Science - USA (2020-05-22)

In a second set of experiments, we used a
similar photonic lattice with maximum prop-
agation distancezmax=1.5z 0 instead of 2z 0
(movie S5). The IPR in this case exhibited
a similar behavior (i.e., delocalization to
localization to delocalization) to Fig. 3A. The
most localized output intensity pattern was
observed nearP=3.3mW(seeFig.3E).In
contrast to Fig. 3C, the brightest site in Fig. 3E
was not located where the light was initially
launched but was in another waveguide that
was directly across a diagonal from the in-
jected site; this constitutes direct evidence
of the cyclotron-like motion of the solitons.
Comparing Fig. 2H and Fig. 3A, the peak of
IPR is experimentally observed at a higher
power (P=0.26mm–^1 ) and the contrast of the
peak is lower than what is expected from
theory, as a result of linear loss. Additionally,
the front and rear tails (in time) of the pulses
behaved linearly, producing a small back-
ground and causing a lower contrast in IPR.
That said, the observed peak in IPR is a clear
signature of the topological bandgap solitons.
Our ability to control the“flatness”of the
bulk band by tailoring the coupling parameter
Lis key to the observation of these solitons.
In solid-state systems, flat bands play an im-
portant role in enhancing the relative strength
of interactions [a recent example is twisted
bilayer graphene ( 35 )]. This is also true in our
case but in a different way: The width of the
linear band sets the power threshold of soli-
tons in two dimensions. In the extreme case
of a perfectly flat band, all linear Bloch states
are degenerate, and thus localized eigenstates
can be constructed as a superposition of these
states, implying that stationary states exist
even in the linear domain. Thus, in this limit,
solitons have zero power threshold. Operat-
ing near the flat band with an appreciable
bandgap allows us to probe the solitons at an
experimentally accessible power value.
The observation of topological solitons opens
a new avenue in the investigation of topological
nonlinear optics, complementing other plat-
forms such as Rydberg polaritons ( 36 )and

nonlinear circuits ( 37 ). Furthermore, non-

linearity can act as a means to modify ( 38 , 39 )

and probe ( 40 , 41 ) topological photonic struc-

tures. There are many open questions; for

example, the degree of robustness and the

stability properties of chiral edge states in the

presence of nonlinearity are unknown. It will

also be of central importance to define new

invariants that characterize the observable

behavior of nonlinear topological systems. The

interplay of nonlinearity and disorder in topo-

logical systems will be necessary if nonlinear

topological devices are to be of technological

use. We expect that these issues, among others,

will dictate how topological states can be in-

corporated in useful devices based on wave

mechanics, whether in the photonic, acoustic/

phononic, optomechanical, atomic, polaritonic,

or other domains.

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ACKNOWLEDGMENTS

WethankH.K.Chandrasekharan,J.Guglielmon,D.Leykam,

and J. Noh for useful discussions and N. C. Giebink for use

of a supercontinuum laser source for directional coupler

characterization.Funding:Supported by Office of Naval Research

award N00014-18-1-2595 (S.M. and M.C.R.) and by Packard

Foundation fellowship 2017-66821 and Kaufman Foundation award

KA2017-91788 (M.C.R.).Author contributions:S.M. designed

and built the waveguide fabrication system as well as characterization

setups and carried out all experiments; S.M. and M.C.R. conceived

the idea, designed the experiment, analyzed data, and wrote the

manuscript; S.M. performed theoretical analysis with input from

M.C.R.; M.C.R. supervised the project.Competing interests:The

authors declare no competing interests.Data and materials

availability:All associated data and materials are available in the

manuscript and supplementary materials.

SUPPLEMENTARY MATERIALS

science.sciencemag.org/content/368/6493/856/suppl/DC1

Supplementary Text

Figs. S1 to S9

Movies S1 to S5

References ( 42 – 46 )

12 January 2020; accepted 14 April 2020

10.1126/science.aba8725

Mukherjeeet al.,Science 368 , 856–859 (2020) 22 May 2020 4of4

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