power (P≡
P
s|fs|(^2) ) are conserved. Nonlin-
earity in the off-diagonal coupling term is
negligible.
Consider a periodically modulated pho-
tonic square lattice ( 24 , 25 , 32 ) with nearest-
neighbor couplingsJm(z)[m=1,...,4]thatare
engineered in a cyclic (spatially andzperiodic)
manner such that every waveguide is coupled
to only one of its nearest neighbors at a given
propagation distancez(Fig.1,AandB).
The driving periodz 0 is split into four equal
steps, and over each quarter of the driving
period, only one of the four couplings is
switched on, with a fixedL=∫dzJm(z)(the
integral is taken over one coupling opera-
tion). In the linear regime, the quasi-energy
spectrum can be obtained by diagonalizing
the Floquet evolution operator over one period,
defined as
U^ðz 0 Þ¼Texp i∫
z 0
0
H^ð~zÞd~z
"#
ð 2 Þ
whereTindicates the“time”ordering and
H^ðzÞ¼H^ðzþz 0 Þis the periodic linear Ham-
iltonian. This driven lattice supports two
ungapped bulk bands (henceforth referred
to as the bulk band), and the bandwidth is
determined byL: ForL=p/2, the bulk band
becomes perfectly flat, and the bandgap closes
atL={p/4, 3p/4}. To experimentally realize
a weakly dispersive bulk band with an appre-
ciable bandgap, we setL= 1.85 ± 0.05. The
ratio of the bulk bandwidth to the maximal
coupling strength max[J(z)] is estimated to
be ~0.25, which quantifies the flatness of the
band (whereas the bandwidth of a standard
static square lattice is eight times the coupling
strength). Figure 1C shows a spectrum calculated
forastripgeometryalignedalongthevertical
direction and periodic along the horizontal
direction. As a result of the periodicity of
quasi-energy, the edge modes can cross the
bandgap, connecting the top and bottom of
the band structure. A single chiral edge mode
exists above and below the bulk band (prop-
agatinginthesamedirectiononagivenedge),
which implies that the Chern number of the
bulk band is zero. For such anomalous Floquet
topological insulators ( 23 , 33 ), the topology
can be captured using a different topological
invariant, the winding number ( 31 ). This scenario
canonlyariseinthepresenceofsuitabletime-
periodic driving; anomalous Floquet topological
insulators have no analog in static systems.
There is only one bandgap in the system
(Fig. 1C), and it is topological.
Using a self-consistency method modified
for Floquet systems ( 26 ), we sought localized
nonlinear solutions (solitons) in this modu-
lated photonic square lattice ( 31 ). The result is
solitons in the Floquet sense: Because of the
z-periodic driving, thesolitons reproduce
themselves after each complete period (up to
a phase factor), although micromotion within
the Floquet cycle is allowed for. The solitons
continuously rotate, performing cyclotron-
like motion (movie S1). Figure 2, B to E, shows
the normalized intensity profile (i.e., |fs|^2 /P)
of a soliton at each quarter-period (i.e.,z=
{0, 1, 2, 3}z 0 /4). Figure 2F shows the var-
iation of normalized intensity at the four
sites (1 to 4 in Fig. 2A) where the maximum
optical power of the soliton is supported
during propagation.
The quasi-energy of a soliton is determined
by the overall phase acquired after the propa-
gation of one driving period (e= phase/z 0 ).
The quasi-energy spectrum, plotted as a func-
tion of the renormalized power (Fig. 2G), shows
a family of bandgap solitons (red circles)
bifurcating from the linear band (blue). The
size (i.e., the spatial extent) of the solitons first
decreases as a function of power, showing
maximal localization near the mid-gap quasi-
energyp/z 0. When the power is further
increased, these Floquet solitons become de-
localized as they approach the band from the
other side ( 31 )—abehaviorunlikethatof
standard lattice solitons. In other words, for
a given dispersion of the linear band, the spa-
tial extent of these solitons is determined by
their quasi-energy; solitons closer in energy to
the linear band have a larger spatial extent
(movie S2). Because the solitons are strongly
Mukherjeeet al.,Science 368 , 856–859 (2020) 22 May 2020 2of4
Fig. 2. Topological bandgap soliton performing cyclotron-like motion.
(A) Schematic showing the four sites (1 to 4) where the maximum optical power
of a topological soliton is contained during propagation. (BtoE) Normalized
intensity profile of a soliton (atP=0.088mm–^1 ) rotating counterclockwise
and cycling back to itself after each complete period of driving; the color
map is in log scale. Here, the soliton profile is shown for each quarter
of a complete period—that is,z={0,1,2,3}z 0 /4. (F) Variation of
normalized intensity at the four sites [1 to 4 in (A)] showing the dynamics
in a complete period. (G) Quasi-energy as a function of renormalized power,
showing the family of bulk solitons (red circles) on both sides of the linear
modes. (H) A signature of topological solitons: When light is coupled to a
single bulk waveguide, the output intensity pattern atz=2z 0 exhibits a
distinct feature: a peak in the inverseparticipation ratio (IPR), which is
detected in experiments (see Fig. 3).
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