of 7
Parto et al.
Light: Science & Applications
(2025) 14:6
Of
fi
cial journal of the CIOMP 2047-7538
https://doi.org/10.1038/s41377-024-01667-z
www.nature.com/lsa
ARTICLE
Open Access
Enhanced sensitivity via non-Hermitian topology
Midya Parto
1,2,3
, Christian Leefmans
4
, James Williams
1
,RobertM.Gray
1
and Alireza Marandi
1,4
Abstract
Sensors are indispensable tools of modern life that are ubiquitously used in diverse settings ranging from smartphones
and autonomous vehicles to the healthcare industry and space technology. By interfacing multiple sensors that
collectively interact with the signal to be measured, one can go beyond the signal-to-noise ratios (SNR) attainable by
the individual constituting elements. Such techniques have also been implemented in the quantum regime, where a
linear increase in the SNR has been achieved via using entangled states. Along similar lines, coupled non-Hermitian
systems have provided yet additional degrees of freedom to obtain better sensors via higher-order exceptional points.
Quite recently, a new class of non-Hermitian systems, known as non-Hermitian topological sensors (NTOS) has been
theoretically proposed. Remarkably, the synergistic interplay between non-Hermiticity and topology is expected to
bestow such sensors with an enhanced sensitivity that grows exponentially with the size of the sensor network. Here,
we experimentally demonstrate NTOS using a network of photonic time-multiplexed resonators in the synthetic
dimension represented by optical pulses. By judiciously programming the delay lines in such a network, we realize the
archetypal Hatano-Nelson model for our non-Hermitian topological sensing scheme. Our experimentally measured
sensitivities for different lattice sizes con
fi
rm the characteristic exponential enhancement of NTOS. We show that this
peculiar response arises due to the combined synergy between non-Hermiticity and topology, something that is
absent in Hermitian topological lattices. Our demonstration of NTOS paves the way for realizing sensors with
unprecedented sensitivities.
Introduction
The ability to accurately and reliably measure physical
quantities is at the heart of modern sensors with appli-
cations ranging from molecular sensing in chemistry
1
and
biology
2
to light detection and ranging (LiDAR)
3
and
observing gravitational waves
4
. Signi
fi
cant efforts have
been made towards enhancing the responses of individual
sensing elements, for instance, by using high-quality
resonators
5
or exploiting quantum effects
6
. A different,
more generic route to achieving higher sensitivities is to
employ a multitude of modes that collectively contribute
to a coherent signal that encapsulates information about
the quantity to be measured. This has led to classical and
quantum sensing networks which allow for an enhance-
ment of
ffiffiffiffi
N
p
and
N
7
in the sensitivity
fi
gure, with
N
denoting the number of independent/entangled entities
contributing to the sensing signal, respectively.
An alternative path to achieve higher sensitivities is to
employ concepts from non-Hermitian physics
8
10
. The
eigenvalues associated with a non-Hermitian system can
respond to perturbations in a remarkably stronger man-
ner compared to its Hermitian counterparts. This reali-
zation is the foundation of a class of sensors that operate
in the vicinity of non-Hermitian degeneracies known as
exceptional points (EPs), where in the presence of an
N
th-
order EP, the response scales as the
N
-th root of the
perturbation
11
16
. In addition, the introduction of non-
Hermiticity to topologically non-trivial lattices is known
to result in an eigenspace that behaves very differently
from that associated with Hermitian topological sys-
tems
17
,
18
. Recent studies have observed the manifestation
of this distinct behavior in the form of a new type of bulk-
boundary correspondence and the non-Hermitian skin
effect
19
25
.
Quite recently, a new class of sensors based on the
synergy between non-Hermiticity and topology has been
© The Author(s) 2025
Open Access
This article is licensed under a Creative Commons Attribution 4.0 Internat
ional License, which permits use, sharing, adaptation, distribution and
reproduction
in any medium or format, as long as you give appropriate credit to the origina
l author(s) and the source, provide a link to the Creative Commons licence,
and indicate if
changes were made. The images or other third party material in this article are included in the article
s Creative Commons licence, un
less indicated otherwise in a credit line to the material. If
material is not included in the article
s Creative Commons licence and your intended use is not permitted by sta
tutory regulation or exceeds the permitted use, you will need to obtain
permission directly from the copyright hol
der. To view a copy of this licence, visit
http://creativecommons.org/licenses/by/4.0/
.
Correspondence: Alireza Marandi (
marandi@caltech.edu
)
1
Department of Electrical Engineering, California Institute of Technology,
Pasadena, CA 91125, USA
2
Physics and Informatics Laboratories, NTT Research, Inc., Sunnyvale, CA 94085,
USA
Full list of author information is available at the end of the article
These authors contributed equally: Midya Parto, Christian Leefmans.
1234567890():,;
1234567890():,;
1234567890():,;
1234567890():,;
proposed
26
28
. Dubbed as non-Hermitian topological
sensors (NTOS), such devices can exhibit a sensitivity that
grows exponentially with respect to the number of lattice
sites. Remarkably, unlike typical non-Hermitian sensing
schemes, this boosted sensitivity can exhibit robustness
against undesirable
fl
uctuations within the underlying
lattice that do not affect the boundary conditions. In
addition, the response of NTOS to its input is in the form
of a linear ampli
fi
cation, in contrast to other non-
Hermitian sensing schemes where the system response
is nonlinear. Despite intense activities in the
fi
eld of non-
Hermitian topology, an experimental observation of this
enhanced sensitivity in photonic arrangements has so far
remained elusive. Here, we experimentally demonstrate
this peculiar behavior in a network of photonic time-
multiplexed resonators. Using non-Hermitian topological
lattices with sizes ranging from
N
=
7to
N
=
23, we
experimentally demonstrate the characteristic exponential
growth of the sensitivity associated with NTOS. In addi-
tion, we show that this extraordinary response arises
exclusively due to the cooperative interplay between non-
Hermiticity and topology, something that is absent in
other Hermitian topological settings.
Results
For our realization of NTOS, we consider the Hatano-
Nelson (HN)
29
model as described by the Hamiltonian:
^
H
HN
¼
X
n
t
R
^
a
y
n
þ
1
^
a
n
þ
t
L
^
a
y
n
^
a
n
þ
1
ð
1
Þ
where
^
a
ðyÞ
n
is the annihilation (creation) operator asso-
ciated with site
n
while
t
R
,
t
L
represent the nonreciprocal
right and left nearest-neighbor couplings within the
lattice. For a
fi
nite lattice, the Hamiltonian of Eq. (
1
)
can exhibit a multitude of spectral behaviors, depending
on the associated boundary conditions (Fig.
1
). In
particular, when the lattice is arranged in a uniform
fashion with periodic boundary conditions (PBC), the set
of eigenvalues form a closed loop in the complex plane
with a nonzero winding around the origin (Fig.
1
), and the
eigenstates are uniformly distributed across the lattice.
We would like to emphasize that here, since the coupling
mechanism between lattice elements are dissipative
30
, the
real part of the system eigenvalues represent dissipation
while the imaginary part corresponds to phase/frequency
shift (see Supplementary part 1). On the other hand, when
the structure is terminated with open boundary
E
= 0

OBC
NTOS
PBC
Im{
E
}
Im{
E
}
Re{
E
}
Re{
E
}
E
=

E
?
Im{
E
}
Re{
E
}
w
=

1

N
–1
N
1
2
NTOS
Fig. 1 Non-Hermitian topological sensors (NTOS)
. Schematic diagram of the NTOS demonstrated here based on the Hatano-Nelson model which
features nonreciprocal couplings between the adjacent elements of the array. Depending on the boundary conditions, this lattice exhibits differen
t
eigenvalue spectra, as shown in the top part of the
fi
gure. This can be represented by the strength
Γ
of the coupling between the
fi
rst and last
resonators in the system. When
Γ
is equal to the other couplings in the array (the rightmost part of the scale), the structure follows periodic boundary
conditions (PBC), where the eigenvalues form an ellipse around the origin in the complex plane. In this case, a nonzero winding number
W
can be
de
fi
ned. On the other hand, when
Γ
=
0, i.e., under open boundary conditions (OBC), all the eigenvalues reside on the real axis, with one eigenvalue
exactly equal to zero
E
=
0 (for odd values of
N
). This eigenvalue tends to shift from its original value by
Δ
E
which is proportional to the strength of
the boundary coupling
Γ
, as long as the coupling is suf
fi
ciently small. This mechanism can be effectively harnessed for sensing any perturbation that
modi
fi
es
Γ
Parto et al.
Light: Science & Applications
(2025) 14:6
Page 2 of 7
conditions (OBC), the resulting spectrum is entirely real
(Fig.
1
). This corresponds to the case where all the
eigenstates become localized near one edge of the system,
known as the non-Hermitian skin effect
31
. Furthermore,
under such OBC, provided that the number of elements in
the lattice is odd
N
=
2
k
+
1, the Hamiltonian
^
H
HN
always
possesses an eigenstate
ψ
0


>
R
with an eigenvalue equal to
zero (i.e., a zero-mode).
To experimentally demonstrate NTOS, we use a time-
multiplexed photonic resonator network depicted sche-
matically in Fig.
2
. The network consists of a main
fi
ber
loop which supports
N
resonant pulses separated by a
repetition period,
T
R
. Here, each individual pulse repre-
sents a single resonator associated with the annihilation
(creation) operators
^
a
ðyÞ
j
in Eq. (
1
). To realize the non-
reciprocal couplings
t
R
and
t
L
, we use two delay lines to
dissipatively couple nearest-neighbor pulses. Each delay
line is equipped with intensity modulators that control the
strengths of such couplings (see Fig.
2
). To induce the
perturbation signal, we consider a change in the lattice of
the form
Δ
^
H
¼
Γ
^
a
y
N
^
a
1
which shows a small deviation
from the OBC con
fi
guration. In response to this pertur-
bation, the unperturbed eigenstate
ψ
0


>
R
will now change
to
ψ
ð
Γ
Þ
j
>
R
associated with a new eigenvalue that shifts
from the zero point by
Δ
E
along the real axis (see Fig.
1
).
To experimentally implement the perturbation
Δ
^
H
we
use a third delay line which couples the
fi
rst pulse to the
last one in a non-reciprocal fashion. The strength of this
coupling is then modulated accordingly to provide dif-
ferent values of the perturbation strength
Γ
. In a practical
scenario, such a perturbation can represent the con-
centration of an absorptive gas inside a cell that is placed
in the path of this third delay line (Fig.
2
). Our NTOS can
then be used to accurately measure small changes in the
concentration of the target gas molecules.
In our experiments, we
fi
rst initialize the system by
shaping the amplitudes and phases of the input pulses to
represent the zero eigenstate
ψ
0


>
R
associated with the
EOM
EDFA
EOM
EOM
EOM
NTOS
Fig. 2 Schematic of the network of time-multiplexed resonators used to demonstrate NTOS
. Synthetic resonators are de
fi
ned by femtosecond
pulses emitted by a mode-locked laser with a repetition rate of
T
R
passing through an electro-optic modulator (EOM) before injection into the optical
fi
ber-based cavity (yellow
fi
bers). An Erbium-doped
fi
ber ampli
fi
er (EDFA) is used in the main cavity to compensate for the losses and increase the
number of measurement roundtrips. Two delay lines with smaller and larger lengths than the main cavity (corresponding to delays of
T
R
and
+
T
R
,
respectively) are utilized to provide nonreciprocal couplings between the nearest-neighbor resonators, necessary to implement the non-Hermitia
n
topological model of Eq. (
1
). In addition, a third delay line with a length that corresponds to an optical delay of
+
(
N
1)
T
R
associated with the
perturbation
Δ
^
H
is also included. In practice, the strength of such a perturbation, i.e.,
Γ
, can be modi
fi
ed by an absorptive gas inside a cell (bottom). In
such a scenario, NTOS can be used to accurately measure the concentration of the target gas molecules
Parto et al.
Light: Science & Applications
(2025) 14:6
Page 3 of 7
Hamiltonian in Eq. (
1
). Figure
3
shows an example of such
pulses in the experiments concerning
N
=
23 time-
multiplexed resonators (the green inset depicts the zero
eigenstate
ψ
0


>
R
). In order to increase the power of the
pulses in the measurements, we repeatedly inject this
pulse pattern into the closed cavity (with closed delay
lines) which results in building up power inside the cavity
(Fig.
3
for times < 10.5
μ
s). After this initialization, the
input to the cavity is blocked so that the pulses start to
circulate through the cavity and the delay lines according
to the discrete-time evolution corresponding to the
Hatano-Nelson model. Subsequently, at each time step
that is de
fi
ned by the integer multiples of the cavity
round-trip time, we project the state of the network on
the left eigenstate of the unperturbed Hamiltonian
ψ
0


>
L
.
The perturbed eigenvalue can now be estimated from the
decay rate of this projection per cavity round-trip (see
Methods and Supplementary parts 2
4). In addition to
the zero-mode, we also inject two control pulses into the
unused time slots of the main cavity (red boxes in Fig.
3
).
The
fi
rst control pulse is left uncoupled to other time slots
of the network and is utilized to accurately measure the
intrinsic cavity decay rate (corresponding to the zero-
mode decay rate of the HN model). Using this as a
reference, we then measure
Δ
E
by calculating the differ-
ence between the perturbed eigenvalue and the unper-
turbed one. We use the second control pulse to accurately
characterize the throughput of the ±1
T
R
delay lines which
set the nearest-neighbor nonreciprocal right and left
couplings
t
R
and
t
L
within our HN lattice (see Methods).
Figure
4
displays experimentally measured changes in
the eigenvalues obtained for HN lattices with different
sizes and for different perturbation strengths together with
simulated results. For perturbations well below a critical
value
Γ
Γ
C
, the NTOS exhibits a linear response with
respect to the input parameter. However, for larger
1.6
Power build-up
Decay
Control pulses
1.2
Intensity [a.u.]
Intensity [a.u.]
0.8
0.4
0
89
10
Time [

s]
11
12
1
0.5
0
3
711
Site number
15
19
23
Fig. 3 Measurement procedure for the time-multiplexed NTOS
.
Experimental time trace showing the pulse patterns at the output of
the time-multiplexed resonator network for
N
=
23. At the beginning
(
t
< 10.5
μ
s) optical pulses representing the zero eigenstate
ψ
0
j
>
R
of
the unperturbed Hamiltonian in Eq. (
1
) (bottom green inset) are
repeatedly injected into the closed cavity (power build-up regime).
After this, the input path to the cavity is blocked while the delay lines
are opened, allowing for the pulses to circulate inside the cavity and
the delay lines. This results in a temporal decay of the input eigenstate
for
t
> 10.5
μ
s. By measuring these pulses and projecting them onto
the left eigenstate of the unperturbed Hamiltonian
ψ
0
j
>
L
,we
experimentally estimate the shift in the zero eigenvalue
Δ
E
associated
with the Hatano-Nelson model resulting from the nonzero
perturbation in the system. In addition to the zero eigenstate, we also
inject two control pulses (shown in the red boxes in the top plot) into
the cavity. We use the
fi
rst control pulse to accurately measure the
intrinsic cavity decay, while the second one is intended to characterize
the ±1
T
R
delay lines (Methods)


E

×
T
RT

×
T
RT
0.12
N
= 23
0.08
0.04
0
0
0.03
0.06
0.09
N
= 17
N
= 13
N
= 7
Fig. 4 Experimental demonstration of NTOS
. Experimentally
measured shifts in the eigenvalue
Δ
E
as the boundary coupling
strength
Γ
is perturbed from zero value (OBC conditions), for different
lattice sizes
N
=
7, 13, 17 and 23. As evident in the
fi
gure, as long as
Γ
is small enough, our NTOS responds linearly to the induced
perturbations. However, as
Γ
passes a threshold which depends on the
size of the non-Hermitian topological lattice
N
, the change in the
eigenvalue is no longer linear. The transition to this nonlinear regime
is marked for each case in the
fi
gure by vertical dashed lines.
Theoretically expected values are shown as solid curves. Here,
T
RT
represents the round-trip time of the optical cavity
Parto et al.
Light: Science & Applications
(2025) 14:6
Page 4 of 7
perturbations, the shifted eigenvalue associated with
ψ
ð
Γ
Þ
j
>
R
is no longer real, signaling a crossover to the PBC
where the sensor response is no longer linear
19
.By
increasing the perturbation further, the non-Hermitian
skin effect breaks down and the eigenstates are no longer
exponentially localized at the edge of the structure. Since
the performance of the NTOS as a sensor is contingent
upon this localization, it is crucial to avoid this non-
Hermitian phase transition. Although in the thermo-
dynamic limit
Γ
C
tends to vanish, our analytical results
show that for
fi
nite lattices its value remains nonzero and
scales exponentially with
N
. In the Supplementary part 5
we evaluate
Γ
C
for different values of the lattice size
N
implemented in our experiments. In order to fully char-
acterize our NTOS, we applied perturbations in a wide
range of strengths spanning both below and above the
aforementioned critical coupling. As shown in Fig.
4
, the
experimentally measured results exhibit a linear system
response to small
Γ
. For larger inputs, the sensor response
eventually becomes nonlinear, hence setting the dynamic
range of our demonstrated NTOS. Hence, there is a fun-
damental trade-off between the enhanced response and
the dynamic ranges attainable by NTOS as the size of the
non-Hermitian lattice grows. For our experiments, the
requirement to reliably generate such small perturbations
limited our demonstration of NTOS to lattices with the
number of sites smaller than
N
23.
Discussion
To evaluate the performance of NTOS, we calculated
the sensitivity de
fi
ned as
S
E
/
Γ
using our measure-
ment data in the small parameter regime
Γ
Γ
C
. Figure
5
shows theoretically expected values along with experi-
mental results for different lattice sizes
N
. As evident in
this
fi
gure, the sensitivity of the NTOS grows exponen-
tially with the size of this non-Hermitian topological
system. These results are consistent with theoretical
predictions based on perturbation theory. As mentioned
earlier, a unique aspect of NTOS is its boosted response
to perturbations in the boundary conditions while sup-
pressing undesirable
fl
uctuations in other parameters
de
fi
ning the lattice. In particular, it can be shown that
random perturbations in the non-reciprocal couplings
t
L
and
t
R
tend to have negligible effects on the output
response (please see Supplementary Section 1). Another
possible disorder in the lattice can arise due to random
changes in the losses associated with different lattice sites.
As shown in the Supplementary Section 1, although their
effect is not completely nulli
fi
ed, the NTOS does not
amplify such perturbations and tends to suppress them as
the number of lattice sites
N
becomes larger. It should be
emphasized that our implementation based on time-
multiplexing of resonators further minimizes these
undesirable loss
fl
uctuations. Remarkably, the exponential
enhancement of the sensitivity is known
27
to arise in
scenarios where the bulk non-Hermitian lattice possesses
a nonzero topological winding number
W
de
fi
ned as
1
2
π
i
Z
π

π
dk
k
log
f
det
½
H
ð
k
Þg
ð
2
Þ
Here,
H
(
k
) denotes the Bloch Hamiltonian associated
with the implemented lattice under PBC conditions. To
corroborate this, we simulated the behavior of other types
of lattices when subjected to the same perturbation
Γ
in
their boundary conditions as the NTOS studied here. We
fi
rst consider the limiting case of the Hamiltonian in Eq.
(
1
) where the nearest-neighbor couplings become reci-
procal
t
R
=
t
L
, resulting in a trivial system
0. As
shown in Fig.
5
, the sensitivity of a sensor implemented
using a uniform lattice tends to deteriorate as 1/
N
with
respect to the number of array elements (see Supple-
mentary part 6). As a second example, we choose a
Hermitian, but topologically non-trivial lattice, namely
that associated with the Su-Schrieffer-Heeger (SSH)
model
32
. When properly terminated, such a lattice also
supports a pair of topological edge states that are localized
in the open ends of the structure, in a way similar to the
N
8
12
16
20
24
10log(S) [dB]
10
NTOS
HN
t
R
t
L
Trivial
Trivial
tt
SSH
SSH
t
2
t
1
Averaging
0
–10
–20
Fig. 5 Exponential enhancement in the sensitivity of the NTOS
.
Experimentally obtained sensitivities
S
of the NTOS for different lattice
sizes
N
are shown as green circles on the left plot. The corresponding
theoretically predicted values are also depicted as orange squares. The
data shows an exponential enhancement in the sensitivity
S
as the
NTOS lattice size grows (green dashed line). For comparison, we
performed similar analysis for other types of lattices including a trivial
lattice with uniform couplings as well as the Hermitian topological
lattice represented by the SSH Hamiltonian (depicted on the right side
of the
fi
gure). As shown in the plot, in sharp contrast to NTOS, such
lattices tend to become less sensitive to their boundary conditions as
the structure grows. The plot also displays the effective enhancement
in the sensitivity resulting from the noise suppression via averaging
N
different measurements
Parto et al.
Light: Science & Applications
(2025) 14:6
Page 5 of 7
NTOS constructed in our experiments. However, unlike
NTOS, the SSH Hamiltonian exhibits a trivial non-
Hermitian winding number according to Eq. (
2
). For this
system, it can be shown (see Supplementary part 6) that
the sensitivity of the eigenvalues associated with such
Hermitian edge states are in fact exponentially
insensitive
to the changes in the boundaries of the array as
N
grows
(Fig.
5
). These results hence con
fi
rm that the unusual
enhancement in the sensing response observed in our
experiments arises uniquely due to the synergy between
non-Hermiticity and topology.
In summary, we have experimentally demonstrated
enhanced sensitivity by non-Hermitian topological ampli-
fi
cation. For HN lattices comprised of different numbers of
elements, we characterized the response of the system as
the shift in one of its eigenvalues as the boundary condi-
tions change. While this response tends to saturate for
perturbations larger than a critical limit, it tends to be linear
forsmallerranges.Thesensitivityparametercalculated
using experimental data clearly exhibits an exponential
growth with the lattice size
N
, in agreement with theoretical
predictions. Using examples of other types of lattices, we
showed that this peculiar enhancement arises due to the
collaborative effect of non-Hermiticity and topology,
something that does not occu
r for instance in Hermitian
topological systems. As indicated in various recent stu-
dies
26
28
,
33
,
34
, another distinguishing feature of NTOS is to
boost the sensitivity without suffering from undesirable
noise enhancements which could notoriously limit the
application of topologically trivial non-Hermitian sensing
schemes (please see Supplementary Section 7). Conse-
quently, our work sheds light on novel avenues where the
synergistic cooperation between non-Hermiticity and non-
trivial topology can unlock functionalities that are otherwise
unattainable in physical systems lacking these combined
features. We would like to emphasize that the underlying
principle of the enhancement in our study is topological
ampli
fi
cation in a classical context, due to the speci
fi
c
implementation in our experimental setup. Demonstrating
similar effects in the quantum regime (for instance using
the model proposed in ref.
26
)isanexcitingprospectthat
could be the subject of future studies.
Materials and methods
Experimental procedure
To realize non-Hermitian topological sensors (NTOS),
we construct the
fi
ber-based time-multiplexed resonator
network shown in Fig.
2
. This network consists of a main
cavity (yellow
fi
ber) and three optical delay lines (blue
fi
ber).
We populate this network with optical pulses separated by a
repetition period
T
R
4ns which are generated by a mode-
locked laser optical frequency comb source (MenloSystems
FC1500-250-WG), and we choose the lengths of the delay
lines to introduce couplings between these pulses. These
delay lines are assembled from optical rails and rail carriers
(Thorlabs RLA2400, RC4) and FC/APC
fi
ber collimation
packages. The ± 1
T
R
delay lines produce couplings between
nearest-neighbor pulses in the cavity, while the
+
(
N
1)
T
R
delay line, which is where we
introduce perturbations,
couples the
fi
rst pulse in our synthetic lattice to the
fi
nal
pulse. While the main cavity can support up to 74 pulses,
we use the
+
(
N
1)
T
R
delay line to set the size of the
lattice under study, and we do not excite the unused time
slots in the main cavity.
Prior to an experiment, we calibrate the electro-optic
modulators (EOMs, IXblue MXAN-LN-10) in the network
using the calibration procedure described in Supplementary
part 3. We calibrate the EOM between the laser and the
main cavity to carve the zero-mode of the unperturbed
Hatano-Nelson lattice from the pulse train of the laser,
whilewecalibratethemodulatorsinthe±1
T
R
delay lines
to implement the Hatano-Nelson model
s asymmetric
couplings. The
+
(
N
1)
T
R
delay line also contains two
EOMs (not shown in Fig.
2
), which control the strength of
the perturbation between the
fi
rst and
fi
nal sites of the HN
lattice. We calibrate the throughput of these modulators to
set the perturbation strength for any given experiment.
After completing our calibration, we begin an experi-
ment by injecting the Hatano-Nelson zero-mode into the
network for 10 roundtrips, which allows the power in the
zero-mode to resonantly build up within the cavity. We
implement a Pound-Drever-Hall (PDH) locking scheme
to lock the cavity and the delay lines to the comb source
by using output taps that are fed into an adjustable gain
fi
ber-optic receiver (New Focus Model 2053), providing
an electronic locking signal input to a proportional-
integral derivative (PID) controller (Red Pitaya STEMlab
125-14). During the injection phase, we leave the IMs in
the ± 1
T
R
delay lines biased to minimum throughput so
that we do not couple neighboring pulses through these
delay lines. After 10 roundtrips, we stop injecting the
zero-mode and we turn on the couplings in the ± 1
T
R
delay lines. We save a trace of the cavity ring-down by
detecting individual pulses using a biased detector
(Thorlabs DET08CFC) read by an oscilloscope (Tektronix
MSO6B), and we repeat this measurement on the order of
100 times to generate statistics for our data analysis. The
control signals for the EOMs and the oscilloscope is
generated by an FPGA (Xilinx Zynq UltraScale
+
RFSoC).
In addition to injecting the zero-mode into our network,
we also inject a single pulse (
fi
rst control pulse in Fig.
3
)
into one of the unused time slots of the main cavity. We
leave this single pulse uncoupled to the surrounding time
slots so that this pulse decays at the intrinsic decay rate of
just the main cavity. In the absence of the perturbation,
this is the same decay rate that we would expect for the
zero-mode of the Hatano-Nelson model. Therefore, this
auxiliary pulse acts as a reference from which we can
Parto et al.
Light: Science & Applications
(2025) 14:6
Page 6 of 7
extract the change in the decay rate of the zero-mode due
to the perturbation.
Finally, we inject a second single pulse (second control
pulse in Fig.
3
) into our time-multiplexed network that we
use for accurately measuring the couplings associated
with the ± 1
T
R
delay lines. For this pulse, we open the
delay lines so that the neighboring time slots next to this
pulse are populated with light pulses with amplitudes
proportional to the nearest-neighbor nonreciprocal right
and left couplings
t
R
and
t
L
within the HN lattice.
Data analysis
As illustrated in Fig.
3
, the state of the time-multiplexed
network in every roundtrip is represented by the ampli-
tudes of the pulses within the cavity time slots which
de
fi
ne different site numbers in the Hatano-Nelson lattice.
In our experiments, we record these time traces for 100
instances and use the average of these traces to measure
the state of the network
ψ
ð
mT
RT
Þ
ji
in successive round-
trips
m
=
1, 2, . . . , where
T
RT
represents the roundtrip
time of the optical cavity. We then project this state into
the left eigenstate of the unperturbed HN model to de
fi
ne
P
(
m
)
=
ψ
0
L
ψ
(
mT
RT
)
. From here, we estimated the
decay rate of
P
(
m
) per cavity roundtrip to measure the
response of NTOS de
fi
ned as the quantity
Δ
E
×
T
RT
that
is reported in Fig.
4
.
Acknowledgements
The authors acknowledge support from ARO Grant W911NF-23-1-0048, NSF
Grants No. 1846273 and 1918549 and the Center for Sensing to Intelligence at
Caltech. The authors wish to thank NTT Research for their
fi
nancial and
technical support.
Author details
1
Department of Electrical Engineering, California Institute of Technology,
Pasadena, CA 91125, USA.
2
Physics and Informatics Laboratories, NTT Research,
Inc., Sunnyvale, CA 94085, USA.
3
CREOL, The College of Optics and Photonics,
University of Central Florida, Orlando, FL, USA.
4
Department of Applied Physics,
California Institute of Technology, Pasadena, CA 91125, USA
Author contributions
All authors contributed to the writing of this manuscript.
Data availability
The data used to generate the plots and results in this paper is available from
the corresponding author upon reasonable request.
Con
fl
ict of interest
A.M. has
fi
nancial interest in PINC Technologies Inc., which is developing
photonic integrated nonlinear circuits. The remaining authors declare no
competing interests.
Supplementary information
The online version contains supplementary
material available at
https://doi.org/10.1038/s41377-024-01667-z
.
Received: 31 March 2024 Revised: 12 October 2024 Accepted: 23 October
2024
References
1. Xue, L. et al. Solid-state nanopore sensors.
Nat. Rev. Mater.
5
,931
951 (2020).
2. Altug, H. et al. Advances and applications of nanophotonic biosensors.
Nat.
Nanotechnol.
17
,5
16 (2022).
3. Zhang, X. et al. A large-scale microel
ectromechanical-sy
stems-based silicon
photonics LiDAR.
Nature
603
,253
258 (2022).
4. Aasi, J. et al. Enhanced sensitivity of t
he LIGO gravitational wave detector by
using squeezed states of light.
Nat. Photonics
7
,613
619 (2013).
5. Zhu, J. et al. On-chip single nanoparticle detection and sizing by mode
splitting in an ultrahigh-q microresonator.
Nat. Photonics
4
,46
49
(2010).
6. Degen, C., Reinhard, F. & Cappellaro, P. Quantum sensing.
Rev. Mod. Phys.
89
,
035002 (2017).
7. Liu, L.-Z. et al. Distributed quantum ph
ase estimation with e
ntangled photons.
Nat. Photonics
15
,137
142 (2021).
8. El-Ganainy, R. et al. Non-herm
itian physics and PT symmetry.
Nat. Phys.
14
,
11
19 (2018).
9. Parto, M. et al. Non-hermitian and topological photonics: optics at an
exceptional point.
Nanophotonics
10
,403
423 (2021).
10. Ashida, Y., Gong, Z. & Ueda, M. Non-hermitian physics.
Adv. Phys.
69
, 249
435
(2020).
11. Hodaei, H. et al. Enhanced sensitivity at higher-order exceptional points.
Nature
548
,187
191 (2017).
12. Chen, W. J. et al. Exceptional points enhance sensing in an optical microcavity.
Nature
548
,192
196 (2017).
13. Zhong, Q. et al. Sensing with exceptional surfaces in order to combine sen-
sitivity with robustness.
Phys. Rev. Lett.
122
, 153902 (2019).
14. Hokmabadi, M. et al. Non-hermitian ring laser gyroscopes with enhanced
sagnac sensitivity.
Nature
576
,70
74 (2019).
15. Lai,Y.H.etal.Observationoftheexce
ptional-point-enhanced sagnac effect.
Nature
576
,65
69 (2019).
16. Kononchuk, R. et al. Exceptional-po
int-based accelerometers with enhanced
signal-to-noise ratio.
Nature
607
,697
702 (2022).
17. Gong, Z. P. et al. Topological phases of non-hermitian systems.
Phys.Rev.X
8
,
031079 (2018).
18. Bergholtz, E. J., Budich, J. C. & Kunst, F. K. Exceptional topology of non-
hermitian systems.
Rev. Mod. Phys.
93
, 015005 (2021).
19. Kunst, F. K. et al. Biorthogonal bulk-boundary correspondence in non-
hermitian systems.
Phys.Rev.Lett.
121
, 026808 (2018).
20. Yao,S.Y.&Wang,Z.Edgestatesandtopologicalinvariantsofnon-hermitian
systems.
Phys.Rev.Lett.
121
, 086803 (2018).
21. Xiao, L. et al. Non-hermitian bulk-boundary correspondence in quantum
dynamics.
Nat. Phys.
16
, 761
766 (2020).
22. Helbig, T. et al. Generalized bulk-boundary correspondence in non-hermitian
topolectrical circuits.
Nat. Phys.
16
,747
750 (2020).
23. Weidemann, S. et al. Topological funneling of light.
Science
368
,311
314
(2020).
24. Liu, Y. G. N. et al. Complex skin modes in non-hermitian coupled laser arrays.
Light Sci. Appl.
11
, 336 (2022).
25. Xue, W. T. et al. Simple formulas of directional ampli
fi
cation from non-bloch
band theory.
Phys. Rev. B
103
, L241408 (2021).
26. McDonald,A.&Clerk,A.A.Exponenti
ally-enhanced quantum sensing with
non-hermitian lattice dynamics.
Nat. Commun.
11
, 5382 (2020).
27. Budich, J. C. & Bergholtz, E. J. Non-hermitian topological sensors.
Phys. Rev. Lett.
125
, 180403 (2020).
28. Koch, F. & Budich, J. C. Quantum non-hermitian topological sensors.
Phys. Rev.
Res.
4
, 013113 (2022).
29. Hatano, N. & Nelson, D. R. Localizatio
n transitions in non-
hermitian quantum
mechanics.
Phys. Rev. Lett.
77
,570
573 (1996).
30. Leefmans, C. et al. Topological dissipation in a time-multiplexed photonic
resonator network.
Nat. Phys.
18
,442
449 (2022).
31. Weidemann, S. et al. Topological triple phase transition in non-hermitian
fl
oquet quasicrystals.
Nature
601
,354
359 (2022).
32. Su, W. P., Schrieffer, J. R. & Heeger, A. J. Solitons in polyacetylene.
Phys. Rev. Lett.
42
,1698
1701 (1979).
33. Bao,L.Y.,Qi,B.&Dong,D.Y.Exponentiallyenhancedquantumnon-hermitian
sensing via optimized coherent drive.
Phys. Rev. Appl.
17
, 014034 (2022).
34. Yuan, H. et al. Non-hermitian topolectri
cal circuit sensor with high sensitivity.
Adv. Sci.
10
, 2301128 (2023).
Parto et al.
Light: Science & Applications
(2025) 14:6
Page 7 of 7