# Nonlinearity-induced Scattering Zero Degeneracies for Spectral Management of
Coherent Perfect Absorption in Complex Systems

Cheng-Zhen Wang  Wave Transport in Complex Systems Lab, Department of Physics,
Wesleyan University, Middletown, CT-06459, USA  John Guillamon  Wave Transport
in Complex Systems Lab, Department of Physics, Wesleyan University,
Middletown, CT-06459, USA  William Tuxbury  Wave Transport in Complex Systems
Lab, Department of Physics, Wesleyan University, Middletown, CT-06459, USA
Ulrich Kuhl  Université Côte d’Azur, CNRS, Institut de Physique de Nice
(INPHYNI), 06200, Nice, France  Tsampikos Kottos<EMAIL_ADDRESS>Wave
Transport in Complex Systems Lab, Department of Physics, Wesleyan University,
Middletown, CT-06459, USA

###### Abstract

We develop a Coherent Perfect Absorption (CPA) protocol for cases where scale
invariance is violated due to the presence of nonlinear mechanisms. We
demonstrate, using a microwave setting that lacks geometrical symmetries, that
the nonlinearity offers new reconfigurable modalities: the destruction or
formation of nonlinear CPAs (NL-CPAs), and their frequency positioning and
bandwidth management using the incident power as a control knob. The latter
occurs via the formation of exceptional point degeneracies of the zeroes of
nonlinear scattering processes. Our results establish NL-CPA protocols as a
versatile scheme for the creation of reconfigurable hot/cold-spots in
complicated enclosures (e.g. buildings or vessels) with applications to next-
generation telecommunications, long-range wireless power transfer, and
electromagnetic warfare.

Introduction -  The wealth of the underlying mathematical structures of non-
Hermitian operators has revealed a variety of new concepts, including loss-
induced transparency Guo  _et al._ (2009) unidirectional invisibility Lin  _et
al._ (2011); Regensburger  _et al._ (2012); Feng  _et al._ (2013); Fleury  _et
al._ (2015), hypersensitive sensing Wiersig (2014); Hodaei  _et al._ (2017);
Chen  _et al._ (2017); Kononchuk  _et al._ (2022); Suntharalingam  _et al._
(2023), self-protecting power limiters Makri  _et al._ (2015); Riboli  _et
al._ (2023), chiral state transfer Doppler  _et al._ (2016); Nasari  _et al._
(2022), enhanced power emission Gupta  _et al._ (2023) and more (see reviews
Miri and Alu (2019); El-Ganainy  _et al._ (2018); Ozdemir  _et al._ (2019)).
Among the various developments, the coherent perfect absorption (CPA) protocol
has attracted considerable attention. It aims to design coherently injected
multiport wavefronts that appropriately interfere to enforce perfect
absorption from an absorption center that is embedded inside a cavity – even
in cases where the losses are very weak Chong  _et al._ (2010); Wan  _et al._
(2011); Baranov  _et al._ (2017).

Formally speaking, a CPA wavefront is associated with the eigenvector of the
scattering matrix that corresponds to a zero eigenvalue. Such zero scattering
mode describes the solution of a wave operator with purely incoming boundary
conditions. When these zeroes occur on the real frequency axis, they define
incident coherent wavefronts that are completely absorbed by a lossy cavity.
This is the concept of CPA Chong  _et al._ (2010); Krasnok  _et al._ (2019),
that has been experimentally tested in a variety of platforms ranging from
acoustics Romero-García  _et al._ (2016), RF Schindler  _et al._ (2012) and
microwaves Chen  _et al._ (2020) to optics Wan  _et al._ (2011). A new kind of
CPAs can emerge when two (or more) purely incoming solutions of the wave
operator coalesce forming an exceptional point degeneracy (EPD); thus leading
to an anomalous line-shape of the absorbance Sweeney  _et al._ (2019); Wang
_et al._ (2021). For example, in the case of two coalescing CPAs, the
frequency dependence of the absorbance around its maximum has a quartic (not
Lorentzian!) line-shape resulting in a broadband absorption Sweeney  _et al._
(2019); Wang  _et al._ (2021); Soleymani  _et al._ (2022).

Most of the CPA demonstrations Baranov  _et al._ (2017) and their EPDs
involved simple cavity geometries Wang  _et al._ (2021); Soleymani  _et al._
(2022). Recently, however, CPAs have been demonstrated in complex environments
Chen  _et al._ (2020) where multiple scattering and the consequent
interference of many photon paths through the medium generate extraordinary
complexity and sensitivity. A reason for this paradigm shift is the
realization that CPA protocols can be used in wireless communications as a
means to create hot spots in reverberate environments Frazier  _et al._ (2020,
2022); del Hougne  _et al._ (2021). Other experimental settings include
chaotic optical and microwave cavities, networks of coaxial cables, and
disordered systems with parity symmetry Li  _et al._ (2017); Jiang  _et al._
(2022); Sol  _et al._ (2023); Pichler  _et al._ (2019); Röntgen  _et al._
(2023); Ferise  _et al._ (2022). Importantly, all these cases assume linear
scattering processes where the superposition principle is applicable.

On the other hand, many realistic scattering scenarios involve nonlinear
processes. In wireless communications, for example, diodes are indispensable
components of wireless circuits. Similarly, optical nonlinearities emerge
naturally in photonics but up to now they have been underutilized as far as
perfect absorbing protocols are concerned. This is surprising since the
incident power might offer additional freedom for externally controlling the
formation of CPAs via a self-induced variation of the impedance of a medium.
The few NL-CPA studies are either theoretical Reddy and Gupta (2013); Konotop
and Zezyulin (2018), or they have been performed using very simple settings
Müllers  _et al._ (2018); Suwunnarat  _et al._ (2022) that do not represent
typical complex scattering processes occurring in realistic scenarios, e.g.,
wireless communications in reverberant cavities. Likewise, EPDs (of any kind!)
are barely discussed in nonlinear frameworks.

In this Letter, we develop a theory for the implementation of nonlinear CPAs
(NL-CPAs) in complex systems that, in general, lack geometrical symmetries. We
demonstrate, using a microwave complex network of coaxial cables, that scale-
invariance offers new modalities: external tunability of the frequencies, and
linewidths (narrow vs broad) of the CPA phenomenon. Specifically, the
electrical permitivities of nonlinear elements are externally tuned by the
incident power levels; thus allowing us to control the formation of NL-CPAs
and even enforcing their coalescence. Although in many cases this coalescence
occurs among complex zeroes, their proximity to the real frequency axis
constitutes these degeneracies practically indistinguishable from the EPD-
CPAs; thus allowing us to witness a broadening of the absorbance which
maintains the quartic lineshape as in the case of EPD-CPAs.

Figure 1:  (a) The input signal from port 1 of vector network analyzer (VNA)
is splitted into two signals by a T- junction and then modulated by IQ
modulators both in amplitude and phase. The two circulators direct the input
signals into the microwave network and the output signals into ports 2 and 3
for measurement. (b) Ring network. (c) Tetrahedron network. The network is
coupled via kink antennas to a resonator which, in turn, is inductively
coupled to a ring antenna which includes a short-circuited nonlinear diode
(see inset of (a)).

NL-CPA protocol – We assume that the nonlinear mechanism is local, i.e., it
depends on the magnitude of the scattering field $|\psi_{n_{0}}|\equiv|\langle
n_{0}|\psi\rangle|$ at a specific position $|n_{0}\rangle$ associated with the
position of a nonlinear defect (we use Dirac notation and represent all
operators in a local orthonormal basis $\\{|n\rangle\\}$). We do not impose
any constraints on the functional form of the nonlinearity which we
generically represent as $\nu=\nu\left(\left|\psi_{n_{0}}\right|^{2}\right)$.
By neglecting excitation of higher harmonics, the wave operator may be
represented in Fourier space by the following system of coupled nonlinear
algebraic equations,

$\left(H(\omega)+\nu\left(\left|\psi_{n_{0}}\right|^{2}\right)\left|n_{0}\right\rangle\left\langle
n_{0}\right|\right)\left|\psi\right\rangle=iD\left|s^{+}\right\rangle\,,$ (1)

where $H(\omega)$ is an $N\times N$ matrix that describes the dynamics inside
the scattering system in the absence of nonlinear mechanisms, the $N\times
L_{0}$ matrix $D$ describes the coupling of the $L_{0}$ transmission lines
(TL) with the system, and $|s^{+}\rangle$ is the $L_{0}-$dimensional vector
that describes the incident wavefront in a flux-normalized basis. Wave-
continuity requires
$\left|s^{-}\right\rangle=D^{T}\left|\psi\right\rangle+K\left|s^{+}\right\rangle$,
where $K=K^{T}$ is the $L_{0}\times L_{0}$ direct scattering matrix satisfying
unitarity $KK^{\dagger}=1$. Imposing zero-outgoing scattering conditions,
$\left|s^{-}\right\rangle=0$, enables us to re-express Eq. (1) as,

${\tilde{M}}\left|\psi\right\rangle=0;\quad{\tilde{M}}=\left(M+\nu\left(\left|\psi_{n_{0}}\right|^{2}\right)\left|n_{0}\right\rangle\left\langle
n_{0}\right|\right),$ (2)

where $M=H(\omega)+iDK^{\dagger}D^{T}$ (see SM SM ). We introduce the
$(N-1)\times N$ filtering matrix, $F$ with its defining properties,
$F^{T}F+\left|n_{0}\right\rangle\left\langle
n_{0}\right|=\mathbb{I}_{N},FF^{T}=\mathbb{I}_{N-1}$ and
$F\left|n_{0}\right\rangle=0$ to decouple the nonlinear element from the
remaining $|n\rangle,n\neq n_{0}$. The NL-CPA algorithm consists of the
following steps:

Step 1: Insert the identity matrix related to $F$ above into Eq. (2) and
isolate the nonlinear terms,

$MF^{T}F\left|\psi\right\rangle=-\psi_{n_{0}}\left(M\left|n_{0}\right\rangle+\nu\left(\left|\psi_{n_{0}}\right|^{2}\right)\left|n_{0}\right\rangle\right)\,.$
(3)

Step 2a: Apply $F$ to Eq. (3) and invert to solve for the field on the linear
elements in terms of the nonlinear contributions,

$F\left|\psi\right\rangle=-\psi_{n_{0}}\left(FMF^{T}\right)^{-1}FM\left|n_{0}\right\rangle\,.$
(4)

Step 2b: Apply $\left\langle n_{0}\right|$ to Eq. (3), to isolate the
nonlinear part of the scattering field,

$\widetilde{M}_{n_{0}n_{0}}\psi_{n_{0}}=\left(\nu(|\psi_{n_{0}}|^{2})+\alpha_{1}\right)\psi_{n_{0}}=-\langle
n_{0}|MF^{T}F|\psi\rangle=\alpha_{0}\psi_{n_{0}}\,$ (5)

where in the third equality we have inserted Eq. (4). Above, we have
introduced the variables $\alpha_{1}\equiv\langle
n_{0}|M\left|n_{0}\right\rangle$ and $\alpha_{0}\equiv\left\langle
n_{0}\right|MF^{T}\left(FMF^{T}\right)^{-1}FM\left|n_{0}\right\rangle$
(involving only linear modeling elements).

Figure 2:  (a) Absorption measurements (color scale) versus frequency
$\textrm{Re}(f)$ and power $P_{1}$. Red circles represent calculated complex
zeros of the nonlinear scattering process as a function of input power
$P_{1}$. (b) The same plot as in (a) for broadband absorbance. The theoretical
analysis of complex zeroes (red circles) indicates the existence of two nearby
NL-CPAs that bifurcate from a complex-zero EPD occurring in proximity to the
real frequency axis. (c) The relative output signal ${\cal O}=1-A$ versus the
frequency detuning $\delta f$ from the maximum absorption conditions
corresponding to case (b). The blue (green) line is the experiment (theory).
The red dashed (dotted) line is the best fit with slope 3 (slope 3.5)
approximating a quartic lineshape expected for an EPD-CPA. The orange dashed
line corresponds to a Lorentzian lineshape (slope 2) and is drawn for
comparison. (d) Parametric evolution of the complex zero modes, versus the
input power $P_{1}$: (d1) the real part $\textrm{Re}(f)$ of complex zero
frequency; (d2) its imaginary part $\textrm{Im}(f)$ (the small values signify
proximity to the real frequency axis); (d3) input power $P_{2}$; and (d4)
relative phase $\phi$. The degeneracy of the complex frequencies and the
corresponding modes occurs at the same value of $P_{1}$ (vertical blue dashed
line) signifying a complex zero NL-EPD.

Step 3: Solve for $\left|\psi_{n_{0}}\right|^{2}\neq 0$ given the functional
form of the nonlinearity,

$\left|\psi_{n_{0}}\right|^{2}=\nu^{-1}\left(\alpha_{0}-\alpha_{1}\right)\,.$
(6)

A critical condition for the physicality of the solution is that
$\left|\psi_{n_{0}}\right|^{2}\in\mathbb{R}^{+}$, identified by a single
parameter sweep in $\omega$, as opposed to the unaided $2L_{0}$ parameter
sweep in relative amplitudes and phases of $\left|s^{+}\right\rangle$ and
$\omega$. Between Eq. (4) and Eq. (6) the CPA scattering field has been
obtained up to an arbitrary global phase, $\left|\psi_{NL-
CPA}\right\rangle=F^{T}F\left|\psi\right\rangle+\psi_{n_{0}}\left|n_{0}\right\rangle$.
The continuity relation and zero-outgoing scattering conditions, immediately
provides the input wave, $\left|s_{NL-
CPA}^{+}\right\rangle=-K^{\dagger}D^{T}\left|\psi_{NL-CPA}\right\rangle$.

Experimental setup – The microwave network is formed by coaxial cables (bonds)
connected by $n=1,\cdots,N$ T-junctions (vertices). A local lossy (saturable)
nonlinearity is incorporated at the $N$-th vertex, by substituting the
T-junction with a dielectric cylindrical resonator. The bonds are inductively
coupled to the resonator by transforming the cable ends into “kink” antennas.
Subsequently, this resonator is inductively coupled to a metallic ring, which
is positioned above its center and is short-circuited to a diode. The kink
antennas generate an EM field that is confined within the resonator and
induces a current in the diode. High currents generated by strong fields,
activate a nonlinear response of the diode which leads to a shift in the
resonator’s frequency.

The excitation signal is injected into port 1 of the VNA, evenly divided into
two channels by a T-junction, and subsequently routed through IQ-modulators
which control the amplitude and phase of the input signals ($I_{1}$, $I_{2}$).
The shaped wavefront is directed towards two circulators which separate the
input and output signals. The latter are directed to port 2 ($O_{1}$) and 3
($O_{2}$) of the VNA for processing. The experimental setup is shown in Fig.
1.

The efficiency of our NL-CPA protocol is evaluated by the measured absorption
$A=1-{\cal O}$, where ${\cal
O}\equiv\frac{|O_{1}|^{2}+|O_{2}|^{2}}{|I_{1}|^{2}+|I_{2}|^{2}}$ is the total
relative output signal. In our experiment, we have used the VNA and the IQ
modulators to tune four external parameters for the design of NL-CPA
wavefronts: the incident frequency $f\in[6.25,6.45]$ GHz, the relative phase
of the injected field amplitudes
$\phi=\phi_{1}-\phi_{2}\in[0^{\circ},360^{\circ}]$ at channels 1 and 2 and
their corresponding input power $P_{1}$ (channel 1) and $P_{2}$ (channel 2)
with $P_{1,2}\in[-45,5]$ dBm.

Implementation – The theoretical modeling of the network platform and the NL-
CPA protocol implementation can be found in the SM SM .

Intensity manipulation of the CPA frequency. We first demonstrate the
nonlinear CPA modality associated with the intensity manipulation of the CPA
frequency. For this purpose, we have used the ring network (N=2) of Fig. 1 (b)
where this modality is accessible in the available power and frequency range
of our VNA. Figure 2(a) reports the parametric evolution of the complex zeroes
of the nonlinear scattering process (red circles) versus the input power
$P_{1}$. The NL-CPA occurs when a complex zero crosses the real frequency axis
at $f^{cpa}\approx 6.397$ GHz, with input power $P_{1}^{cpa}\approx-16.10$
dBm, relative phase $\phi^{cpa}\approx 97^{\circ}$ and power ratio
$R^{cpa}\equiv P_{1}^{cpa}/P_{2}^{cpa}\approx 2.9$.

The experimental investigation, has been guided by the prediction of our NL-
CPA theory which allowed to confine the large parameter space to a smaller
domain. A maximum absorbance of $A^{*}=99.998\%$ occurs at frequency
$f^{*}\approx 6.391$GHz for a wavefront with $P_{1}^{*}\approx-17.67$ dBm,
$R^{*}\equiv P_{1}^{*}/P_{2}^{*}\approx 2.2$, and $\phi^{*}\approx 97^{\circ}$
(For detailed comparison between theory and experiment, see SM Figs. S1,S2 SM
). The small discrepancies between the predicted NL-CPA wavefront and the
experimental results are associated with the measuring precision of the
electrical permittivity of the coaxial cables and the uncertainty in the
parameters used to model the lossy nonlinearity SM . Figure 2(a) reports the
measured absorbance $A$ versus frequency $f$ and injected power $P_{1}$ (for
fixed $R^{*}$ and phase $\phi^{*}$). This example demonstrates a narrowband
$\sim 20$ MHz absorption with $A\geq 95\%$.

Intensity manipulation of absorbance band-width by NL-EPDs. An example where
the injected power levels are employed as an external parameter for
reconfiguring the absorbance bandwidth is shown in Fig. 2(b). For the previous
ring-graph configuration, we find a broadband $\sim 50$ MHz high absorbance
($A>95\%$) for increased injected powers. The experimental parameters of the
wavefront that results in a peak absorbance $A^{*}\approx 99.55\%$ are
$f^{*}\approx 6.34$ GHz with $P_{1}^{*}\approx-3.5$ dBm, $R^{*}\approx 5.5$
and $\phi^{*}\approx 66^{\circ}$. In this parameter range, the NL-CPA protocol
predicted two NL-CPAs (a) at $f^{\text{cpa}}\approx 6.346\,$ GHz with
$P_{1}^{\text{cpa}}\approx-3.6$ dBm, power ratio $R^{cpa}\approx 6.3$ and
relative phase $\phi^{cpa}\approx 120^{\circ}$; and (b) at
$f^{\text{cpa}}\approx 6.327$ GHz with $P_{1}^{\text{cpa}}\approx-2.3$ dBm,
$R^{cpa}\approx 6.6$ and $\phi^{cpa}\approx 136^{\circ}$. The phase
discrepancy between the experimental and the theoretical values is associated
with the precision of the extracted experimental parameters of the network. It
turns out that the wavefront phase is the most sensitive among all other
characteristics of the optimal wavefront (see SM and Figs. S1,S2 SM ).

Figure 3:  Measured absorption (color scale) versus input frequency
$\textrm{Re}(f)$ and power $P_{1}$ for (a) narrowband and (b) broadband NL-
CPA. The red circles in both (a,b) are the theoretically calculated
trajectories of the complex zeroes as $P_{1}$ varies. In (b), two NL-CPAs,
close to one-another, emanate from a complex zero EPD occuring in the
proximity of the real-frequency axis ($\textrm{Im}(f^{EPD})\sim 10^{-4}$). For
all practical purposes, these two NL-CPAs form an EPD-CPA. (c) The relative
output signal (${\cal O}=1-A$) versus the frequency detuning $\delta f$ at
$P_{1}^{cpa}$. The blue (green) line indicates the experimental (theoretical)
scaling of ${\cal O}$ versus detuning from $f^{*}$ ($f^{EPD}$). The red dotted
line indicates the best power-law fit with slope 3.5 (close to quartic
behavior expected for EPD-CPAs). A quadratic (Lorentzian) scaling is shown
(orange dashed line) for comparison.

We have also analyzed the shape of the absorption spectrum in the vicinity of
$f^{*}$. In Fig. 2(c) we report the measured relative outgoing signal ${\cal
O}=1-A$ versus $\delta f\equiv f-f^{*}$ (blue line) in a double-logarithmic
fashion. We find a cubic power-law scaling ${\cal O}\sim(\delta f)^{3}$ (see
dashed red line) which is distinct from the traditional Lorentzian quadratic
behavior. The latter is shown for comparison with a dashed orange line. In the
same figure, we also present the scaling of the theoretical result (green
line) and find that ${\cal O}\sim(\delta f)^{3.5}$ which is reminiscent of the
quartic scaling occurring in the case of EPD-CPAs. In fact, by tracing the
theoretically evaluated NL-CPAs in the complex frequency domain we have
identified an EPD of the scattering zeroes (see red circles in Fig. 2(b))
occurring at $f^{\text{EPD}}\approx(6.335+i\cdot 0.0075)$ GHz with
$P_{1}^{\text{EPD}}\approx-1.18$ dBm, $R^{EPD}\approx 8.6$ and
$\phi^{EPD}\approx 131^{\circ}$. The small imaginary part of the complex
zeroes-EPD signifies its proximity to the real frequency axis and justifies
the fact that the experimental absorption lineshape demonstrates broadband
characteristics similar to the one predicted for an EPD-CPA Sweeney  _et al._
(2019); Wang  _et al._ (2021); Soleymani  _et al._ (2022).

To further examine the nature of the complex-zero degeneracy, we examined in
detail the parametric evolution (versus input power $P_{1}$) of the complex
frequency, amplitudes, and relative phases of the complex-zero modes, see
Figs. 2(d1)-(d4). All these parameters coalesce simultaneously at the same
input power signifying the formation of a complex- zero NL-EPD.

Generalization to Complex Networks. Next, we implement the NL-CPA protocol to
a tetrahedron network $(N=4)$, see Fig. 1 (c). In the absence of
nonlinearities, and for incommensurate bond lengths, this network shows many
typical features of wave chaos Kottos and Smilansky (1997); Kottos and Schanz
(2004); Haake  _et al._ (2018); thus offering a platform for simulating real-
world scattering environments.

Using the NL-CPA protocol we have predicted the formation of a NL-CPA by
tracing the trajectory of the complex zeroes (red circles) of the nonlinear
scattering process and identifying the point at which it crosses the real
axis. The wavefront parameters that we have found are
$(f^{cpa},P_{1}^{cpa})\approx(6.369\,GHz,-6.6\,dBm)$ for input power ratio
$R^{cpa}\approx 5.96$ and relative phase $\phi^{cpa}\approx 191^{\circ}$.
Again, these estimations allowed us to reduce the parameter space in our
experimental implementation. We have found that an optimal absorption of
$A^{*}\approx 99.97\%$ occurs at $f^{*}\approx 6.371$ GHz with injected power
$P_{1}^{*}\approx-5.09$ dBm, power ratio $R^{*}\approx 6.03$ and relative
phase $\phi^{*}\approx 207^{\circ}$ between the two injected wavefronts.
Figure 3(a) shows the measured absorbance (color-grade) versus incident power
$P_{1}$ and frequency $f$ for fixed relative power ratio $R^{*}$ and relative
phase $\phi^{*}$ (see SM Figs. S3(a,b), S4(a1-d1) SM ).

We have also identified broadband absorption scenarios in a tetrahedron
network (bandwidth $\approx 20$MHz as opposed to $\approx 6.3$MHz in the
previous case, for $A>95\%$). An example is reported in Fig. 3(b) where we
show the measured absorbance (color scale) for varying incident frequency and
injected power $P_{1}$ (fixed relative power $R^{*}\approx 0.45$ and relative
phase $\phi^{*}\approx 301^{\circ}$). At $f^{*}\approx 6.39\,$GHz and
$P_{1}^{*}\approx-19.05$ dBm the absorbance acquires its maximum value
$A^{*}\approx 99.93\%$. In comparison, the NL-CPA protocol predicted the
existence of two nearby NL-CPA wavefronts. These correspond to the wavefront
parameters for which the complex zeroes (red circles) cross the real frequency
axis in Fig. 3(b) while varying $P_{1}$: (a) at $f^{cpa}\approx 6.397\,$GHz
with $P_{1}^{cpa}\approx-19.24$ dBm, $R^{cpa}\approx 0.68$, and
$\phi^{cpa}\approx 377.8^{\circ}$; and (b) at $f\approx 6.396\,$GHz with
$P_{1}^{cpa}=-19.13$ dBm, $R^{cpa}\approx 0.71$, and $\phi^{cpa}\approx
376.4^{\circ}$. Further analysis and comparison between theory and
experimental data is provided in the SM SM , see Figs. S3(c,d) and Figs.
S4(a2-d2). The degeneracy occurs slightly above the real plane (imaginary part
of complex EPD zero $\sim 10^{-4}$) which is completely untraceable in the
experiment; thus emulating the behavior of an actual NL-CPA EPD. The exact
parameters of the complex-zero NL-EPD are
$f^{EPD}\approx(6.396+0.00027i)\,$GHz, $P_{1}^{EPD}\approx-19\,$dBm,
$R^{EPD}\approx 0.69$,and $\phi^{EPD}\approx 376.2^{\circ}$. We have confirmed
that at this point the complex-zero modes form an EPD, by analyzing the
parametric evolution of their complex frequencies, $P_{2}$, and relative phase
$\phi$ that characterize them versus the injected power $P_{1}$. All of them
coalesce at $P_{1}^{EPD}$ as expected for an EPD (for a detailed description
of the NL-EPD modes see SM Fig. S5 SM ). Further confirmation of the EPD was
provided by analyzing the lineshape of the absorbance in the proximity of the
(quasi-)degeneracy, see Fig. 3(c). Similar to the previous case of the ring-
graph, we find that the scaling of the relative outgoing signal with respect
to the detuning $\delta f$ is best fitted by a super-Lorentzian power-law
${\cal O}\sim(\delta f)^{3.5}$ which resembles the quartic line-shape expected
for an EPD-CPA Sweeney  _et al._ (2019).

Conclusions -  We have developed a NL-CPA protocol that is generically
applicable to any scattering setup. The scheme utilizes the nonlinear light-
matter interactions as a reconfigurable knob to tailor the features of the
absorption spectrum through external power control. We have implemented our
proposal using microwave networks of coaxial cables and demonstrated a variety
of new modalities due to the nonlinearity: Control in the
creation/annihilation of NL-CPAs, their frequencies, and band-width
manipulation for which high absorption occurs. The latter is achieved in the
proximity of zero-EPDs of the scattering process. Our results establish NL-CPA
protocols as a versatile scheme for the creation of hot (or cold)- spots in
reverberate environments.

Acknowledgements – We acknowledge partial funding from NSF-RINGS ECCS
(2148318), and Simons Foundation (MPS-733698).

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Supplemental Material for manuscript:
Nonlinearity-induced Scattering Zero Degeneracies for Spectral Management of
Coherent Perfect Absorption in Complex Systems

Cheng-Zhen Wang1, John Guillamon1, William Tuxbury1, Ulrich Kuhl2, and
Tsampikos Kottos1

1 Wave Transport in Complex Systems Lab, Department of Physics, Wesleyan
University, Middletown, CT-06459, USA

2 Université Côte d’Azur, CNRS, Institut de Physique de Nice (INPHYNI), 06108
Nice, France, EU

##  I Theoretical Model

###  I.1 Networks

The transport characteristics of the system are modeled by a fusion of two
primary components: the first component is a metric graph model of one
dimensional wires supporting only one propagating mode connected by
vertices–this model demonstrates wave chaotic scattering statistics; the
second component are the steady state solutions of a set of nonlinear coupled
mode theory equations to model the nonlinear cavity mode response of the
resonator. We will consider a microwave graph with $N$ vertices (or nodes), of
which $L_{0}$ of them are connected to transmission lines (TL). The coupling
to the TL is described by the $N\times L_{0}$ matrix $D$ with elements $1$
($0$) when a vertex is connected (not connected) to a TL. For example, in the
case where the 1st and 2nd vertices are connected to TL, with the $N-2$ other
vertices being internal vertices, the transpose coupling matrix takes the form

$D^{T}=\begin{bmatrix}1&0&\cdots&0\\\ 0&1&\cdots&0\\\ \end{bmatrix}\,.$ (S1)

Bonds (or edges) are one-dimensional connections between vertices, where the
position on each bond from vertex $\mu$ to $\beta$ can be defined as
$x_{\mu\beta}$, with $x_{\mu\beta}=0$ on vertex $\mu$ and
$x_{\mu\beta}=L_{\mu\beta}$ on vertex $\beta$, and
$x_{\beta\mu}=L_{\mu\beta}-x_{\mu\beta}$. To account for how the bonds connect
the vertices, we will introduce the symmetric $N\times N$ matrix $A$ which
takes entries of 1 for connected vertices and 0 otherwise, with all diagonal
elements set to zero as we do not allow self-connected vertices. In addition
to the $A$ matrix, we also use another symmetric $N\times N$ matrix, $L$,
which contains the bond lengths for the connections between the vertices. For
a simple fully connected tetrahedron graph, they take the following form:

$A=\begin{bmatrix}0&1&1&1\\\ 1&0&1&1\\\ 1&1&0&1\\\ 1&1&1&0\\\
\end{bmatrix};\quad L=\begin{bmatrix}0&L_{12}&L_{13}&L_{14}\\\
L_{21}&0&L_{23}&L_{24}\\\ L_{31}&L_{32}&0&L_{34}\\\ L_{41}&L_{42}&L_{43}&0\\\
\end{bmatrix}\,.$ (S2)

Note that $L_{\mu\beta}=L_{\beta\mu}$. Finally, throughout the manuscript, we
will be assuming that the nonlinearity is located at vertex $n_{0}=N$.

###  I.2 Linear Wave Network Theory and CPA conditions

First we address the wave propagation at the linear subspace of the networks.
The one-dimensional bonds can model the coax cables where the microwaves
propagate. The wave equation that describes the electric potential difference
$\psi_{\mu\beta}(x_{\mu,\beta})$ at position $x_{\mu,\beta}$ along the
$L_{\mu,\beta}-$bond is given as

$\frac{d^{2}}{d{x_{\mu\beta}}^{2}}\psi_{\mu\beta}\left(x_{\mu\beta}\right)+\frac{\omega^{2}\epsilon}{c^{2}}\psi_{\mu\beta}\left(x_{\mu\beta}\right)=0\,,$
(S3)

where $\epsilon$ is the dielectric constant of the coaxial cables,
$\omega=2\pi f$ is the angular frequency with $f$ the frequency and $c$ is the
speed of light in vacuum. The wave number is $k=\sqrt{\epsilon}\omega/c$. It
is convenient to define the wave at a vertex $\mu$ as
$\psi_{\mu,\beta}(x_{\mu,\beta}=0)=\psi_{\mu}=\phi_{\mu}$ where we have
introduced the vertex field $\Phi=(\phi_{1},\phi_{2},\cdots,\phi_{N})^{T}$.

The solutions of the above Eq. (S3) must be supplemented with the appropriate
boundary conditions at the vertices. First, we consider wave continuity at
each vertex, i.e., $\psi_{\mu\beta}\left(x_{\mu\beta}=0\right)=\phi_{\mu}$ for
any bond that connects the vertex $\mu$ with any vertex $\beta$. Using the
vertex scattering field $\Phi$ we write the wave at each bond as

$\psi_{\mu\beta}\left(x_{\mu\beta}\right)=\phi_{\mu}\frac{\sin{k\left(L_{\mu\beta}-x_{\mu\beta}\right)}}{\sin{kL_{\mu\beta}}}+\phi_{\beta}\frac{\sin{kx_{\mu\beta}}}{\sin{kL_{\mu\beta}}}\,.$
(S4)

Next, we impose current conservation at each vertex. This condition takes the
form

$\displaystyle\sum_{\beta}{\frac{d}{d{x_{\mu\beta}}}\psi_{\mu\beta}\left(x_{\mu\beta}=0\right)+\sum_{\mu=1,2}\delta_{\mu,\alpha}\frac{d}{d{x_{\mu}}}\psi_{\mu}\left(x_{\mu}=0\right)}=0.$
(S5)

Finally, the CPA condition is satisfied by enforcing only incoming wave forms
on the TLs

$\psi_{\mu}\left(x_{\mu}\right)=I_{\mu}\delta_{\mu\alpha}e^{-ikx_{\mu}}\,,$
(S6)

where $I_{\mu}$ is the amplitude of the injected wave at vertex $\mu$ and the
subscript $\alpha=1,2$ indicates the TL. The position $x_{\mu}\in[0,\infty)$
at the TL is $x_{\mu}=0$ at the $\mu$-vertex where the TL is connected. Note
that by excluding a counter-propagating wave
$R_{\mu}\delta_{\mu\alpha}e^{ikx_{\mu}}$, we ensure that there are no outgoing
waves (CPA-condition).

###  I.3 Modeling of nonlinear resonator

Next, we want to evaluate the field at the edge-points of the three kink-
antennas that are closer to the nonlinear resonator. To this end, we first
calculate the scattering field ${\tilde{a}}(t)$ at the nonlinear resonator. We
use a Temporal Coupled Mode Theory modeling to describe the coupling between
the kink antennas and the resonator

$\displaystyle i\frac{d}{dt}\tilde{a}\left(t\right)$
$\displaystyle=\left(\Omega-\frac{i}{2}D_{R}D_{R}^{T}\right)\tilde{a}\left(t\right)+iD_{R}\tilde{S}^{+}$
$\displaystyle\ket{\tilde{S}^{-}}$
$\displaystyle=K\ket{\tilde{S}^{+}}+D_{R}^{T}\tilde{a}(t)\,.$ (S7)

where for simplicity of the modeling we assume that $K=-\mathbb{I}_{3}$ (no
direct coupling between the antennas) and
$D_{R}=(\gamma_{l_{1}},\gamma_{l_{2}},\gamma_{l_{3}})$ describes the coupling
of the resonator to the three kink antennas. Each of these antennas is
connected to a corresponding $l_{n}$-vertex of the remaining graph. The non-
linear resonant frequency
$\Omega=\omega_{0}-\frac{1}{2}\nu(|{\tilde{a}}|^{2})$ models the combined
resonator-diode system, where $\omega_{0}$ is the linear angular frequency of
the resonator in the absence of the coupling with the ring-diode and might
also include radiative (or/and Ohmic) linear losses.

We assume that the scattering process does not excite higher harmonics. In
this respect, the field at the resonator and the three antennas take the form

$\tilde{a}\left(t\right)=\phi_{N}\cdot e^{-i\omega
t};\quad\quad\ket{\tilde{S}^{\pm}}=\ket{s^{\pm}}\cdot e^{-i\omega t}\,,$ (S8)

where $\ket{s^{\pm}}=(s_{l_{1}}^{\pm},s_{l_{2}}^{\pm},s_{l_{3}}^{\pm})^{T}$ is
a three-dimensional vector that describes the amplitudes of incoming (+) and
outgoing (-) waves from the antennas to the resonator. The indices $l_{n}$
label the kink-antenna that is attached to the vertex $l_{n}$ of the graph.
Substituting the above expressions in Eqs. (S7) we get the following equations
that describe the steady-state scattering fields

$\displaystyle\left(\omega-\Omega+\frac{i}{2}D_{R}D_{R}^{T}\right)\
\phi_{N}=iD_{R}\ket{s^{+}},\quad\quad\quad$ (S9)
$\displaystyle\ket{s^{-}}=-\mathbb{I}\ket{s^{+}}+D_{R}^{T}\phi_{N}.$ (S10)

We proceed by considering the coupling strengths between the resonator and the
three kink-antennas as fitting parameters. Using Eq. (S10), we express the
wave at the origin point of the cable antennas (proximity to the resonator),
as

$\phi_{N}^{(l)}=s_{l}^{+}+s_{l}^{-}=\gamma_{l}\phi_{N},$ (S11)

where the index $l=l_{n}$ labels the kink-antenna that is attached to the
vertex $l$ of the graph. The field at a generic position $x_{Nl}\in[0,L_{Nl}]$
at the cables of the kink antennas ($L_{N,l}$ are the lengths of the kink-
antennas) can be generally expressed as a combination of counter-propagating
waves. We have

$\displaystyle\psi_{l,N}(x_{lN})=s_{l}^{+}e^{-ikx_{Nl}}+s_{l}^{-}e^{ikx_{Nl}}$
(S12)

Using Eqs. (S11,S12) we write the field amplitude at the position
$x_{N,l}=L_{N,l}$ (endpoint/vertex of the kink-antenna that connects to the
rest of the graph) as

$\displaystyle\phi_{l}=-2is_{l}^{+}\sin(kL_{Nl})+\gamma_{l}e^{-ikL_{Nl}}\phi_{N},\quad
l\neq N$ (S13)

which allows us to solve for the coefficients $s_{l}^{+}$ in terms of the
field amplitudes $\phi_{l}$. We have:

$\displaystyle
s_{l}^{+}=i\frac{\csc\left(kL_{Nl}\right)}{2}\left[\phi_{l}-\gamma_{l}e^{-ikL_{Nl}}\phi_{N}\right]$
(S14)

###  I.4 Hybrid nonlinear resonator-network modeling

By combining Eqs. (S4)-(S6) from the linear subnetwork with Eq. (S13) from the
coupled mode theory analysis above, we arrive to the following set of
equations

$\left(H^{(N-1)}-iFDD^{T}F^{T}\right)F\Phi=-\phi_{N}\cdot Y\,,$ (S15)

where $D$ is the $N\times L_{0}$ graph-TL coupling matrix Eq. (S1), $F$ is a
$(N-1)\times N$ filtered matrix, with elements $F_{ij}=\delta_{i,j}$ and
$M^{(N-1)}$ is the $(N-1)\times(N-1)$ matrix with elements

$H_{\mu,\beta}^{(N-1)}=\begin{cases}-\sum_{\gamma\neq\mu}A_{\mu\gamma}\cot(kL_{\mu\gamma})&\text{if
}\mu=\beta\\\ A_{\mu\beta}\csc(kL_{\mu\beta})&\text{if
}\mu\neq\beta\end{cases}$ (S16)

where $\mu,\beta=1,\cdots,N-1$ and $\gamma=1,\cdots,N$. The $(N-1)$
dimensional vector $Y$ has elements
$Y_{\mu}=\delta_{\mu,l_{n}}\gamma_{l_{n}}\csc\left({kL}_{Nl_{n}}\right)$. It
should be noted that Eq. (S15) involves only the linear sub-network i.e. it
excludes the nonlinear vertex $N$.

Furthermore, the wave amplitude $\phi_{N}$ at the nonlinear resonator, can be
expressed in terms of the field amplitudes $\phi_{l_{n}}$ at the
$l_{n}$-vertices using Eq. (S9) where the components $s_{l}^{+}$ of the vector
$\left|s^{+}\rangle\right.$ are substituted from Eq. (S14). We have

$\displaystyle\left[\left(\Omega-i\frac{D_{R}^{T}D_{R}}{2}-\omega\right)+\frac{1}{2}\sum_{l_{n}}\gamma_{l_{n}}\csc\left(kL_{N,l_{n}}\right)e^{-ikL_{Nl_{n}}}\right]\phi_{N}$
$\displaystyle=\frac{1}{2}\sum_{l_{n}}\gamma_{l_{n}}\csc\left(kL_{N,l_{n}}\right)\phi_{l_{n}}=\frac{1}{2}Y^{T}\cdot
F\cdot\Phi$ (S17)

In fact, Eqs. (S15,S17) can be combined and written in a compact form that
follows the formulation of the main text (see Eq. (2) in the main text). In
our case the matrix ${\tilde{M}}$ takes the form

${\tilde{M}}=\left(\left[\begin{matrix}H^{(N-1)}&Y\\\
Y^{T}&H_{NN}\end{matrix}\right]-iDD^{T}\right)+\nu(|\phi_{N}|^{2})\left|N\rangle\langle
N\right|$ (S18)

where $(\left|N\rangle\langle N\right|)_{ij}=\delta_{N,j}$, and
$H_{NN}=2(\omega-\omega_{0})+iD_{R}D_{R}^{T}-\sum_{l_{n}}\left[\gamma_{l_{n}}\csc(kL_{Nl_{n}})e^{-ikL_{Nl_{n}}}\right]$.

From Eq. (S15) we can evaluate the field amplitudes at the vertices of the
linear subnetwork. We have

$F\Phi=-\left(H^{(N-1)}-iFDD^{T}F^{T}\right)^{-1}\cdot Y\cdot\phi_{N}\,$ (S19)

in analogy with Eq. (4) of the main text.

By substituting the expression $F\Phi$ from Eq. (S19) to the right side of Eq.
(S17) we come up with the equivalent expression of Eq. (5). Specifically, we
have

$\nu(|\phi_{N}|^{2})=2\left[\left(\omega_{0}-\omega-i\frac{D_{R}^{T}D_{R}}{2}\right)+\frac{1}{2}\sum_{l_{n}}\gamma_{l_{n}}\csc\left(kL_{N,l_{n}}\right)e^{-ikL_{Nl_{n}}}\right]+Y^{T}\left(H^{(N-1)}-iFDD^{T}F^{T}\right)^{-1}\cdot
Y$ (S20)

which allows us to evaluate the field intensity $|\phi_{N}|^{2}$ at the
nonlinear resonator (as a function of the incident wave angular frequency
$\omega=ck$). The accepted solutions have to satisfy the following physical
conditions

$0<|\phi_{N}|^{2}\in\mathbb{R},\quad k\in\mathbb{R}.$ (S21)

Equations (S19,S20) are crucial for evaluating the CPA wavefront.
Specifically, Eq. (S19) allows us to evaluate the relative phase and amplitude
of the incident wavefront that results in a CPA. For example, if the leads are
attached at a set of vertices $\\{\mu\\}$, then from continuity at these
vertices we get that
$I_{\mu}=\phi_{\mu}=\langle\mu|F\Phi\rangle\propto\phi_{N}$ (we always assume
that the leads are attached to linear vertices). Therefore, the relative
amplitude and phase of the injected wavefront are independent of the field
$\phi_{N}$ at the nonlinear resonator. Finally, the amplitude of these
injected wavefronts is also evaluated by Eq. (S19) by substituting the
magnitude of $\phi_{N}$ (associated with a CPA frequency) from the solution of
Eq. (S20). The global (undetermined phase) is irrelevant for the CPA solution.

Up to this point, we did not specify the form $\nu(\cdot)$ of the nonlinear
interaction. There are various options that we can adopt. For example, a Kerr
nonlinearity assumes that $\nu(|\phi_{N}|^{2})=\chi_{k}|\phi_{N}|^{2}$, while
a saturable nonlinearity assumes that
$\nu(|\phi_{N}|^{2})=z_{1}/[1+\alpha|\phi_{N}|^{2}]$ with $\chi_{k}$,
$\alpha$, and $z_{1}$ being parameters that characterize the specific
nonlinear mechanism. We point out that the above form of the saturable
nonlinearity is typical for a variety of wave systems ranging from
microwavesJeon  _et al._ (2020), opticsChristodoulides  _et al._ (1997) to
acousticsBourquard  _et al._ (2021). It turns out that the nonlinear diode
that we have used is best described by the saturable nonlinearity Jeon  _et
al._ (2020) (see last section).

##  II NL-CPA Wavefront Shaping Protocol in Coupled Mode Theory

In the main text of the paper we have experimentally established an efficient
wavefront shaping protocol applicable to achieve CPA using as a testbed,
networks of coupled coaxial cables consisting of a single nonlinear vertex. In
the previous section of the Supplementary Materials, we have detailed the
implemented protocol for the special case of networks. The developed
algorithmic protocol, however, is generic and independent of the specific
system framework. To demonstrate this generality, we turn to Coupled Mode
Theory (CMT) to describe a cavity consisting of weakly coupled resonances, of
which one, $n_{0}$, is endowed with a known nonlinear mechanism
$\nu\left(|\psi_{n_{0}}|^{2}\right)$, where we have used the notation
$\psi_{n_{0}}\equiv\left\langle n_{0}\middle|\psi\right\rangle$. The CMT
equations read

$i\frac{d}{dt}|\Psi\rangle=H_{\text{eff}}|\Psi\rangle+\nu|n_{0}\rangle\langle
n_{0}|\Psi\rangle+iD|S^{+}\rangle$ (S22a)
$\left|S^{-}\right\rangle=D^{T}\left|\Psi\right\rangle-\left|S^{+}\right\rangle$
(S22b)

The first equation describes internal dynamics of the time-dependent
scattering field $|\Psi\rangle$ in the cavity with respect to a time-dependent
incident waveform $|S^{+}\rangle$ injected at $M$ leads whose coupling to the
cavity is encoded in the purely real $N\times M$ coupling matrix $D$. The
$N\times N$ effective Hamiltonian $H_{\text{eff}}=H_{0}+\Sigma$ describes
dynamics of the linear part of the isolated system together with the self-
energy correction due to its coupling with the environment,
$\Sigma=\Delta-i\Gamma$, where the real part $\Delta$ is associated with a
resonant shift and the imaginary part $\Gamma=\frac{1}{2}DD^{T}$ describes
losses from energy leaking out to the leads [1]. The specific form of $\Sigma$
depends on the nature of the leads. Meanwhile, the basis modes of the isolated
resonances are denoted by $|n\rangle$, therefore, the term
$\nu|n_{0}\rangle\langle n_{0}|$ indicates the presence of a local nonlinear
element associated with the amplitude of the $n_{0}^{\text{th}}$ isolated
resonance. The second equation describes continuity of the field at each of
the lead ports with respect to the time-dependent outgoing waveform
$|S^{-}\rangle$. To develop the NL-CPA wavefront shaping protocol in CMT, we
perform a separation of variables on the previous time-domain equations of
motion, neglecting higher-order harmonics, $|\Psi\rangle=e^{-i\omega
t}|\psi\rangle$ and $|S^{\pm}\rangle=e^{-i\omega t}|s^{\pm}\rangle$, where
$\omega$ is the driving frequency of the incident wave. In terms of the time-
independent scattering field and wavefronts, we have the nonlinear CMT
equations of motion in the frequency domain.

$\left(H-\nu|n_{0}\rangle\langle
n_{0}|\right)|\psi\rangle=iD|s^{+}\rangle;\quad
H=\omega\mathbb{I}_{N}-H_{\text{e}ff}$ (S23a)
$|s^{-}\rangle=D^{T}|\psi\rangle-|s^{+}\rangle$ (S23b)

where $\mathbb{I}_{N}$ is the $N\times N$ identity matrix.

Similar to the approach taken in the main text, we introduce the $(N-1)\times
N$ filtering matrix $F$, which serves to extract the linear components of
$|\psi\rangle$, with the following defining properties,

$\displaystyle F^{T}F+|n_{0}\rangle\langle n_{0}|$
$\displaystyle\equiv\mathbb{I}_{N},$ (S24a) $\displaystyle FF^{T}$
$\displaystyle\equiv\mathbb{I}_{N-1},$ (S24b) $\displaystyle F|n_{0}\rangle$
$\displaystyle\equiv 0$ (S24c)

Explicitly, the elements of the filtering matrix may be represented,
$F_{ij}=\left\\{\begin{matrix}\delta_{i,j},\ \ i<n_{0}\\\ \delta_{i,j+1}\ \
i\geq n_{0}\\\ \end{matrix}\right.\ $.

We now proceed with formulating the nonlinear wavefront shaping protocol by
imposing CPA scattering conditions corresponding to zero outgoing wavefront,
$|s^{-}\rangle=0$ in eq. (S23b). Inserting the simplified continuity equation
into eq. (S23a) and re-arranging, we have,

${\widetilde{M}}|\psi\rangle=0;\quad\widetilde{M}\equiv\left(M-\nu|n_{0}\rangle\langle
n_{0}|\right),$ (S25)

where $M=\omega\mathbb{I}_{N}-\widetilde{H}_{\text{eff}}=H-iDD^{T}$ and
$\widetilde{H}_{\text{eff}}\equiv H_{\text{eff}}+iDD^{T}$ is the effective
Hamiltonian with a time-reversed self-energy correction. Notably, Eq. (S25)
has an identical form to Eq. (2) in the main text with the only nonlinear
component of $\widetilde{M}$ being its $n_{0}-{\text{th}}$ diagonal element,
$\langle
n_{0}|\widetilde{M}|n_{0}\rangle=\omega-\widetilde{\varepsilon}_{n_{0}}-\nu\left(|\psi_{n_{0}}|^{2}\right)$,
where $\widetilde{\varepsilon}_{n_{0}}\equiv\langle
n_{0}|\widetilde{H}_{\text{eff}}|n_{0}\rangle$, therefore the sub-matrix
$F\widetilde{M}F^{T}$ contains purely linear elements. Employing property
(S24a) of the filtering matrix, we separate the linear and nonlinear parts of
Eq. (S25),

$MF^{T}F|\psi\rangle=-\psi_{n_{0}}\widetilde{M}|n_{0}\rangle$ (S26)

which is Eq. (3) of the main text. By multiplying Eq. (S26) by
$\left(FMF^{T}\right)^{-1}F$ and re-arranging, we can solve for the linear
part of the field in terms of the nonlinear part of the scattering field (see
Eq. (4) of the main text),

$F|\psi\rangle=-\psi_{n_{0}}\left(FMF^{T}\right)^{-1}FM|n_{0}\rangle$ (S27)

where on the right-hand side we have again involved property Eq. (S24c). Then,
by multiplying Eq. (S26) with the inner product of $\langle n_{0}|$ and using
property (S24c), we are able to isolate the nonlinear part of the scattering
field,

$\langle
n_{0}|MF^{T}F|\psi\rangle+\psi_{n_{0}}(\omega-\widetilde{\varepsilon}_{n_{0}}-\nu)=0\,.$
(S28)

Inserting the result for the linear part of the field from eq. (S27) into this
expression, and assuming $\psi_{n_{0}}\neq 0$ which cancels, we have the
outcome,

$\langle
n_{0}|MF^{T}\left(FMF^{T}\right)^{-1}FM|n_{0}\rangle+(\omega-\widetilde{\varepsilon}_{n_{0}}-\nu)=0\,$
(S29)

which reproduces Eq. (5) of the main text and can be used to solve for the
field intensity corresponding to the nonlinear resonance and, therefore, the
field amplitude up to an arbitrary global phase,

$\nu(|\psi_{n_{0}}|^{2})=\langle
n_{0}|MF^{T}\left(FMF^{T}\right)^{-1}FM|n_{0}\rangle+\left(\omega-\widetilde{\varepsilon}_{n_{0}}\right)\,.$
(S30)

Therefore, NL-CPA scattering conditions can be identified from a single-
parameter search in $\omega$, whenever the solution for $|\psi_{n_{0}}|^{2}$
from Eq. (S30) is real and positive. The result for the nonlinear part of the
field can then be re-inserted into Eq. (S27) to obtain the rest of the NL-CPA
field. Lastly, owing to the simplicity of the CPA scattering conditions, the
incident NL-CPA wavefront can easily be acquired,
$|s^{+}\rangle=D^{T}|\psi\rangle=D^{T}F^{T}F|\psi\rangle+\psi_{n_{0}}D^{T}|n_{0}\rangle$.

##  III General Solution and Evaluation of Absorbance

In the previous section, the generic merit of our NL-CPA protocol was
demonstrated by justifying its applicability in the abstract framework of CMT.
As it turns out, the notation introduced is also useful to establish an
analytic solution to the general scattering problem, irrespective of the
scattering conditions (e.g., CPA), for systems consisting of a single local
nonlinear element. To this end, we again proceed by employing properties of
the filtering matrix to isolate the nonlinear part of the field. Inserting Eq.
(S24a) into Eq. (S23a) and re-arranging the terms, we have,

$\begin{split}\left(\omega\mathbb{I}_{N}-H_{\text{eff}}\right)F^{T}F|\psi\rangle=\\\
\quad\psi_{n_{0}}H_{\text{eff}}|n_{0}\rangle+\psi_{n_{0}}(\nu-\omega)|n_{0}\rangle+iD|s^{+}\rangle\,.\end{split}$
(S31)

Defining the filtered Green function as
$\widetilde{G}\equiv\left[F\left(\omega\mathbb{I}_{N}-H_{\text{eff}}\right)F^{T}\right]^{-1}$,
we multiply Eq. (S31) by $\widetilde{G}F$ and, utilizing property (S24c), we
obtain the linear part of the scattering field,

$F|\psi\rangle=\psi_{n_{0}}\widetilde{G}FH_{\text{eff}}|n_{0}\rangle+i\widetilde{G}FD|s^{+}\rangle\,.$
(S32)

Then, by multiplying Eq. (S31) by $\langle n_{0}|$, we can isolate the
nonlinear part of the field,

$-\langle
n_{0}|H_{\text{eff}}F^{T}F|\psi\rangle=\psi_{n_{0}}(\varepsilon_{n_{0}}+\nu-\omega)+i\langle
n_{0}|D|s^{+}\rangle\,$ (S33)

where $\varepsilon_{n_{0}}\equiv\left\langle
n_{0}\middle|H_{\text{eff}}\middle|n_{0}\right\rangle$. Inserting the linear
part (S32) to this result, we have the succinct relation,

$\beta_{0}=\psi_{n_{0}}(\nu+\beta_{1})\,,$ (S34a) where the coefficients are
defined, $\beta_{0}\equiv-i\langle
n_{0}|\left(H_{\text{eff}}F^{T}\widetilde{G}F+\mathbb{I}_{N}\right)D|s^{+}\rangle$
(S34b) $\beta_{1}\equiv\langle
n_{0}|H_{\text{eff}}F^{T}\widetilde{G}FH_{\text{eff}}|n_{0}\rangle-\left(\omega-\varepsilon_{n_{0}}\right)\,.$
(S34c)

Computing the norm-squared of Eq. (S34a) and solving for zero, denoting
$I\equiv|\psi_{n_{0}}|^{2}$,

$I|\nu(I)|^{2}+2I\text{Re}\left[\beta_{1}^{\ast}\nu(I)\right]+I|\beta_{1}|^{2}-|\beta_{0}|^{2}=0\,.$
(S35)

Which may be solved using Cardano’s formula once an explicit form of the
nonlinearity is specified and Eq. (S35) is expressed as a cubic polynomial in
$I$,

$aI^{3}+bI^{2}+cI+d=0\,.$ (S36)

In case of a Kerr-type nonlinearity of the form $\nu(I)=\chi I$, the
coefficients of Eq. (S36) are given by,

$a=|\chi|^{2};\quad b=2\text{Re}\left[\beta_{1}^{\ast}\chi\right];\quad
c=|\beta_{1}|^{2};\quad d=-|\beta_{0}|^{2}\,.$ (S37a) Meanwhile, in case of a
saturable-type nonlinearity of the form $\nu(I)=z_{1}-\frac{z_{0}}{1+\alpha
I}=\frac{\delta+\chi I}{1+\alpha I}$ with $\delta\equiv z_{1}-z_{0}$,
$\chi\equiv z_{1}\alpha$ and $\text{Re}[\alpha]>0$, the coefficients of Eq.
(S36) take the more complicated form,
$\begin{gathered}a=|\chi|^{2}+2\text{Re}[\beta_{1}^{\ast}\chi\alpha^{\ast}]+|\alpha\beta_{1}|^{2},\\\
b=2\text{Re}[\delta^{\ast}\chi+\beta_{1}^{\ast}(\chi+\delta\alpha^{\ast})]+2\text{Re}[\alpha]|\beta_{1}|^{2}-|\alpha\beta_{0}|^{2},\\\
c=|\delta|^{2}+2\text{Re}[\beta_{1}^{\ast}\delta]+|\beta_{1}|^{2}-2\text{Re}[\alpha]|\beta_{0}|^{2},\\\
d=-|\beta_{0}|^{2}\,.\end{gathered}$ (S37b)

Once the coefficients of the cubic polynomial are obtained, Cardano’s formula
provides the roots $k\in\\{0,1,2\\}$, via

$I_{k}=-\frac{1}{3a}\left(b+\zeta^{k}C+\frac{\Delta_{0}}{\zeta^{k}C}\right)\,.$
(S38a) Where $\zeta\equiv\frac{-1+i\sqrt{3}}{2}=e^{i\frac{2\pi}{3}}$ is the
primitive 3rd root of unity and the constants $\Delta_{0},\Delta_{1}$ and $C$
are provided by, $\begin{gathered}\Delta_{0}\equiv b^{2}-3ac,\\\
\Delta_{1}\equiv 2b^{3}-9abc+27a^{2}d,\\\
C\equiv\left(\frac{\Delta_{1}\pm\sqrt{\Delta_{1}^{2}-4\Delta_{0}^{3}}}{2}\right)\,.\end{gathered}$
(S38b)

Once physical solutions are recovered, corresponding to real and positive
roots $I_{k}$, $\nu(I)$ can be evaluated for the respective incident frequency
$\omega$ and wavefront $|s^{+}\rangle$. With $\nu$ now known, a conventional
linear approach becomes applicable, specifically (S23a) provides the
scattering field,

$\left|\psi\right\rangle=iG_{\\{\omega,|s^{+}\rangle\\}}D\left|s^{+}\right\rangle$
(S39)
$G_{\\{\omega,|s^{+}\rangle\\}}\equiv\left[\omega\mathbb{I}_{N}-H_{\text{eff}}-\nu|n_{0}\rangle\langle
n_{0}|\right]^{-1}$ (S40)

Then from Eq. S39, Eq. S23b provides the corresponding output wavefront,

$\left|s^{-}\right\rangle=S_{\\{\omega,|s^{+}\rangle\\}}\left|s^{+}\right\rangle\,,$
(S41)

where the nonlinear scattering function,

$S_{\\{\omega,|s^{+}\rangle\\}}\equiv-\mathbb{I}_{N}+iD^{T}G_{\\{\omega,|s^{+}\rangle\\}}D$
(S42)

is dependent on the driving frequency of the incident wave as well as its
amplitude and relative phase at each of the ports. Eventually, the absorbance
can be evaluated as:

$A=1-\frac{\langle s^{-}|s^{-}\rangle}{\langle s^{+}|s^{+}\rangle}$ (S43)

##  IV Nonlinear CPA and EP CPA in Ring Graph

Figure S1:  (a) Experimentally measured absorbance as a function of input
frequency and power $P_{1}$ controlled by the IQ modulator 1. The peak
absorption is $A^{*}=99.998$% occurring at $P_{1}^{*}\approx-17.67$ dBm,
$P_{2}^{*}\approx-21.09$ dBm, a relative phase of $\phi^{*}\approx
97^{\circ}$, and a frequency of $f^{*}\approx 6.391$ GHz. The experimental
wavefront maintained the optimal values of the power ratio $R^{*}\approx 2.2$
and relative phase $\phi^{*}\approx 97^{\circ}$. (b) Theoretically calculated
absorbance versus frequency $f$ and input power $P_{1}$ using a fixed power
ratio $R^{cpa}\approx 2.9$ and relative phase $\phi^{cpa}\approx 97^{\circ}$
that corresponds to the optimal wavefront profile evaluated from the NL-CPA
protocol. The other theoretical NL-CPA conditions are
$P_{1}^{\text{cpa}}\approx-16.1$ dBm, $P_{2}^{\text{cpa}}\approx-20.67$ dBm,
and frequency of $f^{\text{cpa}}\approx 6.397$ GHz. (c) Experimentally
measured absorbance of a broadband NL-CPA state, versus frequency and input
power $P_{1}$. The peak absorption is $A^{*}=99.55$% at frequency
$f^{*}\approx 6.34$ GHz, with $P_{1}^{*}\approx-3.5$ dBm,
$P_{2}^{*}\approx-10.90$ dBm, and relative phase $\phi_{1}^{*}=66^{\circ}$.
The associated wavefront used in these measurements is characterized by a
fixed power ratio $R^{*}\approx 5.5$ and relative phase $\phi_{1}^{*}\approx
66^{\circ}$. (d) The same as (b) for a broadband NL-CPA scenario using a
wavefront with fixed power ratio $R^{cpa}\approx 6.3$ and relative phase
$\phi^{cpa}\approx 120^{\circ}$. The NL-CPA state corresponds to input powers
$P_{1}^{\text{cpa}}\approx-3.6$ dBm and $P_{2}^{\text{cpa}}\approx-11.6$ dBm,
and a relative phase of $\phi^{\text{cpa}}\approx 120^{\circ}$ and occur at
frequency $f^{\text{cpa}}\approx 6.34$ GHz. The CPA point in all subfigures is
marked with a red arrow. In subfigure (d) we indicate with an additional green
arrow the position of the second (case (b) of main text) NL-CPA.

In Fig. S1, we present experimental and theoretical plots of absorption as a
function of frequency and input power. Specifically, Fig. S1(a) shows the
experimental results for narrow-band CPA, for fixed injected power ratio
$R^{*}\approx 2.2$ and relative phase $\phi^{*}\approx 97^{\circ}$– see also
Fig. 2(a) of the main text. The associated numerical calculation of absorbance
(see Fig. S1(b)) uses a wave with fixed injected power ratio $R^{cpa}\approx
2.9$ and relative phase $\phi^{cpa}\approx 97^{\circ}$– see also main text –
associated with the NL-CPA wavefront that was evaluated using our protocol.

Similarly, Fig. S1(c) is the experimental results for a wide-band CPA, where
the injected wavefront has fixed injected power ratio $R^{*}\approx 5.5$ and
relative phase $\phi^{*}\approx 66^{\circ}$– see also Fig. 2(b) of the main
text. The analogous theoretical analysis is shown in Fig. S1(d), demonstrating
a similar behavior as our experimental data. The calculation has been
performed using a fixed injected relative power $R^{cpa}\approx 6.3$, and
relative phase $\phi^{cpa}\approx 120^{\circ}$. These values correspond to the
optimal wavefront – see also main text – and have been evaluated using our NL-
CPA protocol.

Figure S2:  (a1)-(d1) Outgoing signal versus frequency (a1), input power
$P_{1}$ (b1), input power $P_{2}$ (c1), and relative phase $\phi_{1}$ (d1) for
the narrow-band CPA. The solid blue lines are the experimental measurements,
while the red dashed lines are the theoretical results. For each plot, we fix
the three remaining wavefront parameters at the NL-CPA conditions (see figure
caption of Fig. S1a-b). (a2)-(d2) The same analysis as in the upper row, but
now for the broadband NL-CPA scenario. The red and green dashed-lines
correspond to different wavefront (fixed) parameters associated with each
(quasi-degenerate) NL-CPAs (see green and red arrows in Fig. S1d) that have
been found using our NL-CPA protocol (see text). The grey-dashed line with
circle markers corresponds to the wavefront of the case (a) of the main text
in case the lengths of the ring graph has varies by less than $\pm 2$mm.
Figure S3:  (a) Experimentally measured absorption profile for narrow-band NL-
CPA as a function of input frequency and power $P_{1}$ from IQ modulator 1,
achieving a peak absorption of 99.97%. The corresponding parameters include
$P_{1}^{\text{cpa}}=-5.09$ dBm, $P_{2}^{\text{cpa}}=-12.90$ dBm, a relative
phase of $\phi_{1}^{\text{cpa}}=207^{\circ}$, and a frequency of
$f^{\text{cpa}}=6.371$ GHz. It should be noted that we maintain a constant
ratio between $P_{1}$ and $P_{2}$ as the power is varied. (The CPA point in
this subfigure, as well as in subsequent subfigures are marked with black
arrows.) (b) Theoretically calculated absorption profile for narrowband NL-CPA
using the hybrid model, achieving a peak absorption of 100%. The theoretical
NL-CPA conditions are characterized by $P_{1}^{\text{cpa}}=-6.6$ dBm,
$P_{2}^{\text{cpa}}=-14.35$ dBm, a relative phase of
$\phi_{1}^{\text{cpa}}=191^{\circ}$, and a frequency of $f^{\text{cpa}}=6.369$
GHz. (c) Experimentally measured absorption profile of a broadband NL-CPA
state, where the peak absorption recorded is 99.93%. The associated wavefronts
have $P_{1}^{*}\approx-19.05$ dBm, $P_{2}^{*}\approx-15.6$ dBm, a relative
phase of $\phi_{1}^{*}\approx 301^{\circ}$, and a frequency $f^{*}\approx
6.39$ GHz. (d) Theoretical analysis of the broadband NL-CPA state, where the
predicted peak absorption achieves 100%, with the corresponding input powers
as $P_{1}^{\text{cpa}}=-19.24$ dBm and $P_{2}^{\text{cpa}}=-17.79$ dBm, a
relative phase of $\phi_{1}^{\text{cpa}}=377.8^{\circ}$, and a frequency of
$f^{\text{cpa}}=6.397$ GHz.

To confirm the existence of NL-CPA and its corresponding wavefront, we
analyzed the outgoing relative signals ${\cal O}=1-A$ against each of the
remaining parameters, fixing the rest of them at their NL-CPA values. Figures
S2(a1)-(d1) show the experimental measurements (blue solid line) of ${\cal O}$
versus frequency $f$, input powers $P_{1}$ and $P_{2}$ from channels 1 and 2
(IQ-modulator 1 and 2, respectively), and the relative phase $\phi$ between
channel 1 and channel 2, respectively. At the same figures, we also report the
theoretical results (red dashed lines) derived using the two NL-CPA wavefronts
predicted by our protocol. The analogous parametric analysis for the broadband
NL-CPA scenario is illustrated in Figs. S2 (a2)-(d2). The experimental deeps
(demonstrating optimal CPA conditions) typically fall within the parameter
domain where the two NL-CPAs are predicted (red and green dashed lines
corresponding to cases (a) and (b), respectively, in the main text). The
relative phase discrepancy between the experimental and the theoretical
results, where the maximum absorption occurs, is associated with the
measurement precision of the measured lengths. For example, a small variation
of the lengths of the network by less than $\pm 2$mm can shift the emitted
signal dip close to the experimental findings without affecting the nice
agreement of the other optimal wavefront parameters (see grey-dotted lines
with circle markers in Fig. S2 (a2)-(d2) where we have used the CPA wavefront
associated with the case (a) of the main text ).

##  V NL-CPA and zero-EPDs in Tetrahedron Graph

We have repeated the same analysis as above for the case of a tetrahedron
graph. Two graph configurations SM  have been used such that a narrow-band and
a broad-band NL-CPA can be examined in detail within the range of frequency
and power variation that is available by our VNA.

The first configuration depicts a narrow-band NL-CPA scenario and it is shown
in Fig. S3(a) (experiment) and Fig. S3(b) (theory). In each of these two
figures, we kept the power ratio $R$ and relative phase $\phi$ of the injected
wavefront fixed and varied the $P_{1}$ and the injected frequency $f$. The
fixed parameters $R,\phi$ have been chosen to match the ones corresponding to
the maximum absorption scenario ($R^{*}\approx 6.03$ and $\phi^{*}\approx
207^{\circ}$ with maximum absorption $A^{*}=99.97\%$ for the experiment and
$R^{\text{c}pa}\approx 5.96$ and $\phi^{\text{c}pa}\approx 191^{\circ}$ for
the theory). The narrow-band NL-CPA occurs at $P_{1}^{*}\approx-5.09$dBm and
frequency $f^{*}\approx 6.371$GHz for the experiment and
$P_{1}^{\text{c}pa}\approx-6.6$dBm and $f^{\text{c}pa}\approx 6.369$GHz for
the theory. The comparison between the theoretical and the experimental
results confirms the efficiency of our NL-CPA protocol and establishes the
fact that the injected power can be used as an external tuning knob for
achieving CPA. The small differences between theoretical and experimental
parameters are associated with the accuracy of the extraction of the
experimental parameters (electrical lengths of the graph and the T-junctions,
evaluation of the electrical permittivity etc.).

The graph configuration in Fig. S3(c), depicts a broad-band NL-CPA scenario
which again highlights power-dependent absorption, peaking at $A^{*}\approx
99.93\%$. The corresponding theoretical results are shown in Fig. S3(d) and
agree well with the observed experimental data. We remind that in this case,
our protocol has been identified two (quasi-degenerate) NL-CPAs (see Fig. 4(b)
and related discussion in the main text). As previously, the wavefront used in
these cases maintains a fixed injected power ratio $R$ and relative phase
$\phi$ corresponding to the values associated with the optimal configuration:
for the experimental analysis we used $R^{*}\approx 0.45$ and $\phi^{*}\approx
301^{\circ}$ while for the theoretical calculations, we have used the
parameters associated with the NL-CPA of case (a) of the main text which
corresponds to $R^{\text{c}pa}\approx 0.68$ and $\phi^{\text{c}pa}\approx
377.8^{\circ}$.

Figure S4:  Relative outgoing signal (${\cal O}=1-A$) versus frequency
(a1,a2), input power $P_{1}$ (b1,b2), input power $P_{2}$ (c1,c2), and
relative phase $\phi_{1}$ (d1,d2) for the narrow-band (upper row) and
broadband (lower row) CPA respectively. For each plot, we fix the three
remaining parameters taken to be at the NL-CPA conditions (see text). The
solid blue lines are the experimental results while the dashed lines are the
NL-CPA wavefronts predicted by our algorithm. The green and red dashed lines
in the lower row are the theoretical results for the absorbance evaluated
using two different wavefronts (case (a) and case (b) of the main text)
associated with each (quasi-degenerate) NL-CPA that have been found using our
algorithm.

To confirm the existence of the NL-CPAs and their corresponding wavefront
against the predictions of our protocol, we analyzed the relative outgoing
signal ${\cal O}=1-A$ against each of the four parameters, maintaining the
remaining parameters at their NL-CPA values. Figs. S4(a)-(d) show the
experimentally measured ${\cal O}$ (blue solid lines) versus frequency $f$,
input powers $P_{1}$ and $P_{2}$ and the relative phase $\phi$ of the injected
waves, from TLs 1 and 2, respectively. At the same figures, we also report the
theoretically evaluated outgoing relative signal ${\cal O}$ using the
parameters (see main text) for each of the two NL-CPAs predicted by our
algorithm (red and green dashed lines). Due to the proximity of the two NL-
CPAs there is hardly visible any difference between them. The excellent
agreement between theoretical results and experimental data reconfirms the
applicability of our NL-CPA algorithm to predict the appropriate optimal
wavefronts that lead to extreme absorption.

We have scrutinized further the complex-zero EPD by analyzing its dependence
on the input wavefront parameters i.e. their real and imaginary frequency,
injected power $P_{2}$, and relative phase of the incident waves at TLs 1 and
2, versus the incident power $P_{1}$ from the TL 1. These results are shown in
Figs. S5(a-d) and illustrate a degeneracy of the two zero modes at the same
$P_{1}$ value. This EPD is displaced from the real frequency axis by ${\cal
I}(f)\sim 10^{-4}$, which for all practical purposes, is insignificant. As a
result, we were able to witness a super-quadratic scaling ${\cal O}\sim(\delta
f)^{\alpha}$ with a best fitting power $\alpha=3.5$ which resembles the
expected quartic line-shape of EPD-CPA Sweeney  _et al._ (2019) (see Fig. 4(c)
of the main text).

Figure S5:  The same plot as in Fig. 2(d) of the main text. The sub-figures
demonstrate the dependence of the theoretically calculated wavefront
parameters associated with a scattering zero, versus the input intensity from
channel 1 ($P_{1}=|I_{1}|^{2}$): (a) the real part of frequency
($\textrm{Re}(f)$), (b) imaginary part of frequency ($\textrm{Im}(f)$), (c)
input wave intensity from channel 2 ($P_{2}=|I_{2}|^{2}$), and (d) relative
phase $\phi$ between the injected waves at the two TLs. The dashed (solid)
lines indicate the same zero-mode in all subfigures. These modes, coalesce at
the same parameter values (light blue dashed line) forming an NL-EPD. Notice
that the formed EPD is displaced from the real frequency axis by ${\cal
I}(f)\sim 10^{-4}$.

##  VI Experimental Implementation of the Lossy Non-Linear Vertex

The nonlinear vertex consists of a cylindrical resonator (ceramics ZrSnTiO
with permittivity $\epsilon\approx 36$, height 5 mm, diameter 8 mm, resonance
frequency around $f_{0}^{(R)}\approx 6.885$ GHz and a line width
$\gamma\approx 1.7$ MHz) which is inductively coupled to a metallic ring
(diameter 3 mm) that is short- circuited to a diode (detector Schottky diode
SMS 7630-079LF, from Skyworks), see inset of Fig. 1(a) of the main text.

The coupling between the ring and the resonator occurs via the resonator’s
$z$-directional magnetic field component that is excited by the injected
signal. The latter induces a current at the ring and subsequently a voltage
across the diode whose magnitude is controlled by the magnitude of the
magnetic field. The magnitude of this voltage defines the state of the diode:
the “on” state is associated with high voltage (high incident field power) and
leads to high nonlinearities; the “off” state is associated with low voltage
(low incident power) and leads to low nonlinearities. The nonlinear resonator
is designed to operate at 6.1-6.5 GHz.

The resonator-ring structure is coupled with the rest of the graph via three
“kink” antennas [see Fig. 1(b)]. This nonlinear construction has been already
used to realize topological limiters in a coupled resonator waveguide array
Jeon  _et al._ (2020), while more recently it has been used to analyze
asymmetric transport due to nonlinearities Wang  _et al._ (2023).

##  VII Modeling and characterization of the Lossy Non-Linear Vertex

We have characterized the nonlinear resonator (i.e. the form of the saturable
nonlinearity $\nu(|\phi_{N}|^{2})$) and its coupling constants with three kink
antennas by comparing the transmission measurements with the corresponding
expressions from coupled-mode theory that describes a three-port scattering
set-up (see inset in Fig. 1a). Below we follow the methodology used in and
also Ref. Wang  _et al._ (2023).

Combining Eqs. (S9) together, we get the $3\times 3$ scattering parameters

$\displaystyle
S=-1+iD_{R}^{T}\frac{1}{\omega-\Omega+\frac{i}{2}D_{R}D_{R}^{T}}D_{R}$ (S44)

where
$\Omega=2\pi\left(f_{0}+z_{0}-\frac{z_{1}}{1+\alpha|\phi_{N}|^{2}}\right)$ and
$f_{0}\approx 6.4$GHz is the linear resonant frequency of the resonator-ring
system. The constant $z_{0}-z_{1}$ accounts mainly for losses (radiative
and/or Ohmic) due to the presence of the ring which might be different for
weak and strong input powers. From the experimental point of view, two of the
TLs have been connected with the VNA while in each measurement the third port
is coupled to a 50 Ohm terminator.

For weak input powers (e.g. $\sim-25$dBm), $\alpha|\phi_{N}|^{2}\approx 0$ and
the transmission from TL $m$ to $n$ $(m,n=1,2,3)$ takes the form

$\displaystyle|S_{nm}|^{2}=\frac{4\gamma_{n}^{2}\gamma_{m}^{2}}{[4\pi(f-f_{0}-\textrm{Re}(z))]^{2}+[\sum_{l=1}^{3}\gamma_{l}^{2}-4\pi\textrm{Im}(z)]^{2}}\,,$
(S45)

where $z=z_{0}-z_{1}$. The maximum value of $|S_{nm}|^{2}$ occurs at $f_{\rm
max}=f_{0}+\textrm{Re}(z)$. Therefore, the measurement of $|S_{nm}(f_{\rm
max})|^{2}$ can be used for extracting $f_{0}$ (we consider $\textrm{Re}(z)=0$
for simplicity) together with the
$\textrm{Im}(z)=\frac{\sum_{l=1}^{3}\gamma_{l}^{2}}{4\pi}\pm\frac{\gamma_{n}\gamma_{m}}{2\pi|S_{nm}(f_{\rm
max})|}$. Substituting these into Eq. (S45), allows us to express the
scattering parameters in terms of $\gamma_{l}$’s. The latter are extracted via
a direct fitting with the measured $|S_{nm}(f)|^{2}$ versus $f$. This analysis
allows us to evaluate also $\textrm{Im}(z)$ which satisfy the constraint
$\textrm{Im}(z)\leq\frac{\left(\sum_{l=1}^{3}\gamma_{l}^{2}\right)-\gamma_{n}^{2}}{4\pi}$.
The latter bound is enforced by the requirement that the reflectance, which in
the weak incident power limit takes the form,

$\displaystyle|S_{nn}|^{2}=1-\frac{4\gamma_{n}^{2}\left[\left(\sum_{l=1}^{3}\gamma_{l}^{2}\right)-\gamma_{n}^{2}-4\pi\textrm{Im}(z)\right]}{[4\pi(f-f_{0}-\textrm{Im}(z))]^{2}+\left[\left(\sum_{l=1}^{3}\gamma_{l}^{2}\right)-4\pi\textrm{Im}(z)\right]^{2}}$
(S46)

must be bounded from above by unity.

The extraction of $z_{0}$ is achieved by performing a similar analysis of the
transmission and reflection spectra for strong input powers. Since now
$\frac{\tilde{z}_{1}}{1+\chi_{s}|a|^{2}}\approx 0$, Eqs. (S45,S46) will need
to be modified by substituting $z\rightarrow z_{0}$. By repeating the same
procedure as previously, we extract $z_{0}$ which together with the previous
knowledge about $z=z_{0}-z_{1}$ allows us to get also $z_{1}$. Finally, the
nonlinear coefficient $\alpha$ has been estimated by a best-fitting analysis
for a set of experimental scattering data that we have collected for
intermediate values of incident power.

The above analysis led to the following best fitting values for the saturable
nonlinearity: $z_{0}=(-87.5-40.8536i)$ MHz, $z_{1}=(-87.5-32.5189)$ MHz,
$\gamma_{1}^{2}=89.5$ MHz,$\gamma_{2}^{2}=88.5$ MHz,$\gamma_{3}^{2}=64.7$ MHz,
$\alpha=(3+1.5i)\cdot 10^{9}$(mW$\cdot$s)-1. And we should note that, we
adjusted these parameters for the ring and tetrahedron graph fittings in order
to get relatively better fitting for specific configuration and frequency
ranges.

Figure S6:  Schematics of (a) ring graph and (b) tetrahedron graph.

##  VIII Experimental configuration of ring network

For the ring network sketched in Fig. S6 (a), the upper arm cable of the ring
has a geometric length $L^{\text{upper}}_{13}=291$ mm, while the lower arm
cable has length $L^{\text{lower}}_{13}=339$ mm. The left cable antenna has
length $L_{23}=277$ mm. Note that in each arm length we have included an
additional geometric length of $9$mm that estimates the effective length
introduced by the T-junction. The measurement precision for the cable lengths
is $\pm 2$mm. The extracted cable refractive index is $n=1.212+0.001i$.

Theoretical modeling parameters for ring network –. The best-fitting
parameters for the ring network configuration were identified by comparing our
experimental scattering parameters with the results from the modeling for the
frequency range of interest. We extracted $L^{\text{upper}}_{13}=286.23$ mm,
$L^{\text{lower}}_{13}=319$ mm, and $L_{23}=282.2$ mm, with cable refractive
index $n=1.212+0.001i$. We have further refined the nonlinear resonator
parameters (see previous section) such that $f_{0}=6.3933$ GHz,
$z_{0}=-87.5-41.5943i$ MHz, $z_{1}=-87.5-33.6641i$ MHz, $\alpha=(2+1.5i)\cdot
10^{9}$(mW$\cdot s)^{-1}$, $\gamma_{1}^{2}=78.4$ MHz, $\gamma_{2}^{2}=89.2$
MHz, $\gamma_{3}^{2}=82.8$ MHz.

Experimental configuration of tetrahedron network –. The tetrahedron network
shown in Fig. S6(b) is characterized by the following geometric length
configuration: $L_{14}=770$ mm, $L_{15}=18$ mm, $L_{24}=18$ mm, $L_{26}=328$
mm, $L_{34}=289$ mm, $L_{36}=339$ mm, $L_{56}=291$ mm, $L_{35}=942$ mm
(including also an estimation for the length of the phase shifter), with cable
refractive index $n=1.212+0.001i$. Note that in each $L_{ij}$ we have included
an additional geometric length of $9$mm that estimates the effective length
introduced by any T-junction attached to a cable. The measurement precision
for the cable lengths is $\pm 2$mm.

Theoretical modeling parameters for tetrahedron network – . Following the same
procedure as for the modeling of the ring network we have identified
$L_{14}=760.2$ mm, $L_{15}=20.2$ mm, $L_{24}=20.2$ mm, $L_{26}=341.2$ mm,
$L_{34}=309$ mm, $L_{36}=334.4$ mm, $L_{56}=291.9$ mm, $L_{35}=969.6$ mm, with
cable refractive index $n=1.212+0.001i$. We have further refined the nonlinear
resonator parameters (see the previous section) such that $f_{0}=6.3933$ GHz,
$z_{0}=-70.73-35.047i$ MHz, $z_{1}=-70.73-27.46i$ MHz,
$\alpha=(0.7558+1.411i)\cdot 10^{9}$(mW$\cdot s)^{-1}$, $\gamma_{1}^{2}=82.0$
MHz, $\gamma_{2}^{2}=78.0$ MHz, $\gamma_{3}^{2}=62.5$ MHz. We have also
allowed for small variations between these parameters and the one used for
narrow-band CPA modeling.