 Iedge/Itot as a function

of the coupling constant c. The error bars are due to slightly uctuating power of the writing laser and the signal to noise ratio of the recorded charge-coupled device (CCD) images. For c 2pT

indeed almost all of the light remains at the edge, as suggested by the edge band structure shown in Fig. 2d. For a decreasing

a b

1

0.90.80.70.60.50.40.30.20.1

0

10.90.80.70.60.50.40.30.20.1 0

c d

1

0.90.80.70.60.50.40.30.20.1

0

1.00.80.60.40.20.0

T]

]

a

I edge/ I tot

]

k a 2

]

b

T]

]

c

]

k a 2

]

1.0 1.5 2.0

c [ / T ]

Figure 5 | Edge state observation for specic momentum excitation. The region where separated chiral edge modes exist in the edge band structure reduces with decreasing coupling strength c. This is shown by exciting the band structure at kx p2a using an appropriately tilted broad beam and

observing the diffraction pattern for (a) c 2pT, (b) c 1:63pT and (c) c 1:2pT. The excited waveguides are marked by red ellipses, the integration area to

calculate the part of the intensity residing at the edge Iedge is surrounded by a white dashed line. (d) The plot of the intensity fraction of the trapped light clearly indicates that at kx p2a the amount of intensity exciting edge states signicantly reduces for decreasing coupling strength. The intensity in each

gure is normalized to its maximum. The respective edge band structures for c 1:63pT, and c 1:2pT are shown as insets. The error bars result due to

uncertainties of the fabrication process and are estimated via linear propagation of errors.

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the logarithm is chosen such that:

log e ieT i0 ieT;

log e ieT i0 ieT 2pi:

Fabrication of the structures. The single-mode waveguides were written27 inside a high-purity 15-cm-long fused silica wafer (Corning 7980) using a RegA 9000 seeded by a Mira Ti:Al2O3 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 mm 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 high-precision positioning stages (ALS130, Aerotech

Inc.). The refractive index increase of each guide is B8 10 4, the mode eld

diameters of the guided mode were 10.4 mm 8.0 mm 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 mm 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 mm to ensure signicant inter-site hopping. For perfect coupling cj 2pT

,

coupling constant the fraction of light that remains at the edge monotonously decreases until at copT no edge states are present, as the trivial phase is reached.

Edge state observation for specic 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 kx 0 such states are always found until

the phase transition occurs, separated edge modes close to the edge of the band structure (around kx p2a) conti

nuously cease to exist for decreasing coupling. This is illustrated when exciting the band structure only at the specic momentum kx p2a by using an appropriately tilted broad beam29.

In Fig. 5ac, it is shown that the state completely remains at the edge of the lattice for c 2pT (Fig. 5a), and partially spreads

into the bulk for c 1:63pT while a signicant fraction is still

trapped at the edge (Fig. 5b). However, for c 1:2pT the light

almost completely diffracts away from the edge as no separated chiral edge modes remain at kx p2a 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 c 1:63pT and c 1:2pT are shown as insets to see

the region in k-space in which the topological edge states exist. For c 2pT, the respective edge band structure is equal

to Fig. 2d.

DiscussionSummarizing our work, the results presented here clearly demonstrate the signicance of the winding number as the appropriate topological invariant characterizing periodically driven systems. Moreover, the chiral edge states in A-FTIs are highly robust to distortions in the lattice structure (including defects and imperfect hopping). Hence, our experimental observation of an A-FTI opens a new chapter in the eld of topological physics. Only recently, a novel topological phase was predicted in disordered A-FTI: the anomalous Floquet-Anderson insulator30. 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 signicant 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 operatorU k; t

P exp i R

t0 dt0Hk; t0

, with k as the momentum and P as the time-ordering operator. In a system exhibiting a at band at quasi-energy e 0, the

winding number W can be calculated as22:

nedge WU

1 8p2

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 inter-site spacing and interaction length; experimental errors arising due to uncertainties in the fabrication and the measurement are B5%. 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 bends28. 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 HTPS-EC-1) was launched into the system using a NA 0.35 objective. Whereas this is sufcient for single-site

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 momentum29.

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 strength31. We calibrated the wavelength-dependent coupling strength for the different interaction lengths of the individual sections in independent directional couplers.

Data availability. The data that support the ndings of this study are available from the corresponding author (A.S.) upon reasonable request.

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Z dtdkxdky Tr U 1@tU U 1@k U; U 1@k U

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Acknowledgements

We gratefully acknowledge nancial support from the Deutsche Forschungsgemeinschaft

(grants SZ 276/7-1, SZ 276/9-1, BL 574/13-1, GRK 2101/1) and the German Ministry for

Science and Education (grant 03Z1HN31).

Author 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 co-wrote the paper.

Additional information

Competing nancial interests: The authors declare no competing nancial

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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).

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r The Author(s) 2017

NATURE COMMUNICATIONS | 8:13756 | DOI: 10.1038/ncomms13756 | http://www.nature.com/naturecommunications

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