This Letter presents an experimental demonstration of tunable exceptional points (EPs) in an electromechanical waveguide. EPs are non-Hermitian singularities typically found in parity-time (PT) symmetric systems with balanced gain and loss. Here, piezoelectric transducers on an aluminum beam (waveguide) are shunted to synthetic impedance circuits that emulate negative and positive resistors (responsible for gain and loss) and inductors (for resonant tunability), whose properties can be programmed digitally. Specifically, an electrical mode is introduced via inductive shunts to electromechanically interact with target structural mode(s) to create degeneracy. While the internal structural damping of the waveguide has the effect of breaking PT symmetry inherently, we show that EPs can still be created by using non-trivial gain and loss combinations. The results in this Letter pave the way for practical realization of EPs in elastic media toward their application in enhanced sensing and asymmetric wave control, among others.

1.
R.
El-Ganainy
,
K. G.
Makris
,
M.
Khajavikhan
,
Z. H.
Musslimani
,
S.
Rotter
, and
D. N.
Christodoulides
, “
Non-Hermitian physics and PT symmetry
,”
Nat. Phys.
14
,
11
19
(
2018
).
2.
R.
Cai
,
Y.
Jin
,
Y.
Li
,
T.
Rabczuk
,
Y.
Pennec
,
B.
Djafari-Rouhani
, and
X.
Zhuang
, “
Exceptional points and skin modes in non-Hermitian metabeams
,”
Phys. Rev. Appl.
18
,
014067
(
2022
).
3.
C. M.
Bender
and
S.
Boettcher
, “
Real spectra in non-Hermitian Hamiltonians having PT symmetry
,”
Phys. Rev. Lett.
80
,
5243
5246
(
1998
).
4.
X.
Zhu
,
H.
Ramezani
,
C.
Shi
,
J.
Zhu
, and
X.
Zhang
, “
PT-Symmetric acoustics
,”
Phys. Rev. X
4
,
031042
(
2014
).
5.
B.-B.
Zhou
,
W.-D.
Liu
, and
L.
Dong
, “
The sensitivity of PT-symmetric LC wireless sensors around an exceptional point
,”
Appl. Phys. Lett.
123
,
164103
(
2023
).
6.
Y.
Duan
,
L.
Geng
,
Q.
Guo
,
J.
Yang
,
G.
Hu
, and
X.
Zhou
, “
Acoustic chiral mode switching by dynamic encircling of exceptional points
,”
Appl. Phys. Lett.
123
,
101701
(
2023
).
7.
Y.
Jian
,
Y.
Wang
,
Z.
Guo
,
S.
Hu
,
B.
Wu
,
Y.
Yang
, and
H.
Chen
, “
External excitation enabled chirality reversal of exceptional points in an effective anti-PT-symmetric non-Hermitian system
,”
Appl. Phys. Lett.
123
,
141702
(
2023
).
8.
C. M.
Bender
,
B. K.
Berntson
,
D.
Parker
, and
E.
Samuel
, “
Observation of PT phase transition in a simple mechanical system
,”
Am. J. Phys.
81
,
173
179
(
2013
).
9.
M. A.
Miri
and
A.
Alù
, “
Exceptional points in optics and photonics
,”
Science
363
,
eaar7709
(
2019
).
10.
M.
Sakhdari
,
M.
Hajizadegan
,
Q.
Zhong
,
D. N.
Christodoulides
,
R.
El-Ganainy
, and
P. Y.
Chen
, “
Experimental observation of PT symmetry breaking near divergent exceptional points
,”
Phys. Rev. Lett.
123
,
193901
(
2019
).
11.
Y.
Zhou
,
Z. Z.
Yang
,
Y. Y.
Peng
, and
X. Y.
Zou
, “
Parity-time symmetric acoustic system constructed by piezoelectric composite plates with active external circuits
,”
Chin. Phys. B
31
,
064304
(
2022
).
12.
R.
Fleury
,
D.
Sounas
, and
A.
Alù
, “
An invisible acoustic sensor based on parity-time symmetry
,”
Nat. Commun.
6
,
1
7
(
2015
).
13.
E.
Riva
, “
Harnessing PT-symmetry in non-Hermitian stiffness-modulated waveguides
,”
Phys. Rev. B
105
,
224314
(
2022
).
14.
L.
Feng
,
Z. J.
Wong
,
R. M.
Ma
,
Y.
Wang
, and
X.
Zhang
, “
Single-mode laser by parity-time symmetry breaking
,”
Science
346
,
972
975
(
2014
).
15.
Z.
Hou
,
H.
Ni
, and
B.
Assouar
, “
PT-symmetry for elastic negative refraction
,”
Phys. Rev. Appl.
10
,
044071
(
2018
).
16.
J.
Wiersig
, “
Sensors operating at exceptional points: General theory
,”
Phys. Rev. A
93
,
033809
(
2016
).
17.
H.
Hodaei
,
A. U.
Hassan
,
S.
Wittek
,
H.
Garcia-Gracia
,
R.
El-Ganainy
,
D. N.
Christodoulides
, and
M.
Khajavikhan
, “
Enhanced sensitivity at higher-order exceptional points
,”
Nature
548
,
187
191
(
2017
).
18.
W.
Chen
,
S.
Kaya Özdemir
,
G.
Zhao
,
J.
Wiersig
, and
L.
Yang
, “
Exceptional points enhance sensing in an optical microcavity
,”
Nature
548
,
192
196
(
2017
).
19.
M. I.
Rosa
,
M.
Mazzotti
, and
M.
Ruzzene
, “
Exceptional points and enhanced sensitivity in PT-symmetric continuous elastic media
,”
J. Mech. Phys. Solids
149
,
104325
(
2021
).
20.
R.
Kononchuk
and
T.
Kottos
, “
Orientation-sensed optomechanical accelerometers based on exceptional points
,”
Phys. Rev. Res.
2
,
23252
(
2020
).
21.
R.
Kononchuk
,
J.
Cai
,
F.
Ellis
,
R.
Thevamaran
, and
T.
Kottos
, “
Exceptional-point-based accelerometers with enhanced signal-to-noise ratio
,”
Nature
607
,
697
702
(
2022
).
22.
J.
Yuan
,
L.
Geng
,
J.
Huang
,
Q.
Guo
,
J.
Yang
, and
G.
Hu
, “
Exceptional points induced by time-varying mass to enhance the sensitivity of defect detection
,”
Phys. Rev. Appl.
10
,
064055
(
2022
).
23.
J.
Doppler
,
A. A.
Mailybaev
,
J.
Böhm
,
U.
Kuhl
,
A.
Girschik
,
F.
Libisch
,
T. J.
Milburn
,
P.
Rabl
,
N.
Moiseyev
, and
S.
Rotter
, “
Dynamically encircling an exceptional point for asymmetric mode switching
,”
Nature
537
,
76
79
(
2016
).
24.
S. K.
Özdemir
,
S.
Rotter
,
F.
Nori
, and
L.
Yang
, “
Parity–time symmetry and exceptional points in photonics
,”
Nat. Mater.
18
,
783
798
(
2019
).
25.
V.
Domínguez-Rocha
,
R.
Thevamaran
,
F. M.
Ellis
, and
T.
Kottos
, “
Environmentally induced exceptional points in elastodynamics
,”
Phys. Rev. Appl.
13
,
014060
(
2020
).
26.
G.
Shmuel
and
N.
Moiseyev
, “
Linking scalar elastodynamics and non-Hermitian quantum mechanics
,”
Phys. Rev. Appl.
13
,
024074
(
2020
).
27.
Y.
Fang
,
T.
Kottos
, and
R.
Thevamaran
, “
Emergence of exceptional points in periodic metastructures with hidden parity-time symmetric defects
,”
J. Appl. Mech.
89
,
1
8
(
2022
).
28.
J.
Schindler
,
A.
Li
,
M. C.
Zheng
,
F. M.
Ellis
, and
T.
Kottos
, “
Experimental study of active LRC circuits with PT symmetries
,”
Phys. Rev. A
84
,
040101(R)
(
2011
).
29.
J.
Lončar
,
J.
Vuković
,
I.
Krois
, and
S.
Hrabar
, “
Stability constraints on practical implementation of parity-time-symmetric electromagnetic systems
,”
Photonics
8
,
56
(
2021
).
30.
J.
Yi
,
Z.
Ma
,
R.
Xia
,
M.
Negahban
,
C.
Chen
, and
Z.
Li
, “
Structural periodicity dependent scattering behavior in parity-time symmetric elastic metamaterials
,”
Phys. Rev. B
106
,
014303
(
2022
).
31.
Q.
Wu
,
Y.
Chen
, and
G.
Huang
, “
Asymmetric scattering of flexural waves in a parity-time symmetric metamaterial beam
,”
J. Acoust. Soc. Am.
146
,
850
862
(
2019
).
32.
Z.
Hou
and
B.
Assouar
, “
Tunable elastic parity-time symmetric structure based on the shunted piezoelectric materials
,”
J. Appl. Phys.
123
,
085101
(
2018
).
33.
A.
Erturk
and
D. J.
Inman
,
Piezoelectric Energy Harvesting
(
John Wiley & Sons, Ltd
,
Chichester, UK
,
2011
), p.
392
.
34.
C.
Sugino
,
S.
Leadenham
,
M.
Ruzzene
, and
A.
Erturk
, “
An investigation of electroelastic bandgap formation in locally resonant piezoelectric metastructures
,”
Smart Mater. Struct.
26
,
055029
(
2017
).
35.
C. D. M.
Junior
,
A.
Erturk
, and
D. J.
Inman
, “
An electromechanical finite element model for piezoelectric energy harvester plates
,”
J. Sound Vib.
327
,
9
25
(
2009
).
36.
R. L.
Thomes
,
J. A.
Mosquera-Sánchez
, and
C.
De Marqui
, “
Bandgap widening by optimized disorder in one-dimensional locally resonant piezoelectric metamaterials
,”
J. Sound Vib.
512
,
116369
(
2021
).
37.
O. N.
Kirillov
, “
Stabilizing and destabilizing perturbations of PT-symmetric indefinitely damped systems
,”
Philos. Trans. R. Soc. A
371
,
20120051
(
2013
).
38.
K.
Ding
,
G.
Ma
,
M.
Xiao
,
Z. Q.
Zhang
, and
C. T.
Chan
, “
Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization
,”
Phys. Rev. X
6
,
021007
(
2016
).
39.
C.
Sugino
,
M.
Ruzzene
, and
A.
Erturk
, “
Digitally programmable resonant elastic metamaterials
,”
Phys. Rev. Appl.
13
,
061001
(
2020
).
40.
M.
Alshaqaq
,
C.
Sugino
, and
A.
Erturk
, “
Programmable rainbow trapping and band-gap enhancement via spatial group-velocity tailoring in elastic metamaterials
,”
Phys. Rev. Appl.
17
,
L021003
(
2022
).
41.
C.
Sugino
,
M.
Alshaqaq
, and
A.
Erturk
, “
Spatially programmable wave compression and signal enhancement in a piezoelectric metamaterial waveguide
,”
Phys. Rev. B
106
,
174304
(
2022
).
42.
M.
Alshaqaq
,
C.
Sugino
, and
A.
Erturk
, “
Digital programming of reciprocity breaking in resonant piezoelectric metamaterials
,”
Phys. Rev. Res.
5
,
043003
(
2023
).

Supplementary Material

You do not currently have access to this content.