Supplementary Information for Enhanced sensitivity
via non-Hermitian topology
Midya Parto
1
,
2
,
3
†
, Christian Leefmans
4
,
†
, James Williams
1
,
Robert M. Gray
1
, Alireza Marandi
1
,
4
,
∗
1
Department of Electrical Engineering, California Institute of Technology, Pasadena, CA 91125, USA.
2
Physics and Informatics Laboratories, NTT Research, Inc., Sunnyvale, California 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.
†
These authors contributed equally
∗
marandi@caltech.edu
1
1 Sensitivity analysis of the NTOS
In this section, we explain the principle of operation of NTOS and its exponentially enhanced
sensitivity.
We consider the Hatano-Nelson lattice shown in Fig. S1. The eigenstates of this lattice can
be found using the non-Hermitian Hamiltonian formalism
−
d
dt
|
ψ
⟩
=
H
HN
|
ψ
⟩
,
(1)
where
H
HN
is defined in the Eq. 1 of the main text and is equivalent to the following
N
by
N
matrix operator
H
HN
=
0
t
L
0
···
0
t
R
0
t
L
···
0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Γ
t
L
0
0
···
0
.
(2)
Note that in Eq. 1 the eigenvalues represent decay rates since the couplings implemented in our
experiments are dissipative couplings [1]. Under open boundary conditions, i.e.
Γ = 0
, this
Hamiltonian supports the zero right and left eigenstates with zero eigenvalue
E
0
= 0
given by
|
ψ
0
⟩
R
=
1
0
α
0
α
2
0
.
.
.
α
k
,
⟨
ψ
0
|
L
=
1 0 1
/α
0 1
/α
2
···
1
/α
k
,
(3)
where
α
=
−
t
R
/t
L
and we are assuming that the number of lattice elements is odd
N
= 2
k
+ 1
.
On the other hand, under perturbed boundary conditions, i.e. when
Γ
̸
= 0
, using biorthogonal
first-order perturbation theory one can find the shift in the eigenvalue associated with the zero
2
Figure 1:
Analytical solution for the perturbed zero state in the Hatano-Nelson lattice.
eigenstate as
∆
E
n
=
⟨
ψ
0
|
L
∆
H
|
ψ
0
⟩
R
⟨
ψ
0
|
L
ψ
0
⟩
R
∝
Γ(
p
1
/α
)
N
.
(4)
This latter equation clearly indicates the exponential enhancement of the sensitivity defined as
S
≡
∂
∆
E/∂
Γ
with respect to the number of lattice elements
N
in a Hatano-Nelson lattice, in
accordance with experimental results presented in Fig. 5 of the main text.
As mentioned in the main text, a remarkable property of the NTOS is the fact that its re-
sponse is robust with respect to unwanted perturbations in the HN lattice, i.e. disorder in the
bulk couplings within the lattice. To show this, we estimate the shift in the eigenvalue of the
zero eigenstate in response to off-diagonal perturbation in the the Hamiltonian of Eq. 1 of the
main text.
∆
E
≈
1
N
1 0 1
/α
0 1
/α
2
···
1
/α
k
0
∆
t
12
0
···
0
∆
t
21
0
∆
t
23
···
0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0
0
0
···
0
1
0
α
0
α
2
0
.
.
.
α
k
= 0
.
(5)
3
Hence, the shift in the eigenvalue of the zero eigenstate tends to remain insensitive with respect
to the unwanted fluctuations of the couplings within the Hatano-Nelson model which do not
affect the boundary conditions.
Let us now consider the sensitivity of NTOS to diagonal elements in the Hamiltonian matrix
which represent fluctuations in the losses associated with different pulses in the cavity. To ex-
amine this, we estimate the shift in the eigenvalue of the zero eigenstate in response to diagonal
perturbations in the the Hamiltonian of Eq. 1 of the main text using first-order perturbation
theory.
∆
E
≈
1
N
1 0 1
/α
0 1
/α
2
···
1
/α
k
0
···
0
···
0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0
···
δγ
m
···
0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0
···
0
···
0
1
0
α
0
α
2
0
.
.
.
α
k
=
[1
−
(
−
1)
m
]
N
δγ
m
.
(6)
In other words, the shift in the eigenvalue of the zero mode in response to a small shift in
the diagonal elements of the Hamiltonian is zero if the lattice site number is even. If the lattice
site number associated with the diagonal perturbation is odd (worst case), then the shift in the
zero eigenvalue will be an attenuated version of the perturbation. Hence, we see that although
the response of NTOS is not completely zero to the loss fluctuations associated with the pulses,
it tends to be suppressed and not amplified by NTOS. To further confirm this response, we
numerically simulated the changes in the eigenvalue of the zero mode of the Hamiltonian
H
HN
in Eq. 1 in response to a change in the diagonal element associated with the first and last lattice
sites with a magnitude of
δγ
. Figure S2 shows the simulation results, which clearly indicate
that the NTOS sensitivity to such perturbations is small and decreases as the size of the lattice
4
Figure 2:
NTOS response to diagonal perturbations in the first/last sites.
N
grows.
Finally, we consider the response of the NTOS to the input (
Γ
) in the presence of diagonal
disorder. In particular, we numerically evaluate the dynamics described by the perturbed Hamil-
tonian
H
HN
+ ∆
H
when a diagonal disorder of
δγ
is also applied to the first/last lattice site.
Figure S3 summarizes these results. As shown here, it is clear that the exponentially enhanced
response of NTOS persisits even in the presence of on-site (diagonal) disorders, as long as these
disorders are smaller than the total response at the output of the NTOS, i.e.
δγ <
∆
E
.
5
Figure 3:
NTOS response in the presence of on-site disorders.
6
2 Experimental Setup
We demonstrate non-Hermitian topological sensors (NTOS) using the time-multiplexed pho-
tonic resonator network shown in Fig. S4. As has been discussed previously, such networks
are excellent architectures for studying lattice models in temporal synthetic dimensions [1], as
they can enable the realization of multidimensional synthetic lattices, long-range couplings,
and tunable boundary conditions. In this work, we leverage the ability to implement long-range
couplings and tunable boundary conditions to realize NTOS in our time-multiplexed network.
At a high level, the time-multiplexed network in Fig. S4 consists of a main cavity and three
optical delay lines. The main cavity can support up to
74
optical pulses separated by a repetition
period
T
R
≈
4 ns
. Meanwhile, the lengths of the three delay lines are designed so that the
±
1
T
R
delay lines implement nearest-neighbor Hatano-Nelson (HN) couplings between the pulses, and
the
+(
N
−
1)
T
R
delay line couples the “first” pulse to the “last” pulse in an
N
site HN lattice.
In our experiments, the coupling produced by the
+(
N
−
1)
T
R
delay line acts as a perturbation
on an
N
-site HN lattice with open boundary conditions (OBCs). Adjusting the length of the
+(
N
−
1)
T
R
delay line enables us to study how perturbations effect HN lattices of different
sizes.
We inject pulses into our time-multiplexed network using the mode-locked laser (MLL)
shown in Fig. S4. This MLL generates femtosecond pulses at a repetition period of
∼
4 ns
,
and we stretch the pulses to widths of
∼
5 ps
using a Channel 34 dense wavelength-division
multiplexing (DWDM) filter. After stretching the pulses, we send them through a 90:10 splitter.
Light from the 10% port goes directly to a
600 MHz
photodetector, and the output of this detec-
tor passes through a
300 MHz
low-pass filter to generate a
∼
250 MHz
sinusoid. This sinusoid
acts as a clock for the FPGA that drives the modulators in our time-multiplexed network. We
find that using a clock signal derived directly from our optical pulses improves the synchroniza-
7
To Oscilloscope
fs 1.55 μm
Laser
FPS
Fiber
Stretcher
Tunable
Free
-
Space Delay
IM
+1
IM
-
1
FPS
FPS
-
1T
R
+1T
R
Filter
Main
Cavity
5 GHz
Detector
IM
00
IM
01
Acronyms and Color Scheme:
IM: Intensity Modulator
FPGA: Field
-
Programmable Gate Array
EDFA: Erbium
-
Doped Fiber Amplifier
FPS: Fiber Phase Shifter
PID: Proportional Integral Derivative Controller
FPGA
Data Acquisition
Module
Amps
Low Pass
Filter
PID +
Dither
PID +
Dither
PID +
Dither
250 MHz
Reference
From PC
From PC
To IM
C
From IM
Photodiodes
To IM Bias
Ports
Amp
Amp
Amp
90:10
To
PID+Dither
kHz
Detector
+T
R
Delay Line
-
T
R
Delay Line
Delay Line Common Path
Main Cavity
±
T
R
Delay Line Common Path
Signals To and From Modulators
Signal For Stabilization Circuits
Signals To the Oscilloscope
Oscilloscope
From
Detector
PID +
Dither
Amp
FPS
IM
N
2
IM
N1
+(N
-
1)T
R
IM
C
From
Amps
50:50
+(N
-
1)T
R
Delay
Lin
e
Signal For FPGA Clocking
EDFA
EDFA
Figure 4:
Schematic of the experimental setup used to realize NTOS.
8
tion between the electronics and photonics in our experiment. Meanwhile, light from the 90%
port passes through two intensity modulators (IMs), labeled
IM
00
and
IM
01
. These modulators
prepare the pulse patterns that we inject into the network, which in this work, are the right zero-
modes of HN lattices of various sizes. We use
IM
00
to carve the desired zero-mode from the
incoming pulse train, while we use
IM
01
to improve the extinction ratio for sites in the pulse
pattern that are supposed to have zero amplitude.
After passing through
IM
00
and
IM
01
, the pulses pass through another 90:10 splitter, and
10% of the light is injected into the network. Within the network, a 50:50 splitter divides the
pulses between a common delay line path and the continuation of the main cavity. We add
erbium-doped fiber amplifiers (EDFAs) to both of these paths. In the main cavity, the EDFA
allows us to partially compensate for the roundtrip losses. In the delay line path, the EDFA
enables us to increase the coupling strengths between the pulses. After each EDFA, we add
Channel 34 DWDM filters to remove amplified spontaneous emission (ASE) noise.
After the filter, the light in the common delay line path is divided at another 50:50 splitter.
Half of the light goes into the
+(
N
−
1)
T
R
delay line, and the other half goes to yet another
50:50 splitter, where it is divided between the
±
1
T
R
delay lines. In each delay line, there is
a tunable free space delay, which enables us to adjust the lengths of the delay lines; at least
one IM, which allows us to control coupling strengths during an experiment; and a fiber phase
shifter (FPS), which we use to stabilize the delay lines relative to the main cavity. After passing
through these elements, we recombine the delay lines with more 50:50 splitters before finally
recombining all of the delay lines with the main cavity at a final 50:50 splitter.
After recombining the delay lines and main cavity, the light passes through a fiber stretcher,
a FPS, and an IM before finally reaching the second 90:10 splitter in the main cavity. The
fiber stretcher and the FPS are used to stabilize the main cavity, while the IM (
IM
C
) is used to
“Q-switch” the cavity during our experiment. In particular, we find that it is easier to stabilize
9
our cavity and delay lines if the finesse of the main cavity is lower, so we use
IM
C
to reduce
the cavity finesse when we lock our system prior to an experiment. During the experiment, we
increase the cavity finesse to lower the decay rate of the cavity. The second 90:10 splitter in
the main cavity outputs 10% of the light in the main cavity. This output is divided with a 50:50
splitter between a slow (kHz) detector, which is used for stabilization, and a
5 GHz
detector,
which is used to capture the relative pulse amplitudes on our oscilloscope. The 90% of the light
that remains in the main cavity returns to the input 90:10 splitter.
In addition to the FPGA, which drives the IMs in our experiment, there are several other
electronic components to our setup. First, we use a dedicated data acquisition module to set
the bias voltages of our modulators. This module sets the voltages of
IM
00
,
IM
±
1
, and
IM
C
.
Due to the limited number of channels on our data acquisition module, we use a separate power
supply (not shown in Fig. S4) to set the biases for
IM
N
1
,
IM
N
1
, and
IM
01
. Second, we use
Red Pitaya STEMLabs to perform lock for the delay lines and the main cavity in phase. These
off-the-shelf modules contain built in Pound-Drever-Hall (PDH) locking capabilities [2], we
use them in conjunction with custom-built PCBs, the FPSs in the delay lines, and the FPS and
fiber stretcher in the main cavity to stabilize our system.
3 Calibration
Here, we briefly describe the calibration procedure used to prepare the input waveforms and the
pulse-to-pulse couplings used in our experiment.
To construct the input waveforms using
IM
00
and
IM
01
, we view the output of these two
IMs directly on an oscilloscope. Leaving
IM
01
biased to maximum throughput, we apply a
voltage ramp to
IM
00
to generate a curve of the output optical power as a function of the ap-
plied voltage. We use this calibration curve to generate a first-pass input waveform for
IM
00
.
Recall that this waveform is supposed to generate the right HN zero-mode from the input pulse
10
train. For our next step in the calibration, we iteratively improve our first-pass waveform. We
drive
IM
00
with the current waveform and average the optical response over several traces. In
software, we compare this average trace to the expected HN zero-mode, and we update our
driving waveform accordingly. We continue this procedure until we attain the desired accuracy
of the input waveform. Note that, during this step in the calibration, we also re-bias
IM
01
to
minimum throughput and apply the waveform that this modulator will see during the experi-
ment. The waveform on
IM
01
opens the modulator to maximum throughput when a site in the
HN zero-mode is supposed to be nonzero, and it leaves the modulator at minimum throughput
when a site in the HN zero-mode is supposed to have zero amplitude.
To calibrate the IMs in the
±
1
T
R
delay lines, we first bias the IMs to minimum throughput.
We then iteratively tune the waveform amplitudes applied the two modulators until the ratio of
the coupling strength produced by the
−
1
T
R
delay line to that produced by the
+
T
R
delay line
is
√
2
. Note that we only drive these modulators when they need to produce couplings between
the pulses. In particular, while our cavity can support up to
74
pulses, the HN lattices studied
in our experiments are much smaller. Therefore, we only drive the IMs when they produce
nearest-neighbor couplings in the HN lattice under study. For other (empty) time slots, we
leave the modulators at minimum throughput so that we reduce spurious couplings between the
lattice and the surrounding time slots.
After calibrating the IMs in the
±
1
T
R
delay lines, we calibrate the
+(
N
−
1)
T
R
delay line to
introduce a perturbation between the first and final sites of the HN lattice under study. Because
the perturbation is small, it is not possible to calibrate the
+(
N
−
1)
T
R
delay line by viewing the
throughput of the delay line on our oscilloscope. Therefore, we resort to measuring the average
power through the
+(
N
−
1)
T
R
delay line relative to the average power through the
+1
T
R
delay
line. To do this, we connect a power meter to one of the unused ports of the 50:50 splitter after
the
±
1
T
R
delay lines recombines with the
+(
N
−
1)
T
R
delay line. We then configure
IM
+1
so
11
that the
+1
T
R
delay line outputs a constant stream of pulses whose amplitudes correspond to
the coupling strength that will be used during the experiment. Measuring the average power of
this pulse train from the
+1
T
R
delay line tells us by how much we must attenuate the power in
the
+(
N
−
1)
T
R
in order to introduce the desired perturbation.
With all of the other delay lines blocked, we next observe the average output power of the
+(
N
−
1)
T
R
delay line on the power meter. We begin to attenuate this delay line by biasing
both
IM
N
1
and
IM
N
2
to maximum throughput then detuning the coupling in the free-space
delay shown in Fig. S4. In this manner, we attenuate the delay line to a point where we can
achieve the desired perturbation by properly reducing the throughput of the IMs but where we
can also still lock the
+(
N
−
1)
T
R
delay line when the IMs are set to maximum throughput.
At this point, we measure the power out of the delay line once more and calculate how much
further we need to attenuate the power to achieve the desired perturbation.
We distribute the remaining attenuation between the
IM
N
1
and
IM
N
2
. With one of the
modulators biased to maximum throughput, we tune the bias of the other modulator to introduce
a certain level of attenuation. We then record the bias voltage at which this attenuation is
achieved. We repeat this procedure for both modulators to achieve the full degree of attenuation.
Although the bias of these modulators is not actively stabilized, we verify that the throughput
of the IMs is stable enough that the degree of attenuation does not vary substantially over the
time scale of our experiments. After determining the proper bias voltage for each modulator,
we set the bias voltages for both modulators. Note that, because we do not actively modulate
the modulators in this delay line, the coupling in the
+(
N
−
1)
T
R
delay line is always on. In
experiment, we find that this fact does not have a substantial effect on the observed results.
12
4 Experimental Procedure and Data Analysis
4.1 Experimental Procedure
After calibrating the IMs in our network, we are ready to begin our experiment. Prior to running
an experiment, we prepare our network in the so-called “locking cycle,” in which we lock
the main cavity and all of the delay lines in phase. Recall that the modulators in the
±
1
T
R
delay lines are biased to minimum, while the modulators in the
+(
N
−
1)
T
R
delay line are
biased to introduce the desired perturbation between the first and final pulses of the HN lattice.
Therefore, to lock the delay lines, we drive the IMs to enable sufficient throughput for our
locking electronics to function properly. Furthermore, we drive the intracavity IM
IM
C
(which
is biased to maximum throughput) to reduce the finesse of the cavity. As was mentioned earlier,
we find that reducing the finesse of our cavity during the locking cycle facilitates locking the
main cavity and the delay lines simultaneously.
After locking the network, we initiate a program that triggers our FPGA and runs our ex-
periment. Upon triggering the FPGA, the output of the FPGA switches the network from the
locking cycle to the experiment cycle. At the beginning of the experiment cycle, we use
IM
C
to suppress any residual light that might be in the cavity from the locking cycle. Then we stop
driving
IM
C
to maximize the finesse of the cavity. After we stop driving
IM
C
, we inject the
right zero-mode of the HN lattice under study into the network for 10 roundtrips, which enables
the zero-mode to build up to a steady state. On the
10
th
roundtrip, we initiate the couplings in
the
±
1
T
R
delay lines. We then stop injecting light into the cavity, and we observe the decay of
the HN zero mode in the presence of the perturbation introduced by the
+(
N
−
1)
T
R
delay line.
Ideally, in the absence of the perturbation, the HN zero-mode would decay at the same rate
as an uncoupled pulse in our network. Therefore, we can measure the effect of the perturbation
by measuring the decay rate of the HN zero-mode relative to the decay rate of an uncoupled
13
pulse. We do this in our experiment by injecting an additional pulse into one of the unused
time slots of our cavity at the same time that we inject the HN zero-mode. Our delay lines
are programmed so that this pulse is uncoupled from the other time slots in the cavity, and,
therefore, this pulse provides a reference for the decay rate of the measured zero mode.
Additionally, we inject a second reference pulse into another unused time slot of the main
cavity, and we allow this reference pulse to couple to its direct nearest-neighbors. This reference
pulse allows us to observe the reliability of our nearest-neighbor couplings and helps to ensure
that the timings of the delay lines and the injected state are properly synchronized.
4.2 Data Analysis
As illustrated in Fig. 3 of the main text, the state of the time-multiplexed network in every
roundtrip is represented by the amplitudes of the pulses within the cavity time slots which
define 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
(
mT
RT
)
⟩
in successive roundtrips
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 define
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 defined as the quantity
|
∆
E
×
T
RT
|
that
is reported in Fig. 4 of the main text.
5 Analytical solution for the perturbed zero state
In this section, we present analytical results to evaluate the exact eigenvalue of the zero eigen-
state associated with a finite Hatano-Nelson lattice with perturbed boundary conditions. This
situation is schematically shown in Fig. S1.
Using coupled mode theory, the evolution of the field amplitudes within different lattice sites in
14
this lattice is governed by:
da
n
dt
=
t
R
a
n
−
1
+
t
L
a
n
+1
,
2
< n < N
−
1
da
1
dt
=
t
L
a
2
,
da
N
dt
=
t
R
a
N
−
1
+ Γ
a
1
.
(7)
Using the ansatz
a
i
(
t
) =
A
i
e
Et
,the solution to these coupled differential equations can be given
by
A
n
=
A
1
m
=
⌈
n/
2
⌉−
1
X
m
=0
n
−
m
−
1
n
−
2
m
−
1
(
−
1)
m
E
n
−
(2
m
+1)
t
m
R
t
n
−
m
−
1
L
,
1
< n.
(8)
Assuming
N
= 2
k
+ 1
and by substituting Eq. 8 into the last equation in Eq. 7 one finds
f
(
E
)
≡
m
=
k
X
m
=0
2
k
+ 1
−
m
2
k
+ 1
−
2
m
(
−
1)
m
E
2
k
+1
−
2
m
t
m
R
t
2
k
+1
−
m
L
−
Γ = 0
.
(9)
Assuming the perturbed eigenvalue is real, Eq. 9 must have a real root such that
f
(
E
(Γ)) = 0
where
−
t
L
< E
(Γ)
< t
L
. Therefore, in accordance to Bolzano’s theorem we expect that
f
(
t
L
)
f
(
−
t
L
)
<
0
.
(10)
Meanwhile,
f
(
t
L
) = (
−
x
)
k
B
k
(
−
1
/x
)
−
Γ
, where
x
=
t
R
/t
L
and
B
k
denote the Morgan-
Voyce polynomials which are related to the Chebyshev polynomials of the second kind via
B
k
(
x
) =
U
k
(1 +
x/
2)
[3]. The conditions defined by the Eq. 11 then translate into
Γ
< x
k
|
U
k
(1
−
1
/x
)
|
= Γ
C
,
(11)
which clearly indicate that the conditions for the perturbed finite HN lattice to have a real root
limits the boundary coupling
Γ
to smaller values that scale exponentially with the size of the
finite lattice.
15
6 Sensitivity analysis of trivial and Hermitian topological lat-
tices
In this section we theoretically analyze the sensitivity of trivial and Hermitian topological lat-
tices with respect to their associated boundary conditions. Let us first consider a trivial lattice
shown in Fig. S5a. Here, the evolution of the field amplitudes within different lattice sites in
this lattice is governed by:
i
da
n
dt
+
ta
n
−
1
+
ta
n
+1
= 0
,
2
< n < N
−
1
i
da
1
dt
+ Γ
a
N
+
ta
2
= 0
,
da
N
dt
+ Γ
a
1
+
ta
N
−
1
= 0
.
(12)
When
Γ = 0
, using the ansatz
a
m
= (
A
′
e
imq
+
B
′
e
−
imq
)
e
−
iEt
one finds the eigenvalues to be
E
=
−
2
tcos
(
q
)
where
q
=
kπ/
(
N
+ 1)
,k
= 1
,
2
,...,N
defines discrete quasi-momenta in the
lattice. From here, the field amplitudes can be obtained as
a
m
=
Asin
(
mq
)
e
−
iEt
. Based on
these results, we can now estimate the change in the eigenvalues as a result of the change in the
boundary conditions
Γ
̸
= 0
:
|
∆
E
n
|
=
|
⟨
ψ
n
|
∆
H
|
ψ
n
⟩
⟨
ψ
n
|
ψ
n
⟩
|
=
|
(
−
1)
k
+1
×
4Γ
/N
sin
2
q
|
<
4Γ
/N.
(13)
This is consistent with the results presented in the Fig. 5 of the manuscript.
Next, we consider a one-dimensional Su-Schrieffer-Heeger (SSH) model as an example of
a Hermitian topological lattice (Fig. S5b). Similar analysis in this case shows that the shift in
the eigenvalue of the defect state can be estimated again using first-order perturbation theory
16
Figure 5:
Sensitivity analysis of trivial and Hermitian topological lattices. a
, Trivial lattice
and
b
, the SSH model as an example of a Hermitian topological lattice.
∆
E
0
=
⟨
ψ
0
|
∆
H
|
ψ
0
⟩
⟨
ψ
0
|
ψ
0
⟩
≈
1
−
e
−
2
α
1
−
e
−
2
Nα
×
1 0
−
e
−
α
0
e
−
2
α
0
···
0
0
···
Γ
.
.
.
.
.
.
.
.
.
Γ
···
0
1
0
−
e
−
α
0
e
−
2
α
0
.
.
.
0
= 0
.
(14)
Here,
e
−
α
=
t
1
/t
2
. This insensitivity is consistent with the results presented in Fig. 5 of the
main text associated with a Hermitian topological lattice.
7 Sensor response and noise
In this section, we first provide theoretical analysis in line with [4, 5] regarding the response of
our implementation of NTOS to an external probe that confirms its exponential sensitivity. Here,
we assume that the dynamics of the time-multiplexed resonators is governed by the perturbed
Hamiltonian
H
=
H
HN
+ ∆
H
:
17
Figure 6:
Theoretical analysis of the NTOS response probed by an appropriate input state.
d
dt
|
ψ
(
t
)
⟩
=
H
|
ψ
(
t
)
⟩
,
(15)
assuming the initial probe injected into the system is equal to
|
ψ
(
t
)
⟩
=
|
ψ
0
⟩
R
, i.e. the
right eigenstate of the unperturbed Hamiltonian. In order to generate a readout signal
S
(
t
)
,
the state of the system is projected into the left eigenstate of the unperturbed Hamiltonian,
i.e.
S
(
t
) =
⟨
ψ
0
|
L
ψ
(
t
)
⟩
. Finally, we consider the logarithm of the change in the signal as the
response of our sensor
∆
S
=
log
(
S
(Γ
,t
))
−
log
(
S
(Γ = 0
,t
))
. Figure S6 summarizes these
results, thus confirming the exponential sensitivity enhancement of the NTOS implemented in
our study.
In order to experimentally characterize the noise in our setup, we run the experiment by
injecting the zero eigenstate of the unperturbed Hamiltonian
H
HN
with
N
= 23
into the cavity
and block the
(
N
−
1)
T
delay line to set the perturbation
Γ = 0
. We then experimentally
measure the fluctuations in the intensity of the last pulse
#23
that represents the last lattice site
in our resonator array (please see Fig. S7). In our experiments, we measure these fluctuations by
18
calculating the standard deviations in the intensities for an ensemble of measurements, resulting
in a value of
δI
23
≈
7
×
10
−
3
[
a.u.
]
. Assuming a conventional linear sensing model, the minimum
detectable perturbation in the coupling
Γ
from pulse
#1
to pulse
#23
would be
2Γ
min
×
T
RT
>
2
δI
23
/I
1
. Hence, using a conventional sensing method the minimum detection limit set by the
thermal noise in our experiments is estimated as
Γ
min
×
T
RT
≈
6
×
10
−
3
. As mentioned in
the main text, our proposed NTOS effectively amplifies small perturbations in the coupling
Γ
,
hence providing an enhanced response. Our experimental results presented in Fig. 4 of the
main text shows that using NTOS with
N
= 23
pulses, we were able to measure perturbations
as small as
Γ
×
T
RT
≈
1
.
5
×
10
−
3
. Such improvements in the detection limit is also consistent
with the sensitivities reported in Fig. 5 of the manuscript.
References
1. Leefmans, C.
et al.
Topological dissipation in a time-multiplexed photonic resonator net-
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Nature Physics
18,
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Am. J.
Phys.
69,
79–87 (2001).
3. Merikoski, J. K. Regular polygons, Morgan-Voyce polynomials, and Chebyshev polynomi-
als.
Notes on Number Theory and Discrete Mathematics
27,
79–87 (2021).
4. Langbein, W. No exceptional precision of exceptional-point sensors.
Physical Review A
98,
023805 (2018).
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19