General Rules Governing the Dynamical Encircling of an Arbitrary Number of Exceptional Points

Abstract

Dynamically encircling an exceptional point in non-Hermitian systems has drawn great attention recently, since a nonadiabatic transition process can occur and lead to intriguing phenomena and applications such as the asymmetric switching of modes. While all previous experiments have been restricted to two-state systems, the dynamics in multistate systems where more complex topology can be formed by exceptional points, is still unknown and associated experiments remain elusive. Here, we propose an on-chip photonic system in which an arbitrary number of exceptional points can be encircled dynamically. We reveal in experiment a robust state-switching rule for multistate systems, and extend it to an infinite-period system in which an exceptional line is encircled with outcomes being located at the Brillouin-zone boundary. The proposed versatile platform is expected to reveal more physics related to multiple exceptional points and exceptional lines, and give rise to applications in multistate non-Hermitian systems.

Full text

General Rules Governing the Dynamical Encircling of an
Arbitrary Number of Exceptional Points
Feng Yu,1,§ Xu-Lin Zhang ,1,*,§ Zhen-Nan Tian ,1,†Qi-Dai Chen,1and Hong-Bo Sun1,2,‡
1State Key Laboratory of Integrated Optoelectronics, College of Electronic Science and Engineering,
Jilin University, Changchun 130012, China
2State Key Laboratory of Precision Measurement Technology and Instruments, Department of Precision Instrument,
Tsinghua University, Haidian, Beijing 100084, China
(Received 23 August 2021; accepted 15 November 2021; published 14 December 2021)
Dynamically encircling an exceptional point in non-Hermitian systems has drawn great attention
recently, since a nonadiabatic transition process can occur and lead to intriguing phenomena and
applications such as the asymmetric switching of modes. While all previous experiments have been
restricted to two-state systems, the dynamics in multistate systems where more complex topology can be
formed by exceptional points, is still unknown and associated experiments remain elusive. Here, we
propose an on-chip photonic system in which an arbitrary number of exceptional points can be encircled
dynamically. We reveal in experiment a robust state-switching rule for multistate systems, and extend it to
an infinite-period system in which an exceptional line is encircled with outcomes being located at the
Brillouin-zone boundary. The proposed versatile platform is expected to reveal more physics related to
multiple exceptional points and exceptional lines, and give rise to applications in multistate non-Hermitian
systems.
DOI: 10.1103/PhysRevLett.127.253901
Non-Hermitian systems and their hallmarks—exceptional
points (EPs) [1–3] have attracted great attention recently
across many disciplines including photonics [4–11], acous-
tics [12–14], electrical circuits [15,16], and other contexts
[17–19]. Fundamentally different from Hermitian systems,
eigenstates in non-Hermitian systems are nonorthogonal in
general [2,3], leading to the occurrence of nonadiabatic
transitions (NATs) among them [20,21]. A scenario is the
dynamical encircling of an EP [22–32], i.e., states evolve
with time in a parameter space following a loop that
encloses the EP. Although the self-intersecting energy
surface around the EP provides a passage for the swapping
of eigenstates [33,34], the inevitable NATs add restrictions
to the swapping such that the output state is independent of
the input one but solely determined by the encircling
direction [22–32]. Such chiral dynamics has been con-
firmed in a variety of experiments [24,26–29,31,32] using
two-state systems where the NAT occurs between two
eigenstates. The efficiency of the dynamics is recently
boosted to approach unity by Chen’s group [31]. Although
NATs and associated dynamics in encircling EPs have been
theoretically studied in three-state and four-state systems
[30,35], their experimental realizations remain out of reach.
At a more fundamental level, the question of whether there
is a general rule governing the dynamics in encircling
multiple EPs in multistate systems, is still open. Multistate
systems typically possess more complex and interesting
topology formed by the EPs so that their experimental
investigations are expected to reveal new physics and
applications.
In this Letter, we propose a systematical study with both
theory and experiment on the dynamical encircling of
multiple EPs in multistate non-Hermitian systems. We
show that the EPs encirclement results in a robust chiral
state-switching behavior due to the multiple NATs among
the eigenstates. The discovered rule applies to periodic non-
Hermitian systems with infinite eigenstates, in which the
outcomes are found to be located at the Brillouin-zone
boundary. The physics is verified experimentally in on-chip
photonic waveguide arrays, which allow us to study the
dynamical encircling of an arbitrary number of EPs.
Device design.—Figure 1(a) illustrates a schematic dia-
gram of our system consisting of periodic waveguide arrays
that were fabricated inside glasses using femtosecond-
laser direct writing techniques [36,37] (Supplemental
Material [38], Sec. I). Each unit cell is composed of a
waveguide Aand a waveguide Bwith length L. The modal
size in waveguide Ais nonuniform along the waveguiding
direction (i.e., zaxis) since we have continuously tuned the
scan velocity of the laser [see Fig. 1(b)] which can strongly
affect the on-site energy (i.e., the propagation constant) of
waveguide A[see Fig. 1(c)]. We define δas the position-
dependent detuning of its on-site energy (denoted by βA)
introduced to that of a reference waveguide (denoted by β0)
fabricated using a scan velocity of 10 mm=s, i.e., δðzÞ¼
βAðzÞ−β0(Supplemental Material [38], Sec. II). The
waveguide Bis uniform with a propagation constant of
β0but a re-exposure technique (Supplemental Material [38],
Sec. I) was applied to introduce point scatterers inside it,
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making the system to be non-Hermitian. The loss of a single
scatterer is estimated to be ∼0.03 dB. The scatterer spacing
along the zaxis is carefully designed [see Fig. 1(b)], leading
to a position-dependent transmission loss in waveguide B
denoted by ΓðzÞ. Then the variation of δðzÞand ΓðzÞform a
closed loop in the δ−Γparameter space, as plotted in
Fig. 1(d). The starting point and end point of the loop is
located at δ¼0and Γ¼0, corresponding to the two end
facets of the device where the cross section of waveguide A
and waveguide Bare identical [see the inset of Fig. 1(a)].
Photon transmissions from the left-hand side to the right-
hand side correspond to a clockwise (CW) loop in the
parameter space, and a right-to-left process is described by a
counterclockwise (CCW) loop.
Multiple exceptional points encirclement.—We show
that the designed loop will enclose multiple EPs, of which
the number is equal to that of the unit cells. Here each
waveguide works on the single-mode condition with work-
ing wavelength of 808 nm. The wave dynamics in the
system with Nunit cells can be captured by a Schrödinger-
like equation i∂zjØðzÞi þ HðzÞjØðzÞi ¼ 0, where jØðzÞi is
the state vector and His a 2N×2Nnon-Hermitian
Hamiltonian. The elements are H2k−1;2k−1¼β0þδ,H2k;2k¼
β0þiγ,H2k−1;2k¼H2k;2k−1¼κ1and H2k;2kþ1¼
H2kþ1;2k¼κ2, where k¼1∼N,γis the loss term related
to Γ,andκ1(κ2) is the intracell (intercell) coupling
coefficient determined by the gap distance g1(g2)(see
Supplemental Material [38], Sec. II for fitting details). We
emphasize that in order to restrict our problem to a synthetic
2D parameter space (i.e., δ−Γ) for the ease of study, we treat
κ1and κ2as constants extracted from g1¼8μmandg2¼
10 μm in experiment. We have discussed in Supplemental
Material [38] (Sec. II) that in the real case that they are
varying along the zaxis, the physics and phenomena are
the same.
We first investigate a multistate system with N¼3and
show the calculated eigenvalues of the Hamiltonian (i.e., the
propagation constants of the waveguide arrays β)intheδ−Γ
parameter space in Figs. 2(a) (real part) and 2(b) (imaginary
part). The system supports six eigenstates (represented by
different colors according to their mode losses) which form
three EPs [inset of Fig. 2(b)]. Therefore, photon transmission
in a device following the designed loop corresponds to the
dynamical encircling of three EPs. Figures 2(c) and 2(d) plot,
respectively, the calculated normalized light intensity in
such a device for CW and CCW loops with incidence via
waveguide 1B(see Supplemental Material [38], Sec. III
for calculation methods). Shortly after excitations, a discrete
wave diffraction phenomenon can be found because of
evanescent couplings [39], which, however, does not last
for the whole process as does that in Hermitian systems
FIG. 1. System under investigation. (a) Schematic diagram of
non-Hermitian waveguide arrays fabricated in boroaluminosili-
cate glasses. (b) The position-dependent scan velocity for
fabricating waveguide A(left axis) and position-dependent point
spacing between adjacent scatterers inside waveguide B(right
axis). (c) The measured modal diameter (along yaxis) of the
fundamental mode in waveguide A(left axis) and the fitted
detuning parameter δ(right axis) as a function of the scan
velocity, where k0is the vacuum wave vector. (d) The variation of
δand Γin the δ−Γparameter space, which form a closed loop
enclosing multiple EPs in multistate systems or an EL in periodic
systems.
FIG. 2. Numerical demonstration of the chiral dynamics in
multistate systems via encircling multiple EPs. (a),(b) Calcu-
lated real part (a) and imaginary part (b) of the eigenvalues of a
non-Hermitian system with three unit cells. The inset of
(b) shows the position of the three EPs and the encircling
loop in the δ−Γparameter space. (c) Numerically simulated
wave scatterings in the three-unit-cell device with L¼50 mm,
where waveguide 1Bis excited from the left-hand port,
corresponding to the dynamical encircling of the three EPs
in a CW loop. (d) Results of a CCW loop.
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(Supplemental Material [38], Fig. S5). Instead, the power
distribution becomes stable after some time, indicating that
from there on only one eigenstate survives. The physics
behind this non-Hermitian-unique phenomenon is the exist-
ence of NATs (i.e., state jumps from higher-loss sheet to
lower-loss one) [20,21], which was first confirmed in two-
state systems with a single EP [22–32].
To study the role of NATs in multistate systems, we check
the output states from the viewpoint of symmetry. The
output state for the CW-loop device is an antisymmetric
mode [see Fig. 2(c)] as the eigenfields in waveguide kA and
waveguide kB are out of phase (k¼1∼3), whereas the
CCW loop results in a symmetric output [see Fig. 2(d)] with
eigenfunctions being in phase in the same unit cell. Besides
the different symmetry, these two outcomes share a common
feature that the eigenfields in adjacent unit cells (e.g., in
waveguides 1Aand 2A) exhibit a phase difference of π,
which will be explained later. We mark the two output states
on the energy surface in Fig. 2(a) (see the two circles). The
physics behind the dynamics then becomes clear. No matter
how many energy sheets there exists, only one sheet can
exhibit the lowest loss [i.e., the light-blue one, also see
Fig. 2(b)] and the state would jump from other sheets to it
via NATs. Different from two-state systems where only one
NAT can occur [24], starting from different states in multi-
state systems would see the NAT at different time steps
and in most cases more than one NAT can appear [see
Supplemental Material [38], Sec. IVand Figs. S6 and S7 for
detailed results on the NATs]. In spite of this, the statewould
still be on the lowest-loss sheet when it approaches the end
point, where the energy sheets are interlaced [see Fig. 2(a)]
since this point lies in the exact phase of the non-Hermitian
system. Therefore, processes following CCW and CW
loops will arrive at different final states [see the arrows in
Fig. 2(a)], resulting in the chiral dynamics.
The underlying physics indicates that the chiral dynam-
ics is topologically protected by the energy surface around
the EPs as well as the NAT dynamics. Therefore, the chiral
phenomenon is robust to the input state, the device length,
and other nontopological disturbances (Supplemental
Material [38], Figs. S8, S9, and S16). This is in contrast
to topological systems where the immunity to disturbances
typically results from the inhibition of the backward
propagating modes [40]. Moreover, the conclusion is
applied to multistate systems with an arbitrary number
of unit cells (see Supplemental Material [38], Fig. S10 for
the results with N¼5).
Exceptional line encirclement in periodic systems.—We
now consider a periodic system with Nbeing infinite.
The dynamics can be captured by a non-Hermitian
Hamiltonian [41]
H0¼β0þδκ
1þκ2expð−iKxÞ
κ1þκ2expðiKxÞβ0þiγ;ð1Þ
where Kxis the Bloch wave vector. Figure 3(a) plots the
calculated real part of the eigenvalues of H0in the first
Brillouin zone by fixing δ¼0, where Λ¼g1þg2is the
lattice constant. By increasing γ, EPs can first emerge
at γ¼2ðκ1−κ2Þand disappear when γ>2ðκ1þκ2Þ
(Supplemental Material [38], Sec. V). The system therefore
supports infinite EPs which form an exceptional line (EL)
[Fig. 3(b)]. We plot in Figs. 3(c) and 3(d) the eigenvalues in
the δ−Γparameter space. The infinite energy sheets form
energy bulks but there is still one surface exhibiting the
lowest loss (marked by blue). In this sense, the NAT physics
would still govern the chiral state-switching dynamics by
encircling the EL, with the two outcomes for CCW and CW
loops marked by the circles in Fig. 3(c). The eigenfield
patterns are also shown in the inset. From the viewpoint of
the phase in the same unit cell, the final Bloch state for the
CCW loop is a symmetric modewhereas the CW output is an
antisymmetric mode, which are in accordance with the
results in systems with finite unit cells. We also mark these
two Bloch states in Fig. 3(a) (see the case with γ¼0), where
they are found to be located at the Brillouin-zone boundary
with KxΛ¼π. Therefore, the eigenfield changes signs
FIG. 3. Numerical demonstration of the chiral dynamics in
periodic systems via encircling an EL. (a) Calculated band
structure of the periodic non-Hermitian system with different γ
where δ¼0. (b) Calculated positions of the EPs in multistate
systems and the EL in a periodic system. (c),(d) Calculated real
part (c) and imaginary part (d) of the eigenvalues of the periodic
non-Hermitian Hamiltonian in the δ−Γparameter space. (e),(f)
Calculated evolution of the Bloch wave in the periodic system
where an EL is dynamically encircled in a CW direction. (g),(h)
The Bloch wave results for a CCW loop.
PHYSICAL REVIEW LETTERS 127, 253901 (2021)
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[i.e., expðiπÞ¼−1] in adjacent unit cells [see Fig. 3(c)], and
this Bloch wave vector is also the origin of the phase
distributions in finite unit-cell systems.
We performed numerical simulations by injecting a
series of Bloch states with arbitrarily chosen amplitudes
to investigate the dynamics when encircling the EL (see
Supplemental Material [38], Sec. VI for calculation meth-
ods). For a given Kxand z, there are two Bloch states [e.g.,
see Fig. 3(a)]. The amplitude of the upper-band Bloch state
with a larger βis denoted by CUwhile CLrepresents the
amplitude of the lower-band state. Figures 3(e) and 3(f)
show the evolution of the amplitude of the instantaneous
Bloch states for the CW-loop device. We find that the
injected Bloch wave is bended and compressed, and the
final state is located at KxΛ¼πwith jCLjbeing domi-
nated. This outcome is the antisymmetric mode, which
confirms the deduction in Fig. 3(c). The results for the
CCW-loop device are given in Figs. 3(g) and 3(h), where in
contrast, the output state is the symmetric mode as defined
in Fig. 3(c). This is the band-structure demonstration of the
chiral dynamics in periodic systems via EL encirclement.
The phenomenon is also robust and independent of the
input Bloch wave (Supplemental Material [38], Fig. S11).
Experimental results.—We performed experiments to
verify our theory. A laser light at 808 nm (CNI, MDL-III-
808L) was used as the source and the diffraction pattern of
light at the output facet of the sample was measured using a
CCD (XG500, XWJG). Figure 4(a) shows the measured
output images of CW-loop devices (L¼50 mm) with
different numbers of unit cells, where all of them were
excited via waveguide 1B. The calculated eigenfield
patterns using
COMSOL
[42] are also given for comparison.
Since the measured diffraction patterns only contain the
information of light intensities, we apply the gap field as a
means to analyze the phase. When electric fields in adjacent
waveguides are in phase, there would be a great field
enhancement inside their gap. In contrast, an out-of-phase
configuration would result in field destructions within the
gap. These “bright”and “dark”fields inside each gap are
used to recognize the symmetry of the measured patterns.
For the CW cases, the antisymmetric output indicates that
there exist “dark”fields in the gap between waveguide kA
and waveguide kB, while the Bloch wave vector gives rise
to “bright”fields in the gap between waveguide ðkþ1ÞA
and waveguide kB.WefindinFig.4(a) that all the measured
diffraction patterns conform to this principle (see the arrows
for gap field enhancement). The situation is just opposite
for CCW loops as shown in Fig. 4(b), where only the gap
within the same unit cell exhibits a field enhancement. The
phenomena are also observed in a device with up to 16
waveguides [Figs. 4(c) and 4(d) with L¼75 mm]. These
results clearly demonstrate the general rule that governs the
multi-EP encirclement process, which shows a chiral mode
switching behavior with the two outcomes exhibiting a
Bloch wave vector Kx¼π=Λ.
FIG. 4. Experimental demonstration of the chiral dynamics in
multistate systems. (a),(b) Measured output light intensities
(lower panel) and simulated electric field distributions of the
final state (upper panel) in multistate systems where NEPs
(N¼2, 3, 4) are encircled in a CW (a) and CCW direction (b).
The arrows mark the gap within which a field enhancement
occurs since the electric fields in the bilateral waveguides are in
phase. (c),(d) The measured results with N¼8.
FIG. 5. Experimental demonstration of the robustness of the
non-Hermitian dynamics. (a) Measured output light intensities
for CW-loop four-unit-cell non-Hermitian devices with different
lengths. (b) Same as (a) except that the device is lossless.
(c) Results for CW-loop four-unit-cell non-Hermitian devices
which are excited via different waveguides. (d) Same as
(c) except that the device is lossless. The measured power
of the output state is indicated in the inset, where the input
light power is fixed at 16 mW.
PHYSICAL REVIEW LETTERS 127, 253901 (2021)
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Finally, we compare the robust dynamics in our non-
Hermitian devices with Hermitian devices. We fabricated
various CW-loop devices with N¼4but with different
lengths. Figure 5(a) shows the measured output patterns
which are indeed independent of the device length. We also
fabricated corresponding Hermitian devices by removing
the re-exposure process, leaving the waveguide Bto be
lossless. The measured output fields in Fig. 5(b) are found
to highly depend on the device length. This is because
without the NAT dynamics, all the eigenstates in Hermitian
systems evolve independently and interfere to form differ-
ent patterns at different positions. The robustness of the
non-Hermitian phenomenon to the input port is demon-
strated in Fig. 5(c), where the output fields are similar.
We should emphasize that the output power depends on the
excited port (see the inset value), i.e., an input state
exhibiting a larger overlap with the lowest-loss eigenstate
can lead to a higher-power output (Supplemental Material
[38], Fig. S13). In contrast, the patterns are entirely
different in Hermitian devices [Fig. 5(d)].
Discussion.—The proposed general rule applies to
multistate non-Hermitian systems possessing one lowest-
loss energy sheet, which is the case of most non-Hermitian
systems. If the system possesses multiple degenerate
lowest-loss sheets in the parameter space, the final state
could be a superposition of these eigenstates whose ratios
depend on the input (see Supplemental Material [38],
Fig. S15 for an example). The chiral behavior results from
the location of the starting point which lies in the
symmetry-unbroken phase. The dynamics can be nonchiral
if we move the starting point to the symmetry-broken
phase ([27], Supplemental Material [38], Fig. S17).
Conclusion.—To conclude, we have experimentally
proposed a photonic platform which is capable of inves-
tigating the dynamical encircling of an arbitrary number of
EPs in multistate non-Hermitian systems. We have revealed
a general rule governing the multi-EP encirclement process
and expanded the results to periodic systems that possess an
EL. We expect the versatile and expandable platform to
explore more physics in multistate non-Hermitian systems
with even more complex topologies formed by EPs. We
also expect the EP encirclement process and energy surface
engineering in future works to be applied for robust
generation of desired eigenstates in multistate systems.
This work was supported by National Natural Science
Foundation of China (NSFC) under Grants No. 61590930,
No. 11974140, No. 61805098, No. 61825502 and
No. 61827826, and China Postdoctoral Science
Foundation (2019M651200 and 2019T120234). X. L. Z.
thanks C. T. Chan for the support in
COMSOL
simulations.
*Corresponding author.
xulin_zhang@jlu.edu.cn
†Corresponding author.
zhennan_tian@jlu.edu.cn
‡Corresponding author.
hbsun@tsinghua.edu.cn
§These authors contributed equally to this work.
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10.1103/PhysRevLett.127.253901 for details, including I.
Experimental details, II. Fitting the model Hamiltonian, III.
Calculation method to obtain the light diffraction patters, IV.
Detailed results of the NATs, V. Exceptional line in the
periodic system, VI. Calculation method to obtain the
evolution of the Bloch wave, and VII. Supplemental figures.
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PHYSICAL REVIEW LETTERS 127, 253901 (2021)
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