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Photonic Floquet skin-topological effect
Yeya ng S un 1#, Xiangrui Hou1#, Tuo Wan1, Fangyu Wang1, Shiyao Zhu1,2,
Zhichao Ruan1* and Zhaoju Yang1*
1School of Physics, Interdisciplinary Center for Quantum Information, Zhejiang Province Key
Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou 310027, Zhejiang
Province, China
2Hefei National Laboratory, Hefei 230088, China
#These authors contributed equally to this work
*Email: zhichao@zju.edu.cn; zhаojuyang@zju.edu.cn.
Abstract
Non-Hermitian skin effect and photonic topological edge states are of great
interest in non-Hermitian physics and optics. However, the interplay between them is
largly unexplored. Here, we propose and demonstrate experimentally the non-
Hermitian skin effect that constructed from the nonreciprocal flow of Floquet
topological edge states, which can be dubbed ‘Floquet skin-topological effect’. We first
show the non-Hermitian skin effect can be induced by pure loss when the one-
dimensional (1D) system is periodically driven. Next, based on a two-dimensional (2D)
Floquet topological photonic lattice with structured loss, we investigate the interaction
between the non-Hermiticity and the topological edge states. We observe that all the
one-way edge states are imposed onto specific corners, featuring both the non-
Hermitian skin effect and topological edge states. Furthermore, a topological switch for
the skin-topological effect is presented by utilizing the gap-closing mechanism. Our
experiment paves the way of realizing non-Hermitian topological effects in nonlinear
and quantum regimes.
Topological insulators are a new phase of matter that is constituted by insulating
bulk and conducting edges. They have been extensively explored in condensed matter
physics [1,2], photonics [3–11], phononics [12–14], and so on. In photonics, shortly
after the observation of topologically protected edge states in microwaves [4], the
topological states in the optical frequency range relying on artificial gauge fields were
experimentally realized [5,6]. One paradigmatic example is the photonic Floquet
topological insulator [5] consisting of a honeycomb array of helical optical waveguides.
The periodic driving force results in the artificial gauge field and Floquet topological
phases. The one-way topological edge states that are immune to backscattering were
predicted and observed. The realizations of the topological states in classical-wave
systems show potential in lasing [15–18] and quantum sources [19,20].
Characterized by complex eigenenergies and nonorthogonal eigenstates, non-
Hermitian physics [21–23] governing systems interacting with the environment has led
to many frontiers, such as PT symmetric physics [24–34] and non-Hermitian
topological phases [35–44]. Recently, the non-Hermitian skin effect (NHSE) [45–58]
has drawn a lot of attention both in theory and experiment. The NHSE features the
coalescence of the extended bulk states into the edges of 1D systems, which can be well
described by the non-Bloch band theory [48,51]. The interplay between the NHSE and
aforementioned topological states brings us a new concept of the hybrid skin-
topological effect [59–61]. Different from the higher-order non-Hermitian skin
effect [62–66], the action of the coalescence of extended eigenstates for the skin-
topological effect is only on the topological edge modes [60,61]. Therefore, the number
of the skin-topological modes is proportional to the length size of the system. So far,
such an effect has only been realized by introducing asymmetric coupling into higher-
order topological insulators [66], whereas the interaction between the one-way
propagating topological edge states and NHSE remains less explored.
In this work, we bridge this gap by adopting a 2D optical array of lossy helical
waveguides and observe the photonic Floquet skin-topological effect. First, by
introducing staggered loss into a 1D optical array of helical waveguides, we realize the
Floquet non-Hermitian skin effect. Next, we pile up the 1D lattice and arrive at a 2D
non-Hermitian Floquet topological insulator. The complex spectrum of the non-
Hermitian system shows that the gapless unidirectional edge states spanning across the
topological band gap (corresponding to a nonzero Chern number of 1) can acquire the
non-trivial point-gap winding topology [45,60,61,67], which indicates the existence of
the NHSE that induced by the nonreciprocal flow of these edge states. The sign of the
winding number determines which corner of the sample is the topological funnel of
light. Moreover, by introducing a large enough on-site energy difference and closing
the Floquet topological band gap, a topological switch [61,68] for the Floquet skin-
topological effect is demonstrated.
We start from a 1D optical array consisting of helical waveguides, as shown in Fig.
1a. This 1D optical array contains two sublattices A and B, and the sublattice B is
endowed with considerable loss, as marked in blue. The paraxial propagation of light
in this non-Hermitian system can be described by the tight-binding equation [5]:
i=()
=() (1)
where is the electric field amplitude in the nth waveguide, c is the coupling
strength between the two nearest waveguides, is the displacement pointing from
waveguide m to n. This z-dependent equation describing paraxial light propagation can
be mapped to a time-dependent Schrodinger equation and the z axis plays the role of
time. The periodic driving is equivalent to adding a time-dependent vector potential
()=((),(), 0) to the optical array. The distance between the
two nearest waveguides is =15 m and the lattice constant is =153 m. The
helix radius is = 8 m and the period is = 1 .
For the above tight-binding model with z-periodic Hamiltonian, its eigenstates can
be calculated from the equation of =, where =
()
is the
effective Hamiltonian for wavefunctions over one period and is the quasienergy for
Floquet systems. The complex quasienergy spectrum of this 1D optical array under
periodic boundary condition (PBC) and open boundary condition (OBC) is shown in
Fig. 1b, as labeled by grey dotted curves and black dots, respectively. We can see that
the complex spectrum under PBC forms a closed loop and drastically collapses into a
line under OBC. The eigenfunctions displayed in Fig. 1b reveal that the eigenstates all
localize at the left boundary, which is the direct result of the NHSE. The closed loop in
the complex spectrum results in the non-trivial point-gap topology, which can be
characterized by the winding number [45,69] for Floquet systems
=
arg[() ]
(2)
where is a reference quasienergy for numerical calculations. The winding number
for the 1D optical array is = 1 indicating the existence of the NHSE.
In experiments, we fabricate the 1D optical array of helical waveguides by
utilizing the femtosecond laser writing method [5]. The optical loss in sublattice B is
introduced by setting breaks periodically into the waveguides [70] (see methods for
more details). To observe the 1D Floquet NHSE, a laser beam with a wavelength of 635
nm is initially launched into the center waveguide of the 1D array. We perform a series
of measurements at different propagation lengths of z=2, 4, 6, 8 and 10 cm. The results
are shown in Fig. 1d. The white dashed circles mark the location of the input waveguide.
As we can see, the light propagates continuously to the left, which indicates the collapse
of the extended eigenstates onto the left boundary. The overall reduced light intensity
as increasing the propagation length is due to the passive setting of the optical array.
This observation unravels the existence of 1D NHSE induced by pure loss as well as
periodic driving and provides a cornerstone for the next exploration of the interplay
between the NHSE and photonic topological edge states.
Having observed the Floquet NHSE in a 1D array, we investigate the interaction
between the non-Hermiticity and topological edge states. We pile up the 1D Floquet
lattice composed of helical waveguides and arrive at a 2D photonic non-Hermitian
Floquet topological insulator with the nonzero non-Hermitian Chern number [35,67] of
= 1. The structured loss is introduced into one sublattice of the honeycomb lattice.
The non-trivial topology guarantees the existence of the topological boundary states
when we consider a finite sample. The complex spectra for two cases of x-PBC/y-OBC
(periodic boundary along the x-axis and open boundary in the y-direction, as pointed
out in panel d) and x-OBC/y-OBC are shown in Fig. 2a,c. Apart from the bulk states
marked by grey dotted points, there exist two counter-propagating edge states localized
at the opposite edges spanning across the topological band gap in the complex spectrum,
as displayed in Fig. 2a, b. The blue (red) dots correspond to the right (left)-propagating
edge mode with relatively larger (negligible) loss. The complex spectral loop in the
topological band gap reveals the non-trivial point-gap topology characterized by the
winding number of = 1, which results in the NHSE induced by the one-way edge
flow of light. For the case of double OBC (x-OBC/y-OBC), the closed loop in the
complex spectrum drastically collapses (Fig. 2c) and all non-Hermitian topological
edge modes localize at the upper-left corner (labeled as corner I in Fig. 2d), whereas
the bulk states stay extended. The solid (dashed) white arrows pointing to the
propagating direction of the edge states correspond to the flow of light with negligible
(large) loss. As a result, all the energy of the topological edge states accumulates at
corner I, which indicates the existence of the so-called hybrid skin-topolgoical modes.
Note that if we add the loss configuration on the sublattice B, the point-gap winding
will change to =1, which gives rise to the skin-topological modes at corner III.
To experimentally study the Floquet skin-topological effect, a 2D non-Hermitian
honeycomb lattice of helical waveguides is fabricated. The breaks that can generate
relatively larger optical loss are introduced into the waveguides of sublattice A. To
excite the topological edge modes, a broad tilted Gaussian beam with the momentum
of = is initially launched into the outer perimeter of the sample. The white
ellipse indicates the position and shape of the launched beam. The total propagation
length in the sample is 10 cm (z-axis). By moving the injected beam along the outer
perimeter, we can see in Fig. 3a,b that the input light can propagate unidirectionally
along the edge of IV-III, circumvent corner III (panel a), and propagate along the edge
of III-II, circumvent the sharp corner II (panel b) without backscattering. However, the
situation changes drastically when the input light encounters corner I. By moving the
input Gaussian beam leftwards along the edge, as shown in Fig. 3c, we observe that the
injected light propagates along the upper edge and stays trapped at corner I without
penetration into the bulk, which is the direct evidence of the existence of the hybrid
Floquet skin-topological modes. The observations in Fig. 3 together elucidate that the
Floquet skin-topological effect features both the NHSE and topological protection of
the propagating light.
For comparison, we fabricate a 2D Hermitian photonic Floquet topological
insulator with no structured loss. We reproduce the similar experiments and observe
that the initial tilted wave packet can propagate counterclockwise along the outer
perimeter and bypasses corner I (see methods for more details).
A topological switch [59,68] manifests that topology can provide a switch for the
NHSE through topological phase transitions. In our model, by introducing a large
enough on-site energy difference between the two sublattices (see methods for more
details), the topological phase transition occurs and the Flouqet topological band gap
closes. Without the topological edge states providing asymmetric coupling for the
NHSE, the complex spectrum shows no point-gap winding and therefore no Floquet
skin-topological modes can be found. We adopt the same experimental philosophy as
shown in Fig. 3. As we can see in Fig. 4a-c, the injected light penetrates into bulk and
cannot accumulate at corner I. Therefore, the skin-topological effect is switched off.
In conclusion, we have experimentally demonstrated the Floquet NHSE in a 1D
lossy optical array and Floquet skin-topological effect in a 2D non-Hermitian photonic
Floquet topological insulator. By introducing structured loss into the periodically driven
optical waveguides, we have found the point-gap topology in the complex spectra of
the 1D and 2D non-Hermitian photonic systems. In experiments, we have fabricated
the optical lattices by the standard femtosecond laser-writing method and observed the
topological funnel of light at the left boundary and corner I of the 1D and 2D optical
arrays, respectively. Moreover, a key to switching on/off the skin-topological effect has
been realized by utilizing the topological phase transition. Our work investigates the
interaction between the NHSE and photonic topological edge states and provides the
first example of the NHSE in an optical Floquet topological insulator, which may pave
the way for further exploration of non-Hermitian topological effects [71,72] in
nonlinear [73–75] and quantum regimes [76,77].
Acknowledgements
This research is supported by the National Key R&D Program of China (Grant No.
2022YFA1404203, 2022YFA1405200), National Natural Science Foundation of China
(Grant No. 12174339, 12174340), Zhejiang Provincial Natural Science Foundation of
China (Grant No. LR23A040003) and Excellent Youth Science Foundation Project
(Overseas).
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Figure 1. Floquet NHSE in a 1D optical array. a, Schematic of a 1D non-Hermitian
optical array consisting of helical waveguides. The color grey (blue) represents normal
(lossy) waveguides. The optical loss is introduced by setting breaks, as can be seen in
the blue waveguides. b, Complex quasi-energy spectrum of the 1D optical array. The
spectral loop under PBC indicates the existence of the point-gap topology. c, Floquet
skin eigenmodes (grey curves) and the summing eigenmodes (red curve). We set 200
sites in the array for these numerical results. The parameters for simulations are:
coupling strength = 1.5 cm , optical loss = 0.2 cm , lattice constant =
153 m, helix radius = 8 m and the period = 1 cm . d, Experimental
observation of the Floquet NHSE. The light shifts continuously to the left indicating
the collapse of the eigenstates into the left boundary.
Figure 2. Hybrid skin-topological effect in a non-Hermitian 2D photonic Floquet
topological insulator. a, b, Complex quasienergy spectrum under x-PBC/y-OBC. The
blue (red) dots correspond to the right (left)-propagating edge mode with relatively
larger (negligible) loss. The spectral loop in the topological band gap reveals the non-
trivial point-gap topology indicating the existence of the NHSE of light at the boundary.
c, d, Complex quasienergy spectrum under x-OBC/y-OBC. The skin-topological modes
(black dots) emerge and localize at the upper-left corner of the sample (corner I).
Figure 3. Experimental observation of the Floquet skin-topological effect. By moving
the input tilted Gaussian beam along the outer perimeter, we observe the output light
distribution at the end facet of the sample after 10 cm long propagation. a, The injected
light propagates along the edge of IV-III and bypasses corner III without backscattering.
b, The light propagates along the edge of III-II and bypasses the sharp corner III without
backscattering. c, The light propagates along the edge of II-I and accumulates at corner
I, which confirms the existence of the Floquet skin-topological modes.
Figure 4. Topological switch for the skin-topological effect. a-c, Introducing large
enough on-site energy difference of = 3 results in the topological phase
transition and closes the Flouqet topological band gap. In experiments, by moving the
input tilted Gaussian beam leftwards along the upper edge, we can see that the injected
light penetrates into the bulk. The skin-topological effect is switched off by the gap-
closing mechanism.
