# Linear response theory of open systems with exceptional points

A. Hashemi1,2,∗ K. Busch3,4 D. N. Christodoulides5 S. K. Ozdemir6 and R. El-
Ganainy1,2,† 1 Department of Physics, Michigan Technological University,
Houghton, Michigan, 49931, USA  2 Henes Center for Quantum Phenomena, Michigan
Technological University, Houghton, Michigan, 49931, USA  3 Humboldt-
Universität zu Berlin, Institut für Physik, AG Theoretische Optik & Photonik,
D-12489 Berlin, Germany  4 Max-Born-Institut, Max-Born-Straße 2A, 12489
Berlin, Germany  5 CREOL/College of Optics and Photonics, University of
Central Florida, Orlando, Florida 32816, USA  6 Department of Engineering
Science and Mechanics, and Materials Research Institute, The Pennsylvania
State University, University Park, Pennsylvania 16802, USA  ∗ Corresponding
author<EMAIL_ADDRESS>† Corresponding author<EMAIL_ADDRESS>
###### Abstract

Understanding the linear response of any system is the first step towards
analyzing its linear and nonlinear dynamics, stability properties, as well as
its behavior in the presence of noise. In non-Hermitian Hamiltonian systems,
calculating the linear response is complicated due to the non-orthogonality of
their eigenmodes, and the presence of exceptional points (EPs). Here, we
derive a closer form series expansion of the resolvent associated with an
arbitrary non-Hermitian system in terms of the ordinary and generalized
eigenfunctions of the underlying Hamiltonian. This in turn reveals an
interesting and previously overlocked feature of non-Hermitian systems, namely
that their lineshape scaling is dictated by how the input (excitation) and
output (collection) profiles are chosen. In particular, we demonstrate that a
configuration with an EP of order $M$ can exhibit a Lorentzian response or a
super-Lorentzian response of order $M_{s}$ with $M_{s}=2,3,\ldots,M$,
depending on the choice of input and output channels.

INTRODUCTION
Resonance is a universal physical phenomenon that takes place in a large
variety of systems across a wide range of spatial and time scales. In optics,
the rapid progress in modeling and fabrication has enabled the realization of
several photonic resonator structures that can trap light for very long times
(high quality factors) Vahala (2003) or confine it in smaller domains compared
with the free-space wavelength (nano-scale mode volumes) Gramotnev and
Bozhevolnyi (2010). These devices have become indispensable components in
almost every field of optical science and engineering including but not
limited to lasers, nonlinear optics, optical communication, quantum optics,
and biophotonics. While the notion of a completely closed resonator is
sometimes used as an idealization to simplify the analysis and to gain
intuition, it is neither realistic nor desirable. Quite the opposite, it is
necessary to have open channels between the interior of the resonator and its
surrounding in order to facilitate the input/output coupling of light. Thus,
even in the absence of material loss, optical resonators are fundamentally
non-Hermitian— a fact that is often overlooked despite some early works that
considered non-Hermitian effects in optical systems Petermann (1979); Siegman
(1989). These studies demonstrated how non-Hermitian effects in lasers leave
their fingerprint on emission linewidth. Those early works, however, focused
on situations where the system does not exhibit spectral singularities known
as exceptional points (EPs) since this scenario was not relevant to the
experimental setups under study at that time.

In recent years, the interest in non-Hermitian optical structures has acquired
a new dimension following a number of theoretical studies of parity-time (PT)
symmetry in optics El-Ganainy  _et al._ (2007); Musslimani  _et al._ (2008);
Makris  _et al._ (2008) and its first experimental demonstrations Guo  _et
al._ (2009); Ruter  _et al._ (2010). This in turn has initiated intense
theoretical Schomerus (2010); Wiersig  _et al._ (2011); Schomerus (2013);
Schomerus and Wiersig (2014); Jing  _et al._ (2014); Wiersig (2014); Zhong
_et al._ (2016); Wiersig (2016); Lin  _et al._ (2016); Lü  _et al._ (2017);
Pick  _et al._ (2017); Zhong  _et al._ (2018); Jiang  _et al._ (2018); Zhong
_et al._ (2019); Arkhipov  _et al._ (2019); Bliokh  _et al._ (2019); Franke
_et al._ (2019); Rivero and Ge (2019); Reséndiz-Vázquez  _et al._ (2020);
Zhong  _et al._ (2020a, b, 2021); Tzortzakakis  _et al._ (2020a); Komis  _et
al._ (2020); Tzortzakakis  _et al._ (2020b); Kristensen  _et al._ (2020);
Khanbekyan and Wiersig (2020); Hadad and Engheta (2020); Hashemi  _et al._
(2021); Rivero  _et al._ (2021), and experimental Feng  _et al._ (2013); Peng
_et al._ (2014a); Hodaei  _et al._ (2014); Feng  _et al._ (2014); Chang  _et
al._ (2014); Peng  _et al._ (2014b); Brandstetter  _et al._ (2014); Gao  _et
al._ (2015); Zhen  _et al._ (2015); Peng  _et al._ (2016); Xu  _et al._
(2016); Doppler  _et al._ (2016); Chen  _et al._ (2017); Yoon  _et al._
(2018); Zhang  _et al._ (2018); Zhao  _et al._ (2018); Lai  _et al._ (2019);
Chen  _et al._ (2020a); Xia  _et al._ (2021) investigations of non-Hermitian
effects in photonic platforms. In contrast to earlier studies, the notion of
EPs is at the heart of these recent works. This in turn has initiated feverish
efforts seeking to explore the exotic features of wave dynamics in waveguide
and resonator geometries that exhibit EPs. For recent reviews, see El-Ganainy
_et al._ (2018); Feng  _et al._ (2017); Özdemir  _et al._ (2019); Miri and Alu
(2019); El-Ganainy  _et al._ (2019); Wiersig (2020a). Despite these intense
activities on the one hand, and the fact that the mathematics of non-self
adjoint operators is well developed on the other, some of the basic features
of complex photonic structures that are pertinent to their non-Hermitian
nature are either underestimated or misunderstood. Particularly, on one side
of the spectrum, the prevailing traditional point of view treats openness only
as a source of energy loss or gain, and hence relies completely on Hermitian
intuition to analyze the system. On the other side, some works that deal with
non-Hermitian systems exhibiting EPs tend to assume that the linear response
associated with a defective Hamiltonian (i.e. a Hamiltonian whose spectrum has
an EP or more) can be studied only within the context of perturbation theory
which is conceptually misleading even when the final results are
mathematically correct.

Figure 1: Perturbation analysis of defective Hamiltonians ($\hat{H}_{EP}$) and
its subtleties at exceptional surfaces. Two crucial steps are involved: (1)
finding a perturbation ($\hat{H}_{pt}$) that removes the degeneracy, and (2)
obtaining the limit when the perturbation vanishes. $\hat{G}(\epsilon)$ is the
Green operator. The first step can be straightforward in simple systems. b In
systems that exhibit exceptional hypersurfaces in the parameter space Zhong
_et al._ (2019); Zhou  _et al._ (2019); Zhang  _et al._ (2019a); Soleymani
_et al._ (2022); Qin  _et al._ (2021), finding such a perturbation can be a
very complex task since any perturbation that shifts the system along the
surface will fail. In addition, taking the limit when $\epsilon\rightarrow 0$
involves the cancellation of several singular terms with opposite signs. For
complex geometries with a large number of degrees of freedom, this can be a
daunting task. In general, it is not possible a priori to confirm if this
approach gives exact answer or approximate response function.

Beyond this fundamental issue, the perturbative analysis can be cumbersome and
complex for non-Hermitian arrangements with large number of resonant elements,
particularity when the spectrum contains several EPs, some of which exhibit
higher order. This situation becomes relevant for example when studying non-
Hermitian topological arrangements Gong  _et al._ (2018); Ge  _et al._ (2019);
Liu  _et al._ (2019, 2020), non-Hermitian disordered media Tang  _et al._
(2020); Tzortzakakis  _et al._ (2020c, 2021), as well as non-Hermitian spin
systems where the number of EPs scale exponentially with the system’s size
Luitz and Piazza (2019). In addition, the recent discovery of non-Hermitian
systems that exhibit exceptional surfaces (ESs) rather than EPs introduces
another hurdle for applying perturbation expansions. Figure 1b illustrates
this point. The perturbation analysis used to study a defective Hamiltonian
$\hat{H}_{EP}$ typically starts by introducing a perturbation Hamiltonian
$\epsilon\hat{{H}}_{pt}$ that removes the non-Hermitian degeneracy. The
resolvent of the resultant non-defective Hamiltonian
$\hat{\mathcal{H}}_{tot}\equiv\hat{H}_{EP}+\epsilon\hat{{H}}_{pt}$, defined as
$G(\omega;\epsilon)\equiv\left(\omega\hat{I}-\hat{\mathcal{H}}_{tot}\right)^{-1}$
where $\omega$ is the frequency and $\hat{I}$ is the unit operator/matrix, can
be then obtained using the left and right eigenstates. The resolvent of the
defective Hamiltonian is obtained by evaluating
$G_{EP}(\omega)=\lim_{\epsilon\rightarrow 0}G(\omega;\epsilon)$ . Thus, in
systems exhibiting exceptional surfaces, one first has to identify the
hypersurface of all EPs and carefully identify perturbation Hamiltonians that
forces the system out of this hypersurface, otherwise the perturbation
analysis will fail since $\hat{\mathcal{H}}_{tot}$ will be also defective (see
Fig. 1b). This task is highly non-trivial since, for systems with large
degrees of freedom, the exceptional surface is embedded in a space of high
dimensionality. Importantly, taking the above limit involves cancellations of
several infinite terms. As a result, any finitely small approximation in
applying the perturbation analysis can lead to inconsistent results.

In addition to the above mathematical difficulties in using perturbative
expansions to analyze defective Hamiltonians, the outcome of this analysis
does not provide much insight into the role of non-Hermiticity in shaping the
linear response of the system, specially when the latter exhibits many degrees
of freedom and several input and output channels. In order to illustrate some
of the subtleties arising in such systems in an intuitive way, and in doing so
motivates our work, we consider the example shown in Fig. 10. It consists of
three microring resonators that are coupled sequentially via horizontal
waveguides. An additional vertical waveguide provides access to selectively
excite the second resonator. We neglect the cross talk between the horizontal
and vertical waveguides since it can be minimized using various design
strategies Daly  _et al._ (1996); Mingaleev  _et al._ (2004); Kobayashi and
Tsuda (2005); Longhi (2015). A similar system was considered in Zhong  _et
al._ (2020a) and shown to exhibit a third-order EP in the subspace spanned by
the CW, CCW and CW modes of the resonators $\text{R}_{1,2,3}$, respectively.
When the input/output channels are selected as shown in Fig. 10a, light will
cross only cavity $R_{3}$ and hence the response features a Lorentzian
function. On the other hand, for the channels depicted in Fig. 10b, input
light will interact with both cavities $R_{2,3}$ in a series fashion before it
exists. One thus expects a super-Lorentzian response of order two. Finally,
for the input/output choice shown in Fig. 10c, light will traverse all three
cavities in series which results in a super-Lorentzian response of order three
(see supplementary note 1 for detailed analysis of this example). This rather
intuitive example reveals that a system with EP of order three can exhibit
very different response lineshapes depending on the choice of the input/output
channel configuration- a feature that to the best of our knowledge has not
been identified in non-Hermitian systems. Needless to say that in more complex
structures, identifying the response corresponding to a given input/output
channel configuration is not a trivial task.

Figure 2: Subtleties of linear response of non-Hermitian systems having EPs:
an illustrative example. a An excitation ($P_{3}$) and collection ($Q_{3}$)
scheme that leads to linear responses featuring Lorentzian lineshape. b and c
Excitation ($P_{2,1}$) and collection ($Q_{2,1}$) schemes featuring super-
Lorentzian responses of order two and three, respectively. Intuitively the
order of the super-Lorentzian response depends on how many cavities the wave
has traversed between the input and output ports. $a_{1,2,3}$ are cavity modes
for ring resonators $R_{1,2,3}$, respectively.

The lesson learned from the above simple example is of extreme importance
since many of the exotic and useful features of non-Hermitian systems with EPs
arise due to the modified spectral lineshape. For instance, the recent work on
EP-based optical amplifiers shows that the gain-bandwidth product of a
resonant optical amplifier can be enhanced by operating at an EP, provided
that the response lineshape features a super-Lorentzian response, with better
results obtained for higher-order super-Lorentzians Zhong  _et al._ (2020a).
In other settings, super-Lorentzian response could lead to narrower
lineshapes, and eventually resulting in to a stronger light-matter coupling,
which can be utilized to enhance spontaneous emission Pick  _et al._ (2017);
Zhong  _et al._ (2021); Ren  _et al._ (2021) and energy harvesting Fernández-
Alcázar  _et al._ (2021) among other potential applications. At the
fundamental level, probing the quantum noise in non-Hermitian systems requires
a proper characterization of the lineshape response at various output ports
due to vacuum fluctuations-induced noise at the input channels, including loss
and gain ones Yoo  _et al._ (2011).Thus, in light of the above observation, it
will be useful to develop a systematic, generic approach that establishes a
universal relation between the response function and the input/output channel
configuration.

Figure 3: Non-Hermitian resonant systems and their responses. a A schematic of
a resonant open system, where non-Hermiticity arises due to coupling to
external channels and the presence of optical loss or gain. As summarized in
b, the linear response of this system, as described by its resolvent operator
$\hat{G}(\omega)$ can be readily obtained in terms of the right/left
eigenvectors of the associated non-Hermitian Hamiltonian when the spectrum of
the latter does not contain any EPs. On the contrary, to date, a general
expression for $\hat{G}_{EP}(\omega)$ in the presence of EPs has not been yet
developed.

In this work, we bridge this gap and present a clear and general analysis of
the linear response of any non-Hermitian resonant system. Figure 3a presents a
schematic of one such a generic system, which consists of large number of
coupled non-Hermitian resonators subject to an arbitrary specific choice of
input/output channels. It is important to reiterate here that we focus on
situations where the eigenvalue spectrum of the system exhibits EPs. While
this situation represents a subset of the more general non-Hermitian family of
Hamiltonians (see Fig. 3b), it is by now understood that most of the novel
behavior of non-Hermitian systems arise when the system is at or near these EP
singularities. In addition, in the absence of EPs, the system’s response can
be easily obtained in terms of the left and right eigenvectors of the
Hamiltonian. It is in the case when EPs present that the dimensionality of the
eigenspace collapses which complicates the analysis, thus prompting several
authors to use perturbation methods as we described earlier. The main results
of this work can be summarized as follows: (1) The linear response of
resonator geometries that exhibit EPs can be obtained exactly without the need
for perturbative expansions; (2) The Green’s function expansion can be used to
tailor the response of the system by carefully selecting the input/output
channels; (3) The excitation channels can be classified based on their
interference properties; (4) The most efficient drive of the signal does not
necessarily correspond to mode matching between the input signal and resonant
modes. Importantly, we emphasize that even though we focus here on optical
setups, due to the well-established mathematical analogy between this latter
and other physical systems our results will be useful in analyzing and
understanding other non-Hermitian platforms such as electronic Schindler  _et
al._ (2011, 2012); Chen  _et al._ (2018); Xiao  _et al._ (2019); Sakhdari  _et
al._ (2019), acoustic Pannatoni (2011); Fleury  _et al._ (2016); Esmann  _et
al._ (2018); Ma  _et al._ (2019); Chen  _et al._ (2020b), mechanical Süsstrunk
and Huber (2016); Schomerus (2020); Gupta  _et al._ (2022), and thermal
Fernández-Alcázar  _et al._ (2021) systems.

RESULTS

High-$Q$ resonators can be strongly non-Hermitian — Before we proceed to the
main topic of this work, it is instructive to first emphasize an important
point that is sometimes overlooked in the literature, namely that non-
Hermitian effects can be significant even in optical resonators with high
quality ($Q$) factors (see for instance Peng  _et al._ (2014a); Wiersig  _et
al._ (2008)). In general, the degree of non-Hermiticity can be assessed only
by comparing the spatial scale of any non-Hermitian perturbation and its
strength to the wavelengths of the resonant modes and their spectral
separation, respectively. More specifically, if the non-Hermitian perturbation
is spatially inhomogeneous and varies at a length scale comparable or smaller
than the resonant wavelength, it can result in a measurable discrepancy
between the lifetimes associated with different modes. In addition, the
perturbation can also introduce hybridization between the bare modes. If such
a modal mixing is induced by a coupling strength that is comparable to the
frequency difference between the otherwise unperturbed modes, it could bring
the system close to an EP. In that case, two or more of the eigenvectors of
the system become very close (almost parallel). Obviously, in such a case,
non-Hermitian effects beyond linewidth broadening cannot be neglected. A
simple example that illustrates this point is the archetypal dimer
configuration composed of two resonators having identical resonant frequency
$\omega_{o}$ and two different loss factors $\gamma_{1,2}$. This system is
described by a $2\times 2$ Hamiltonian $H$ with
$H_{11,22}=\omega_{o}-i\gamma_{1,2}$ and $H_{12}=H_{21}=\kappa$ (we assume
real coupling). If $\kappa=\Delta\gamma/2$ with
$\Delta\gamma=(\gamma_{2}-\gamma_{1})$, the dimer will exhibit an EP. The
quality factor associated with this degenerate mode is
$Q=\frac{\omega_{o}}{2\pi\bar{\gamma}},$ where
$\bar{\gamma}=(\gamma_{1}+\gamma_{2})/2$. The above formulas do not impose any
restrictions on $Q$, which can be engineered to be very large. Yet the system
is highly non-Hermitian. The key observation here is that the perturbation
$\Delta\gamma$, which introduces additional loss, is spatially inhomogeneous
(affects only one site) and is comparable to the energy splitting of the
unperturbed system $2\kappa$. Examining the limit $\Delta\gamma=0$ reveals
that the eigenstates are orthogonal and the system does not exhibit any
drastic non-Hermitian effects even if $Q$ is designed to have a very low
value. Thus, the important message here is that the degree of openness alone
is insufficient to quantify if the system is highly non-Hermitian or not. This
in turn highlights the need to exercise extreme caution when dealing with non-
Hermitian systems.

Model and preparatory comments— Within the context of temporal coupled mode
formalism Fan  _et al._ (2003), a complex resonant photonic structure (see
Fig. 4) under linear conditions can be modeled by the following set of
equations:

$\begin{split}i\frac{d\ket{a(t)}}{dt}&=\hat{\mathcal{H}}\ket{a(t)}+i\hat{\Gamma}\ket{b(t)}\\\
\ket{v(t)}&=\hat{Y}\ket{b(t)}-\hat{\Gamma}^{T}\ket{a(t)},\end{split}$ (1)

where the kets $\ket{a(t)}=[a_{1}(t),a_{2}(t),\ldots,a_{N}(t)]^{T}$,
$\ket{b(t)}=[b_{1}(t),b_{2}(t),\ldots,b_{L}(t)]^{T}$ and
$\ket{v(t)}=[v_{1}(t),v_{2}(t),\ldots,v_{L}(t)]^{T}$ represent the modal
amplitudes of the resonant modes and the input and the output channels,
respectively. The $N\times N$ time-independent non-Hermitian matrix
Hamiltonian $\hat{\mathcal{{H}}}$ characterizes coupling between the different
resonant states, whereas the $L\times L$ matrix $\hat{Y}$ quantifies the
direct scattering between incoming and outgoing channels. Finally, the
$N\times L$ matrix $\hat{\Gamma}$ describes the coupling between the $N$
resonant modes and the $L$ input/output channels. To simplify the notations,
we also define $\ket{f(t)}\equiv i\hat{\Gamma}\ket{b(t)}$. The general
solution to Eq.(1) that takes into account the transient response can be
obtained by using Laplace transform. Here, however, we are interested in the
steady state response $\ket{A(\omega)}$, which can be expressed in terms of
the frequency domain resolvent (sometimes also called Green’s operator or
function) $\hat{G}(\omega)\equiv(\omega\hat{I}-\hat{\mathcal{H}})^{-1}$ (i.e.,
$\hat{I}$ is the unit operator) as:

$\ket{A(\omega)}=\hat{G}(\omega)\ket{F(\omega)},$ (2)

where $\ket{A(\omega)}\equiv\mathcal{FT}(\ket{a(t)})$ and
$\ket{F(\omega)}\equiv\mathcal{FT}(\ket{f(t)})$ with $\mathcal{FT}(\cdot)$
denoting the Fourier transform. Before we proceed, we emphasize that the
existence of a finite response (i.e. non-diverging resolvent) is not always
guaranteed as we will discuss in more detail later.

Figure 4: A schematic of an open resonant structure with an EP and multiple
input/output channels. As indicated by the structure of the eigenvectors
$\\{\ket{\psi_{n}}\\}$, the spectrum of this resonant systems is assumed to be
of dimension $N$ and having an EP of order $M$ that arises due to
gain/absorption or coupling to $L$ input/output ports. $\ket{b_{i}}$ is the
input signal from $i^{th}$ input port, and $\ket{v_{i}}$ is the output signal
from $i^{th}$ output port.

To keep the discussion focused, we will only consider an excitation vector
$\ket{f(t)}$ with separable time dependence and spatial profile, i.e.
$\ket{f(t)}=s(t)\ket{u}$ where $s(t)$ is a scalar function of time and
$\ket{u}=[u_{1},u_{2},\ldots,u_{N}]^{T}$ is a time independent excitation
profile vector. In the frequency domain, the excitation vector thus takes the
form $\ket{F(\omega)}=S(\omega)\ket{u}$. The eigenvectors of the non-Hermitian
$\hat{\mathcal{H}}$ defined by the set
$\\{\ket{\psi^{r}_{n}}:\hat{\mathcal{H}}\ket{\psi^{r}_{n}}=\Omega_{n}\ket{\psi^{r}_{n}},\braket{\psi^{r}_{n}}{\psi^{r}_{n}}=1\\}$
have the following two important properties: (1) The eigenvalues $\Omega_{n}$
are in general complex; and (2) The eigenvectors $\ket{\psi^{r}_{n}}$ do not
need to be orthogonal. In the absence of EPs, the eigenvectors form a complete
basis, hence we can represent any input profile using the expansion
$\ket{u}=\sum_{n=1}^{N}c_{n}\ket{\psi_{n}}$, which in turn reduces Eq.(2) to:

$\frac{\ket{A(\omega)}}{S(\omega)}=\sum_{n=1}^{N}\frac{c_{n}}{\omega-\Omega_{n}}\ket{\psi^{r}_{n}}.$
(3)

Note, however, that the expansion coefficients $c_{n}$ cannot be calculated
using the usual projection $\braket{\psi_{n}^{r}}{u}$. Instead, one has to
employ the left eigenvectors of $\hat{\mathcal{H}}$, defined by
$\\{\bra{\psi^{l}_{n}}:\bra{\psi^{l}_{n}}\hat{\mathcal{H}}=\Omega_{n}\bra{\psi^{l}_{n}},\braket{\psi_{m}^{l}}{\psi_{n}^{r}}=\delta_{n,m}\\}$,
to obtain $c_{n}=\braket{\psi^{l}_{n}}{u}$ (see supplementary note 2 for more
details). This is known as bi-orthogonal projection. The alerted reader will
notice that we dropped the normalization condition
$\braket{\psi^{l}_{n}}{\psi^{l}_{n}}=1$ from the definition of
${\bra{\psi^{l}_{n}}}$. In fact, it can be shown that, for non-normal
matrices, the conditions $\braket{\psi_{m}^{l}}{\psi_{n}^{r}}=\delta_{n,m}$,
$\braket{\psi_{n}^{r,l}}{\psi_{n}^{r,l}}=1$ cannot be satisfied simultaneously
Siegman (1989) (see supplementary note 3).
By using
$c_{n}\ket{\psi_{n}^{r}}=\braket{\psi_{n}^{l}}{u}\ket{\psi_{n}^{r}}=\ket{\psi_{n}^{r}}\braket{\psi_{n}^{l}}{u}$,
and by assuming that the non-orthogonal basis are complete, we can express the
resolvent as Pick  _et al._ (2017); Schomerus (2020):

$\hat{G}(\omega)=\sum_{n=1}^{N}\frac{\ket{\psi^{r}_{n}}\bra{\psi_{n}^{l}}}{\omega-\Omega_{n}}.$
(4)

Even without considering the effect of EPs, the above results already lead to
counterintuitive results. For instance, energy transfer from an excitation
signal to a particular eigenmode of the system can still take place even when
their Hermitian overlap vanishes. In order to illustrate this, consider an
input signal
$\ket{u}=\frac{\ket{\psi^{r}_{m}}}{\braket{\psi^{r}_{n}}{\psi^{r}_{m}}}-\ket{\psi^{r}_{n}}$
with $\braket{\psi^{r}_{m}}{\psi^{r}_{n}}\neq 0$ (due to non-Hermiticity).
Clearly, $\braket{\psi^{r}_{n}}{u}=0$ yet the signal $\ket{u}$ will excite the
state $\ket{\psi^{r}_{n}}$. The converse is also true. An input given by
$\ket{u}=\ket{\psi^{r}_{m}}$ will excite only the state $\ket{\psi^{r}_{m}}$
even though it may have a finite overlap with other states.

Defective spectrum does not mean defective response— We now consider the case
when the spectrum of the Hamiltonian $\hat{\mathcal{H}}$ contains an EP. For
generality, we assume that the EP is of order $M$, i.e. formed by the
coalescence of $M$ eigenstates. Such a Hamiltonian operator is said to be
defective (i.e., the eigenstates of the Hamiltonian do not form a complete
basis). In the literature of non-Hermitian optics, the response of a system
described by a defective $\hat{\mathcal{H}}$ is sometimes studied using
perturbative approaches Mailybaev and Seyranian (2004) (see for instance Pick
_et al._ (2017)). While these perturbative expansions eventually lead to
correct conclusions, they complicate the analysis and give the impression that
the resultant formulas are only approximations. As we describe below, this is
actually not the case. On the contrary, one can use exact, non-perturbative
expansions to arrive at the same final results in a straightforward fashion.
In this section, we focus on the case when the system exhibits only one EP we
discuss the more general case in the supplementary material.

To proceed, let us denote the eigenvalue and eigenvector corresponding to an
EP by $\Omega_{EP}=\Omega^{R}_{EP}+i\Omega^{I}_{EP}$ and
$\ket{\psi^{r}_{EP}}$, respectively. In order to avoid any confusion, we will
also refer to the Hamiltonian and its corresponding resolvent in this case by
$\hat{H}_{EP}$ and $\hat{G}_{EP}(\omega)$. Note that, since the Hamiltonian
$\hat{H}_{EP}$ is defective, $\hat{G}_{EP}(\omega)$ cannot be expressed using
a simple expansion similar to that of Eq.(4). Perturbative methods such as
Newton-Puiseux series Mailybaev and Seyranian (2004) have been used in the
literature to mitigate this problem and obtain an expansion for
$\hat{G}_{EP}(\omega)$. What is often overlooked, however is that even for a
defective $\hat{H}_{EP}$, the operator $(\omega\hat{I}-\hat{H}_{EP})$ can
still be invertible. This key observation can tremendously simplify the
analysis in many situations. Particularly, when an optical resonator system
operates in the passive mode (i.e., no optical gain) or even in active mode
below the lasing threshold (i.e., as in amplifiers), the complex eigenvalue
lies in the lower half of the complex plane, i.e. $\Omega^{I}_{EP}<0$. On the
other hand, probing the response of such a system is done by using external
excitation that has a real frequency, i.e. along the real axes. In this case,
the corresponding eigenvalue of $\hat{G}_{EP}(\omega)$, which is given by
$\mu_{EP}=\left[\omega-(\Omega^{R}_{EP}+i\Omega^{I}_{EP})\right]^{-1}$, does
not diverge even at resonance when $\omega=\Omega^{R}_{EP}$. As a result, one
can evaluate $\hat{G}_{EP}(\omega)$ by direct matrix inversion for any value
of the real frequency $\omega$. In principle, this information is sufficient
for characterizing the linear response of the resonator, but it does not
provide much insight into the interplay between the excitation profile and the
response. In addition, one may wonder about the fate of the expansion in (4).
As we mentioned earlier, Newton-Puiseux series is often used to generalize
this expression. We now show that this generalization is in fact exact and
does not employ any perturbation analysis. To do so, we first note that the
nondegenerate eigenvectors of an $N\times N$ matrix $\hat{G}_{EP}$ that has an
EP of order $M$ span only a reduced $N-M$ dimensional space, call it
$\mathcal{D}_{\phi}$. We will denote the missing domain by $\mathcal{D}_{EP}$.
In order to form a complete basis, we follow the standard Jordan chain
procedure Shafarevich and Remizov (2013), defined by the set of vectors that
satisfy the following recursive equations:

$\displaystyle(\hat{H}_{EP}-\Omega_{EP}\hat{I})\ket{J_{1}^{r}}$
$\displaystyle=0$ (5)
$\displaystyle(\hat{H}_{EP}-\Omega_{EP}\hat{I})\ket{J_{2}^{r}}$
$\displaystyle=\chi_{2}\ket{J_{1}^{r}}$ $\displaystyle\vdots$
$\displaystyle(\hat{H}_{EP}-\Omega_{EP}\hat{I})\ket{J_{M}^{r}}$
$\displaystyle=\chi_{M}\ket{J_{M-1}^{r}}.$

In the above, we used the notation $\ket{J_{1}^{r}}\equiv\ket{\psi_{EP}^{r}}$
for clarity. The constants $\chi^{\prime}s$ are introduced to ensure the
consistency of the physical dimensions, and their values are chosen to achieve
normalization, i.e. $\braket{J_{n}^{r}}{J_{n}^{r}}=1$ for any integer
$n=1,2,\ldots,M$. The vectors $\ket{J_{n}^{r}}$ are generalized right
eigenvectors of the operator $\hat{H}_{EP}$, i.e. they satisfy the eigenvalue
problem $(\hat{H}_{EP}-\Omega_{EP}\hat{I})^{n}\ket{J_{n}^{r}}=0$ which implies
that $\braket{\psi_{m}^{l}}{J_{n}^{r}}=0$ for any $n$ and
$\bra{\psi_{m}^{l}}\neq\bra{\psi_{EP}^{l}}$ (see supplementary note 2). Since
the vectors $\ket{J_{n}^{r}}$ are linearly independent by construction (see
supplementary note 2 for a brief proof), it follows that they span the domain
$\mathcal{D}_{EP}$, and thus complete the basis. Note however that while
$\ket{J_{1}^{r}}$ is unique (up to a constant), there is a freedom in choosing
the set of other vectors $\ket{J_{n}^{r}}$ for $n>1$. Intuitively, this
situation is similar to fixing the $z$ axis in three dimensions and rotating
the $x$ and $y$ axes in the $x-y$ plane around the origin. Thus, in general
extra normalization conditions are required in order to fix the choice of the
vectors $\ket{J_{n}^{r}}$. Here, we will not be concerned with the exact
orientation of $\ket{J_{n}^{r}}$. Another important observation is that some
of the $\ket{J_{n}^{r}}$ vectors are self-orthogonal (see supplementary note
2). Thus, while any arbitrary input can be decomposed according to
$\ket{u}=\sum_{n=1}^{N-M}c_{n}\ket{\psi_{n}^{r}}+\sum_{m=1}^{M}d_{m}\ket{J_{m}^{r}}$
with the constants $c_{n}=\braket{\psi_{n}^{l}}{u}$ evaluated using
biorthogonality as before, the coefficients $d_{m}$ cannot be obtained
directly using the same strategy. However, as we show in supplementary note 2,
one can use the vectors of ($\bra{J_{m}^{l}}$) to define another set of
vectors ($\bra{\tilde{J}_{m}^{l}}$) that satisfy the relation
$\braket{\tilde{J}_{m}^{l}}{J_{n}^{r}}=\delta_{m,n}$ and hence
$d_{m}=\braket{\tilde{J}_{m}^{l}}{u}$.

Figure 5: Structure of the eigenspace and its fingerprint on the linear
response. a The underlying vector space associated with the non-Hermitian
Hamiltonian $\hat{\mathcal{H}}$ can be divided into two subspaces:
$\mathcal{D}_{\phi}=\\{\ket{\psi_{n}^{r}}:n=1,2,\ldots,N-M\\}$ and
$\mathcal{D}_{EP}=\\{\ket{J_{m}^{r}}:m=1,2,\ldots,M\\}$. The former is spanned
by the non-degenerate right eigenvectors of $\hat{\mathcal{H}}$, while the
latter is spanned by the right generalized eigenvectors. Note that in this
classification, the degenerate (or exceptional) vector belongs to
$\mathcal{D}_{EP}$. The dual spaces $\mathcal{D}_{\phi}^{d}$ and
$\mathcal{D}_{EP}^{d}$ are defined in a similar fashion for the left ordinary
and generalized eigenvectors. b Pictorial representation of the linear
response associated with a non-Hermitian system having an EP. Typical
Lorentzian response arises due to coupling between an input and an output
channel that belong to the same modal class. On the other hand, super-
Lorentzian responses emerge when the input signal matches a particular
generalized eigenmodes of certain order while the output signal matches a
lower order generalized eigenvector (including the exceptional vector)
according to Eq. 6. The arrows, together with the legends, illustrate a few
possible responses explicitly. The symbol $\text{L}^{m}$ indicates a super-
Lorentzian response of order $m$, i.e. a Lorentzian raised to the power $m$.

Next, given the above input signal $\ket{u}$, we seek a similar expansion of
the output, i.e. in the form:
$\hat{G}_{EP}\ket{u}=\sum_{n=1}^{N-M}\tilde{c}_{n}\ket{\psi_{n}^{r}}+\sum_{m=1}^{M}\tilde{d}_{m}\ket{J_{m}^{r}}$,
or equivalently
$(\omega\hat{I}-\hat{H}_{EP})\left\\{\sum_{n=1}^{N-M}\tilde{c}_{n}\ket{\psi_{n}^{r}}+\sum_{m=1}^{M}\tilde{d}_{m}\ket{J_{m}^{r}}\right\\}=\sum_{n=1}^{N-M}c_{n}\ket{\psi_{n}^{r}}+\sum_{m=1}^{M}d_{m}\ket{J_{m}^{r}}$.
By applying the operator $\omega\hat{I}-\hat{H}_{EP}$ to each term inside the
summation and rearranging, we obtain a sum of the form
$\sum_{n=1}^{N-M}C_{n}\ket{\psi_{n}^{r}}+\sum_{m=1}^{M}D_{m}\ket{J_{m}^{r}}=0$.
By noting that all the vectors $\ket{\psi_{n}^{r}}$ and $\ket{J_{m}^{r}}$ are
linearly independent, we find that the above relation can be satisfied if and
only if all the coefficients $C_{n}=D_{m}=0$ for every $n$ and $m$. This
finally leads to the expansion (see supplementary note 4 for details):

$\displaystyle\hat{G}_{EP}=\sum_{n=1}^{N-M}\frac{\ket{\psi^{r}_{n}}\bra{\psi_{n}^{l}}}{\omega-\Omega_{n}}+\sum_{m=1}^{M}\sum_{k=m}^{M}\alpha_{k}^{(m)}\frac{\ket{J^{r}_{m}}\bra{\tilde{J}^{l}_{k}}}{(\omega-\Omega_{EP})^{k-m+1}},$
(6)

where the coefficients $\alpha$’s are defined by the recursive relations
$\alpha_{m}^{(m)}=1,\ \alpha_{k}^{(m)}=\alpha_{k-1}^{(m)}\chi_{k}$ for
$m<k\leq M$ (see supplementary note 4).

We now pause to make several comments on the above expression. Firstly, the
expansion series is finite and hence no convergence analysis is needed.
Secondly, due to the same reason (finite terms in the series), its
completeness is guaranteed. Importantly, the above expression is exact and
non-perturbative despite the fact that the spectrum of $\mathcal{\hat{H}}$
contains an EP. Another important observation is that the expansion in Eq. (6)
is not unique: a different choice of the vectors $\ket{J_{m}^{r}}$ with $m>1$
will lead to a different series (though more complicated one). Thirdly, unlike
previous works that considered small systems Sunada (2018), expressions for
the resolvent that applies only in the vicinity of EPs Heiss (2015) or
expansions of the resolvent as a power series of the Hamiltonian itself Kato
(1995), the above expansion is valid everywhere in the frequency domain for
any system with arbitrary size and is expressed in terms of the eigenvector
and canonical vectors. Finally, we emphasize a crucial point: expression (6)
for the resolvent is evaluated for a system “with” an EP not “at” an EP. This
semantic difference has caused confusion in the literature, mainly conveying
an impression that these systems can be studied only within the context of
perturbation analysis. An EP is a characteristic of the system itself, not the
probe. The former can contain an EP in its spectrum that lies in the complex
plane away from the real axes and still be probed with a signal that has a
real frequency to obtain finite response without any singularities or
divergences. On the other hand, probing the system at an EP entails either
using a probe with complex frequency Baranov  _et al._ (2017); Trainiti  _et
al._ (2019); Zhong  _et al._ (2020c) or supplying enough gain to bring the
complex exceptional eigenvalue to the real axis. In both cases, the resolvent
will diverge, which corresponds to the fact that the amplitude of the
oscillations will grow indefinitely. In reality however, this does not happen
because nonlinear effects regulate the dynamics (think of gain saturation in
laser systems for instance).

Super-Lorentzian frequency response— In realistic configurations, a resonant
structure interacts with its environment via certain scattering or coupling
channels defined by the geometry. One can either probe the system via
individual channels or by excitation profiles that are superpositions of
several channels. At the abstract level, the concepts of channel and
excitation profile are the same since they are just related by unitary
transformations. In this section, we will focus on channels that directly
excite the vectors $\ket{\psi_{n}^{r}}$ and $\ket{J_{m}^{r}}$ and analyze
their scattering/coupling characteristics. We begin by rewriting the
expression in Eq.(6) after unfolding the double summation:

$\hat{G}_{EP}(\omega)=\sum_{n=1}^{N-M}\frac{\ket{\psi^{r}_{n}}\bra{\psi_{n}^{l}}}{\omega-\Omega_{n}}+\sum_{k=1}^{M}\frac{\alpha_{k}^{(1)}\ket{J_{1}^{r}}\bra{\tilde{J}_{k}^{l}}}{(\omega-\Omega_{EP})^{k}}+\sum_{k=2}^{M}\frac{\alpha_{k}^{(2)}\ket{J_{2}^{r}}\bra{\tilde{J}_{k}^{l}}}{(\omega-\Omega_{EP})^{k-1}}+\cdots+\frac{\ket{J_{M}^{r}}\bra{\tilde{J}_{M}^{l}}}{\omega-\Omega_{EP}}.$
(7)

Clearly, an input signal with a profile $\ket{u}=\ket{\psi_{n}^{r}}$ will only
excite the mode $\ket{\psi_{n}^{r}}$ with a Lorentzian response centered at
$\Omega_{n}$ as expected. Similarly, an input that matches the exceptional
vector $\ket{u}=\ket{J_{1}^{r}}$ will excite only the mode $\ket{J_{1}^{r}}$,
also with a Lorentzian response. A more interesting situation arises when the
excitation profile coincides with a higher order Jordan vector, say
$\ket{u}=\ket{J_{k}^{r}}$. In that case, according to Eq. (7), the exceptional
eigenvector will be excited with a frequency response that features an
$k^{th}$ order super-Lorentzian lineshape (i.e. a Lorentzian function raised
to the power $k$). Additionally, each of the states $\ket{J_{m}^{r}}$ with
$1<m\leq k$ (which are not eigenstates of $\hat{H}_{EP}$) will be also excited
with a frequency response that corresponds to a super-Lorentzian of order
$k-m+1$. At this point, it is important to reiterate our previous comment on
the freedom of choosing the set $\ket{J_{m}^{r}}$. While $\ket{J_{1}^{r}}$ is
unique, the vectors $\ket{J_{m}^{r}}$, $1<m\leq M$ are not. The consequences
of this observation are not trivial. For instance, in order to excite the
exceptional vector with a certain super-Lorentzian response, one can choose
from a continuous manifold of excitation profiles. To illustrate this, we
consider the excitation of $\ket{J_{1}^{r}}$ with a second order super-
Lorentzian response. This can be done by using the input
$\ket{u}=\ket{J_{2}^{r}}+x\ket{J_{1}^{r}}$, where $x$ is a free parameter. We
anticipate that these general results, which are illustrated schematically in
Fig. 5b, will be instrumental in developing a more comprehensive understanding
of the quantum limits of several non-Hermitian optical devices (such as
lasers, amplifiers, and sensors) operating at EPs. Equally important, the
above analysis provides a complete picture of how the frequency response of
non-Hermitian systems with EPs scales as a function of the input profile,
which can be of great utility in engineering devices that rely on this
feature, such as optical amplifiers with relaxed gain-bandwidth restrictions
Metelmann and Clerk (2014); Zhong  _et al._ (2020a). In addition, our analysis
also can be used to classify the input channel according to their interference
effects inside the resonant system as we explain in details in supplementary
note 5 and 6. Finally, we note that, so far we have focused on situations
where the system has only one EP. Supplementary note 7 discusses the case when
the spectrum of the relevant Hamiltonian has multiple EPs.

Connection with experiments—Here we present illustrative realistic examples
that demonstrate some of the results discussed in this work as well as the
power and insight provided by our formalism which expresses the resolvent
operator as an exact expansion series of the right/left canonical vectors.

Our first example is depicted in Fig. 6a. It consists of a microring resonator
evanescently coupled to two identical waveguides, with one port of, say,
$\text{W}_{1}$ terminated by a mirror. This geometry was introduced in Zhong
_et al._ (2019) and shown to exhibit an exceptional surface with potential
applications for sensing and controlling spontaneous emission Zhong  _et al._
(2021). In addition, it has been recently implemented in microtoroid
resonators with the feedback realized using a fiber loop mirror Soleymani  _et
al._ (2022). It is described by the following set of equations:

$\displaystyle i\frac{d}{dt}\left[\begin{matrix}a_{1}(t)\\\ a_{2}(t)\\\
\end{matrix}\right]$
$\displaystyle=\left[\begin{matrix}\omega_{o}-2i\gamma&0\\\
\kappa&\omega_{o}-2i\gamma\\\
\end{matrix}\right]\left[\begin{matrix}a_{1}(t)\\\
a_{2}(t)\end{matrix}\right]+i\sqrt{2\gamma}\left[\begin{matrix}b_{1}(t)\\\
b_{2}(t)\\\ \end{matrix}\right]$ $\displaystyle Q_{1}$
$\displaystyle=-\sqrt{2\gamma}\ a_{2}$ $\displaystyle Q_{2}$
$\displaystyle=P_{1}-\sqrt{2\gamma}\ a_{2},$ (8)

where, $\omega_{o}$ is the resonant frequency, $\gamma$ is the decay rate of
the resonant mode into each waveguide. In addition,
$\kappa=-2i\gamma|r|e^{i\phi}$, where $|r|$ is the absolute value of the
mirror amplitude reflection coefficient, and $\phi\equiv 2\beta L+\phi_{r}$ is
a phase factor that quantifies the phase of the mirror reflectivity,
$\phi_{r}$, and its distance from the resonator, $L$, where $\beta$ is the
propagation constant in the waveguide.

The resolvent in this case, which we will denote by $G_{2}$ can be evaluated
in closed form and is given by:

$G_{2}(\omega)=\left[\begin{matrix}\frac{1}{\Delta\omega+2i\gamma}&0\\\ &\\\
\frac{\kappa}{\left(\Delta\omega+2i\gamma\right)^{2}}&\frac{1}{\Delta\omega+2i\gamma}\end{matrix}\right],$
(9)

where $\Delta\omega=\omega-\omega_{o}$. It is straightforward to show that
$\ket{\psi_{EP}^{r}}\equiv\ket{J_{1}^{r}}=[0,1]^{T}$ and
$\ket{J_{2}^{r}}=\frac{\chi}{\kappa}[1,0]^{T}$. Note that
$\braket{J_{1}^{r}}{J_{1}^{r}}=1$, while $\braket{J_{2}^{r}}{J_{2}^{r}}\neq
1$, i.e. the latter is not normalized. If we now consider the two normalized
inputs $\ket{b_{1,2}}=\alpha_{1,2}\ket{J_{1,2}^{r}}e^{-i\omega t}$ with
$\alpha_{1}=1$ and $\alpha_{2}=\frac{\kappa}{\chi}$ we find that the
normalized stored energies are given by
$\braket{E_{1}}=\frac{2\gamma}{\Delta\omega^{2}+4\gamma^{2}}$ and
$\braket{E_{2}}=\frac{2\gamma}{\Delta\omega^{2}+4\gamma^{2}}\left[1+\frac{4\gamma^{2}|r|^{2}}{\Delta\omega^{2}+4\gamma^{2}}\right]$.
Thus,
$\eta\equiv\frac{\braket{E_{2}}}{\braket{E_{1}}}=1+\frac{4\gamma^{2}|r|^{2}}{\Delta\omega^{2}+4\gamma^{2}}>1$.
At resonance when $\Delta\omega=0$, we find that $\eta=2$ when a completely
reflecting mirror, $r=1$, is used.

Figure 6: Optical energy inside a photonic resonator with a second order EP. a
A schematic of a photonic structure that exhibits a second order EP: it
consists of a microring resonator evanescently coupled to two identical
waveguides, one of which is terminated with a mirror at port $\text{W}_{1}$.
The exceptional eigenstate $\ket{J_{1}^{r}}=\ket{\psi_{EP}^{r}}$ can be
excited through port $\text{P}_{1}$, while the generalized eigenstate
$\ket{J_{2}^{r}}$ can be excited via port $\text{P}_{2}$. b Plot of $\eta$
(enhancement factor of stored energy in the microring) when the system is
excited by an input that matches $\ket{J_{2}^{r}}$ compared to an excitation
matching $\ket{\psi_{EP}}$ as a function of frequency detuning near resonance
as obtained by full-wave simulations (black dots) and the closed form
expression (red line). c and d Distributions of the electric field amplitudes
under excitation either from ports $\text{P}_{1}$ or $\text{P}_{2}$ are
plotted. As expected from the analysis, the case when
$\ket{u_{1}}=\ket{\psi_{EP}^{r}}$ leads to energy storage in both the CW and
CCW modes as evidenced by the standing wave pattern in d. All simulations were
performed by using the finite element method available from the COMSOL
software package. The optical parameters of the structure and the simulations
details are discussed in supplementary note 8.

To confirm these predictions, we consider a realistic photonic implementation
of the structure of Fig. 6a as explained in detail in supplementary note 8. It
is straightforward to show that excitations from ports $\text{P}_{1,2}$ are
mode-matched with $\ket{J_{1,2}^{r}}$, respectively. Figure 6b plots the value
of $\eta$ as a function of the frequency detuning as obtained from the full-
wave simulations (black dots) as well as by direct substitution in the closed
form expression for $\eta$ described above (red line). On the other hand,
Figs. 6c and d depict the steady state field distribution as obtained by full-
wave analysis (see supplementary note 8 for the details of the optical
parameters of the structure used in the simulations) inside the system under
the two different excitations at resonance. In order to study the spectral
response of this configuration, we plot the scattering coefficients
$|\frac{Q_{1}}{P_{1}}|^{2}$ and $|\frac{Q_{2}}{P_{2}}|^{2}$ in Figs. 7a and b,
where the black dots represent data obtained from full wave simulations and
red line indicate theoretical results. As predicted by our analysis [see Eq.
(Linear response theory of open systems with exceptional points)],
$|\frac{Q_{1}}{P_{1}}|^{2}$ which corresponds to an excitation and collection
from the exceptional vector features a Lorentzian response, while
$|\frac{Q_{2}}{P_{2}}|^{2}$ which corresponds to an excitation matching
$\ket{J_{2}^{r}}$ and collection matching $\ket{\psi_{EP}^{r}}$ follows a
super-Lorentzian of order two. This simple but intuitive example demonstrate a
subtle property of open systems, namely that their response lineshape is not
unique but rather depends on the input/output channel configuration. Not only
this feature arises naturally from our analysis, but the exact expansion
series for resolvent operator [see Eq. (6)] allows us to tailor this response
at will by selecting the appropriate coupling channels. Importantly, we stress
that this scenario is different from the previous works in which a control
parameter, such as coupling strength, frequency detuning, or loss imbalance
between resonance modes, is tuned to feature a Fano, EIT, or an ATS lineshape
Li  _et al._ (2011); Peng  _et al._ (2014c); Wang  _et al._ (2020a); Ma  _et
al._ (2021). In contrast, for the system studied here, there is no such
control parameter: one can obtain a Lorentzian or a squared-Lorentzian
lineshape by choosing the appropriate input and output channel pairs.

Figure 7: Illustration of the Lorentzian and square-Lorentzian responses of
the system. a An input signal is launched form port $\text{P}_{1}$ (see Fig.
6a) and the normalized output power $|\frac{Q_{1}}{P_{1}}|^{2}$ with
Lorentzian distribution is observed at port $\text{Q}_{1}$. b For an input
signal from port $\text{P}_{2}$ a square-Lorentzian response for the
normalized output power $|\frac{Q_{2}}{P_{2}}|^{2}$ is collected at port
$\text{Q}_{2}$. Red solid line shows the theoretical results while black dots
show results obtained through simulations. Figure 8: Spontaneous emission at
exceptional surfaces. a A photonic system consisting of a quantum dot embedded
in a microring resonator evanescently coupled to a waveguide with an end
mirror that induces a unidirectional coupling between these modes. The
emission from the quantum dot couples to the CW and CCW modes ($a_{1,2}$) of
the ring and eventually detected by the detectors $D_{1,2}$. b The ratio
$\eta_{c}$ of the stored energy in the CCW mode to the stored energy in the CW
wave is plotted for the resonant frequency $\omega=\omega_{o}$. c The
normalized power detected by $D_{1}$ has a Lorentzian distribution. d The
normalized power detected by $D_{2}$ for $\Delta\phi=0$ (red curve) and
$\Delta\phi=\pi$ (blue curve) are shown for $|r|=1$. As explained in the text,
the detected power spectrum at $D_{2}$ features an interference between a
Lorentzian and a square-Lorentzian terms, with the final outcome strongly
depending on the relative position between the mirror and the quantum dot as
quantified by the parameter $\Delta\phi$.

Our analysis extends beyond the simple input/output evanescently coupled
channels and can be useful also in understanding light-matter interaction at
EPs. As an example, consider the geometry depicted in Fig. 8a which consists
of a quantum dot embedded inside a microring-waveguide-mirror system
implementing an exceptional surface. This system was investigated recently and
it was shown that it can be used to control the rate of spontaneous emission
and Purcell factor Zhong  _et al._ (2021); Khanbekyan and Wiersig (2020); Ren
_et al._ (2021). Let us now assume that two single-photon detectors, $D_{1,2}$
are located as shown in the figure. What would be the spectrum of the detected
photon as measured by each detector after repeating the experiment a large
number of times? This question can be answered only by calculating the Green’s
operator matrix elements. While this task can be done easily by a matrix
inversion in this simple case (see Zhong  _et al._ (2021) for more details),
this approach does not give any insight into the physics of the system. On the
other hand, by applying our formalism, It is straightforward to see, based on
the photon trajectories that detector $D_{1}$ will report a Lorentzian
response while $D_{2}$ will report a superposition of Lorentzian and super-
Lorentzian of order two. More rigorously, the system here is described by
equations similar to that presented in Eq. (Linear response theory of open
systems with exceptional points) (see Zhong  _et al._ (2021) for details),
where here $\ket{b_{1,2}}$ represent the emission from the quantum dot and are
given by:
$\left[b_{1}(t),b_{2}(t)\right]^{T}=\frac{J_{e}e^{-i\omega_{e}t}}{i\sqrt{2\gamma}}\left[e^{-i\phi_{E}},e^{i\phi_{E}}\right]^{T}$,
where $\omega_{e}$ is the transition frequency of the QE, the phase
$\phi_{E}=\beta d$, and $\beta$ is the propagation constant of the modes (see
Zhong  _et al._ (2021) for a detailed derivation). By using the resolvent in
Eq. (9), the response of the system is calculated as:
$G_{2}(\omega_{e})[J_{e}e^{-i\phi_{E}},J_{e}e^{i\phi_{E}}]^{T}=\frac{J_{e}}{\Delta\omega+2i\gamma}[e^{-i\phi_{E}},e^{i\phi_{E}}+\frac{\kappa}{\Delta\omega+2i\gamma}e^{-i\phi_{E}}]^{T}$.
The above expression indicates that the power spectrum of CW mode associated
with the microring structure in Fig. 8a is Lorentzian while that of the CCW
mode features a superposition of Lorentzian and square-Lorentzian terms. In
addition, by recalling that $\kappa$ is a function of the mirror reflectivity
and location, it is straightforward to see that the interference effects
associated with the mixed term will vary as a function of the optical path
between the mirror and the emitter $\Delta\phi\equiv
2\phi_{E}-\phi=2\beta(d-L)+\phi_{r}$. Figure 8b depicts the ratio of the
stored energy in the CCW mode to that in the CW mode
$\eta_{c}\equiv\frac{\braket{E_{2}}}{\braket{E_{1}}}=|1-\frac{2i\gamma|r|e^{-i\Delta\phi}}{\Delta\omega+2i\gamma}|^{2}$
as a function of the mirror reflectivity $|r|$ and $\Delta\phi$, which clearly
highlights how the response of the intra-cavity modes depends on the relative
position of the mirror with respect to the emitter. For completeness, we also
plot the normalized output power spectra
$P_{D1,D2}=|\frac{\sqrt{2\gamma}\>Q_{1,2}}{J_{e}}|^{2}$ detected by $D_{1,2}$
as a function of the normalized frequency detuning
$\frac{\Delta\omega}{\gamma}$ as shown in Figs 8c and d, respectively. As
expected, $P_{D1}=\frac{4\gamma^{2}}{\Delta\omega^{2}+4\gamma^{2}}$ is
Lorentzian, while
$P_{D2}=\frac{4\gamma^{2}}{\Delta\omega^{2}+4\gamma^{2}}|1-\frac{2i\gamma|r|e^{-i\Delta\phi}}{\Delta\omega+2i\gamma}|^{2}$
features an interference between a Lorentzian and square-Lorentzian terms.
While the above example considers realistic scenarios of a quantum dot having
a small size compared to the wavelength of light, the notion of controlling
spontaneous emission using exceptional points could be also relevant to other
setups involving artificial atoms with larger size where the dipole
approximation fails Kannan  _et al._ (2020), as well as systems consisting of
multiple quantum emitters El-Ganainy and John (2013).

Having demonstrated the utility of our approach and the insight it provides
using a simple system with one resonant element, we now consider a more
complex system that consists of a PT symmetric arrangement and an asymmetric
delayed feedback as shown in Fig. 9a. In the absence of the mirror, this
structure exhibits two independent EPs, each of which of order two. The
introduction of the mirror causes these two EPs to coalesce, forming an EP of
order four Zhong  _et al._ (2020b). Despite the fact that this system consists
of only two ring resonators, the presence of inhomogeneous gain and loss
distribution makes it difficult to trust intuition in this case. On the other
hand, brute-force analysis by evaluating the scattering coefficients one at a
time does not provide much insight into the scattering profile. To analyze
this system using our formalism, we first consider the Hamiltonian matrix
describing this system as written in the natural basis of the individual,
isolated ring resonators $[a_{CW},b_{CCW},a_{CCW},b_{CW}]^{T}$:

$\hat{H}_{4}=\left[\begin{smallmatrix}\omega_{o}-i\gamma-iJ&J&0&0\\\
J&\omega_{o}-i\gamma+iJ&0&0\\\ \kappa&0&\omega_{o}-i\gamma-iJ&J\\\
0&0&J&\omega_{o}-i\gamma+iJ\\\ \end{smallmatrix}\right],$ (10)

Here, $\omega_{o}$ is the resonant frequency, $\gamma$ is the decay rate of
the resonant mode into each waveguide, $J$ is the coupling rate between the
two rings and also the gain/loss factors in the yellow/green rings
respectively, i.e. the system respects PT symmetry. In addition,
$\kappa=-2i\gamma|r|e^{i\phi}$, where $|r|$ is the absolute value of the
mirror reflection coefficient, and the phase factor $\phi$ quantified the
phase of the mirror reflection coefficient and its distance from the adjacent
resonator. The right (canonical) eigenvectors associated with this Hamiltonian
are given $\ket{J_{1}^{r}}=[0,0,-i,1]^{T}$, $\ket{J_{2}^{r}}=[0,0,1,0]^{T}$,
$\ket{J_{3}^{r}}=[1,i,0,0]^{T}$, and $\ket{J_{4}^{r}}=[0,1,0,0]^{T}$. These
vectors are calculated based on the following choice for constant coefficients
$\chi^{\prime}s$ [see Eq. (5)]: $\chi_{2}=\chi_{4}=J,\ \chi_{3}=\kappa$. The
corresponding left bi-orthogonal vectors, which satisfy the relation
$\braket{\tilde{J}^{l}_{n}}{J^{r}_{m}}=\delta_{nm}$, are then given by:
$\bra{\tilde{J}_{1}^{l}}=[-i,1,0,0]$, $\bra{\tilde{J}_{2}^{l}}=[1,0,0,0]$,
$\bra{\tilde{J}_{3}^{l}}=[1,i,0,0]$, and $\bra{\tilde{J}_{4}^{l}}=[0,0,0,1]$.
The scattering profile and spectral response of the system can be then
evaluated by using Eq. (6) and using the following systematic steps: (1) Write
down the excitation profile in the basis of the bare eigenmodes $a_{CW}$,
$a_{CCW}$, $b_{CW}$, and $b_{CCW}$; (2) Express this vector in the basis of
the right (canonical) eigenvectors; (3) Obtain the response using Eq. (6); (4)
Assign a projection operator to each output channel. For instance, a unit
input signal from port $P_{1}$ will directly couple only to the field
amplitude $b_{CCW}$(see Fig. 9a), and hence we have the input vector
$\ket{b_{1}}=[0,1,0,0]^{T}=\ket{J^{r}_{4}}$. Similarly, the vectors
$\ket{b_{2}}=[0,0,0,1]^{T}=\ket{J^{r}_{1}}+i\ket{J^{r}_{2}}$ and
$\ket{b_{3}}=[1,0,0,0]^{T}=\ket{J^{r}_{3}}-i\ket{J^{r}_{4}}$ correspond to
unit input signals from ports $\text{P}_{2}$ and $\text{P}_{3}$, respectively.
By using Eq. (6), we can now obtain the Green’s operator, and hence the linear
response for each different excitation. For instance, in the case of
$\text{P}_{1}$, we obtain
$G_{4}(\omega)\ket{f_{1}}=G_{4}(\omega)i\hat{\Gamma}\ket{b_{1}}=i\sqrt{2\gamma}\left[\frac{J^{2}k}{(\Delta\omega+i\gamma)^{4}}\ket{J^{r}_{1}}+\frac{Jk}{(\Delta\omega+i\gamma)^{3}}\ket{J_{2}^{r}}+\frac{J}{(\Delta\omega+i\gamma)^{2}}\ket{J_{3}^{r}}+\frac{1}{(\Delta\omega+i\gamma)}\ket{J_{4}^{r}}\right]$,
where $\ket{f_{1}}=i\hat{\Gamma}\ket{b_{1}}$ and $\sqrt{2\gamma}$ corresponds
to the element of the coupling matrix $\hat{\Gamma}$ between the input
channels and the interior of the resonator system. Note that this expression
describes the field amplitude inside the resonators system. To calculate the
output, one must project this field amplitude on the output channel. For
illustration purpose, let us consider the output channel $\text{Q}_{1}$, which
is directly coupled to the $b_{CW}$, i.e. it corresponds to the vector
$[0,0,0,1]^{T}=\ket{J_{1}^{r}}+i\ket{J_{2}^{r}}$. Thus one can assign the
projection operator
$\mathbb{P}_{1}=\ket{J_{1}^{r}}\bra{\tilde{J}_{1}^{l}}+i\ket{J_{2}^{r}}\bra{\tilde{J}_{2}^{l}}$
to the output channel $\text{Q}_{1}$. Similarly, the projection operators
$\mathbb{P}_{2}=\ket{J_{4}^{r}}\bra{\tilde{J}_{4}^{l}}$ and
$\mathbb{P}_{3}=\ket{J_{2}^{r}}\bra{\tilde{J}_{2}^{l}}$ describe output
channels $\text{Q}_{2}$ and $\text{Q}_{3}$, respectively. Finally, the output
signal is given by:
$\mathbb{P}_{i}\left[\hat{Y}\ket{b_{j}}-\hat{\Gamma}^{T}\hat{G}\ket{f_{j}}\right]$,
where the matrix $\hat{Y}$ quantifies the direct coupling between the input
and output channels whereas the matrix $\hat{\Gamma}^{T}$ describes the
coupling between the interior of the resonators and the output channels. By
applying the above-sketched recipe for an input/output from ports $P_{1}$ and
$\text{Q}_{3}$, we find: $Q_{3}=-i\frac{2\gamma
Jk}{(\Delta\omega+i\gamma)^{3}}$. In other words, counter-intuitively the
response features a super-Lorentzian of order three, despite the fact that the
system exhibits an EP of order four. This is, however, fully consistent with
our theoretical analysis since $P_{1}$ corresponds to $\ket{J_{4}^{r}}$ and
$Q_{3}$ corresponds to $\ket{J_{2}^{r}}$ as demonstrated by mathematical
structure of Eq. (6), and also pictorially in Fig. 5b. Figure 9b summarizes
the steps descried in this section.

Figure 9: A PT symmetric system with unidirectional coupling between the
CW/CCW modes. a A schematic of the studied system, which was introduced in
Zhong  _et al._ (2020b) and shown to exhibit an EP of order 4. $J$ is the
gain/loss factor. $P_{1,2,3}$ are input signals and $Q_{1,2,3}$ are output
signals. $a_{CW,CCW}$ and $b_{CW,CCW}$ are CW and CCW modes of the cavities. b
Pictorial depiction of the systematic approach to obtain the system’s linear
response using the formalism presented in this work. $\ket{b_{1}}$ is the
excitation profile expanded with respect to the right eigenvectors
$\\{\ket{J_{n}^{r}}\\}$. $\\{\bra{\tilde{J}_{n}^{l}}\\}$ are the corresponding
left bi-orthogonal vectors. $\hat{G}$ is the resolvent and $\mathbb{P}_{1}$ is
the projection operator assigned to the output channel. $\hat{Y}$ quantifies
the direct scattering between incoming and outgoing channels and
$\hat{\Gamma}$ describes the coupling between the resonant modes and the
input/output channels.

DISCUSSION
In this work, we have presented a detailed analysis of the linear response
associated with non-Hermitian systems having EPs and showed that a non-
diverging resolvent associated with the system’s Hamiltonian can be expressed
as an exact series expansion of the ordinary and generalized eigenfunctions of
the Hamiltonian, i.e. without resorting to any perturbation approximation.
Importantly, our formalism revealed a feature that escaped attention in
previous studies, namely that the response lineshape scaling can be engineered
by a judicious choice of the input and output channels. This observation is
crucial for tailoring light-matter interactions at EPs. In order to emphasize
this point and also clarify the application of our formalism, we have
considered and analyzed several realistic photonic examples and we found
excellent agreement between results obtained from full-wave simulations and
our formulas. In doing so, we have also demonstrated an interesting effect
analogous to adjoint coupling but rather in microring cavity setups. In other
words, we have shown that more optical energy can be stored in a microring
cavity system having an EP when the channel associated with the input signal
matches the generalized eigenmode rather than the actual ordinary eigenmode of
the structure. We emphasize that although the examples in this manuscript are
chosen from the optical domain, our results are general and applicable to
other physical systems that can be described by similar coupled mode
formalism. These include electronic, acoustic, mechanical, and thermal
systems. In addition, our framework provides a powerful tool for understanding
the complex interplay between non-Hermiticity and other physical effects such
as topological invariants Hafezi  _et al._ (2011); Schomerus (2013); Zhao  _et
al._ (2018); Reséndiz-Vázquez  _et al._ (2020); Gong  _et al._ (2018); Zhao
_et al._ (2019), optomechanical coupling Jing  _et al._ (2014, 2015, 2017);
Zhang  _et al._ (2018); Schönleber  _et al._ (2016) as well as quantum
statistics Schomerus (2010); Arkhipov  _et al._ (2019); Minganti  _et al._
(2019); Metelmann and Clerk (2014); Luitz and Piazza (2019); Quiroz-Juárez
_et al._ (2019), to just mention a few examples. This in turn may enable the
engineering of more elaborate schemes for controlling energy and information
flow in complex non-Hermitian systems. Finally, we remark that understanding
the linear response of non-Hermitian systems is a very crucial step towards
studying their noise. In this regard, we expect our formalism to provide more
insight into the noise behavior in non-Hermitian systems and play a positive
role in the active debate on signal to noise ratio of EP-based sensors
Langbein (2018); Lau and Clerk (2018); Zhang  _et al._ (2019b); Wang  _et al._
(2020b); Wiersig (2020b); Smith  _et al._ (2020); Kononchuk  _et al._ (2022).
We plan to explore some of these interesting directions in future works.

## Data availability

The data that support the findings of this study are available from the
corresponding authors upon reasonable request.

###### Acknowledgements.

This work is supported by Air Force Office of Scientific Research (AFOSR)
Multidisciplinary University Research Initiative (MURI) Award on Programmable
systems with non-Hermitian quantum dynamics (Award No. FA9550-21-1-0202). R.E.
also acknowledges support from ARO (Grant No. W911NF-17-1-0481), NSF (Grant
No. ECCS 1807552), and the Alexander von Humboldt Foundation. S.K.O.
acknowledges support from NSF (Grant No. ECCS 1807485) K.B. acknowledges
support by the Leibniz Association within the SAW-project LAPTON and from the
Alexander von Humboldt Foundation.

## Author contribution

R.E. conceived the project. R.E. and A.H. performed the theoretical analysis
with feedback from S.K.O, K.B., and D.N.C. The numerical simulations was
performed by A.H. All authors contributed to the manuscript writing.

## Competing interests

The authors declare no competing interests

Supplementary Information:

Linear response theory of open systems with exceptional points

A. Hashemi, K. Busch, D. N. Christodoulides, S.K. Ozdemir, and R. El-Ganainy

## SUPPLEMENTARY NOTES

###  .1 Subtleties arising in resonant systems with EPs: an example

Here we provide a detailed analysis for the example shown in Fig. 2 in the
main text, which is also shown here for convenience. It consists of three
identical microring resonators that are coupled sequentially via horizontal
waveguides. An additional vertical waveguide provides access to selectively
excite the second resonator. We neglect the cross talk between the horizontal
and vertical waveguides since it can be minimized using various design
strategies Daly  _et al._ (1996); Mingaleev  _et al._ (2004); Kobayashi and
Tsuda (2005); Longhi (2015). A similar system was considered in Zhong  _et
al._ (2020a) and shown to exhibit a third order EP in the subspace spanned by
the CW, CCW and CW modes of the resonators $\text{R}_{1,2,3}$, respectively.
In this subspace, and by allowing excitations only from ports $P_{1,2,3}$ with
the collection output ports $Q_{1}$ and $Q_{2}$ (see supplementary Fig. 10)
the system is described by the coupled equations:

$\displaystyle i\frac{d}{dt}\left[\begin{matrix}a_{1}\\\ a_{2}\\\
a_{3}\end{matrix}\right]=$
$\displaystyle\left[\begin{matrix}\omega_{o}-3i\gamma&0&0\\\
\kappa&\omega_{o}-3i\gamma&0\\\
0&\kappa&\omega_{o}-3i\gamma\end{matrix}\right]\left[\begin{matrix}a_{1}\\\
a_{2}\\\ a_{3}\end{matrix}\right]+i\sqrt{2\gamma}\left[\begin{matrix}P_{1}\\\
P_{2}\\\ P_{3}\end{matrix}\right],$ $\displaystyle Q_{1}=$ $\displaystyle
P_{3}-\sqrt{2\gamma}\>a_{3},$ $\displaystyle Q_{2}=$ $\displaystyle
P_{I}-\sqrt{2\gamma}\>a_{3},$ (11)

where $a_{1,2,3}$ are modal amplitudes of the CW, CCW and CW modes associated
with resonators $\text{R}_{1,2,3}$, respectively. The resonant frequency and
decay rate of each resonator are given by $\omega_{o}$ and $3\gamma$. For
resonator $\text{R}_{2}$ the decay rate is due to equal coupling to three
waveguides. On the-other hand, for $\text{R}_{1,3}$ the coupling to waveguides
contributes only $2\gamma$ and we assume that an additional loss of $\gamma$
is intentionally introduced for example by depositing a metal layer on top of
the ring or introducing an auxiliary waveguide next to each existing
waveguide. The ports $P_{1,2,3}$ serve as excitation channels for the modes
$a_{1,2,3}$, respectively, while the output signal is collected from ports
$Q_{1}$ and $Q_{2}$ as depicted in supplementary Fig. 10. In addition, $P_{I}$
represents the signal in the waveguide between $\text{R}_{2}$ and
$\text{R}_{3}$ (see supplementary Fig. 10b). Within this modal subspace, there
is unidirectional coupling from modes $a_{1}\rightarrow a_{2}$ and
$a_{2}\rightarrow a_{3}$. The exact details of this waveguide-mediated
indirect coupling depends on the evanescent tunneling between the rings and
waveguides as well as the distances between the rings. Here we assume that
unidirectional coupling coefficients are identical and denote them by
$\kappa$.

Figure 10: Subtleties of linear response of non-Hermitian systems having EPs:
an illustrative example. A photonic system that exhibits an EP of order three.
As described in the main text, it can have very different linear responses
based on the input/output channel configuration. Here we present the detailed
mathematical analysis of this structure.

In this case, the resolvent, $G_{3}$, can be evaluated in closed form:

$G_{3}=\left[\begin{array}[]{ccc}\vspace*{2mm}\mu_{3}&0&0\\\
\vspace*{2.5mm}\kappa\mu_{3}^{2}&\mu_{3}&0\\\
\kappa^{2}\mu_{3}^{3}&\kappa\mu_{3}^{2}&\mu_{3}\\\ \end{array}\right],$ (12)

where $\mu_{3}\equiv 1/(\omega-\omega_{o}+3i\gamma)$. One can easily confirm
that the exceptional eigenvector of $G_{3}$ is $\ket{K_{1}^{r}}=[0,0,1]^{T}$,
while the generalized eigenvectors forming the Jordan chain are give by
$\ket{K_{2}^{r}}=[0,\frac{\xi}{\kappa\mu_{3}^{2}},0]^{T}$ and
$\ket{K_{3}^{r}}=\left[(\frac{\xi}{\kappa\mu_{3}^{2}})^{2},-\frac{\chi_{o}^{2}}{\kappa\mu_{3}^{3}},0\right]^{T}$
(see Eq.(5.1) of supplementary note 6).

Next, we consider the above system when it is excited from one of the ports
$P_{1,2,3}$. These signal pathways inside the structure for each of these
distinct situations are depicted in supplementary Figs. 10a, b, and c. It is
straightforward to show that the scattering coefficient between the input and
output ports in each case is given by:

$\displaystyle\begin{split}\frac{Q_{2}}{P_{3}}&=-2i\gamma\mu_{3}=-\frac{2i\gamma}{(\omega-\omega_{o}+3i\gamma)},\\\
\frac{Q_{1}}{P_{2}}&=-2i\gamma\kappa\mu_{3}^{2}=-\frac{2i\gamma\kappa}{(\omega-\omega_{o}+3i\gamma)^{2}},\\\
\frac{Q_{1}}{P_{1}}&=-2i\gamma\kappa^{2}\mu_{3}^{3}=-\frac{2i\gamma\kappa^{2}}{(\omega-\omega_{o}+3i\gamma)^{3}}.\end{split}$
(13)

These results can be intuitively understood by referring again to
supplementary Fig. 10, which depicts the designated input and output ports,
and the actual paths taken by the excitation signal. In the first case, the
input signal $P_{3}$ crosses only the ring resonator $\text{R}_{3}$ before it
couples to the output $Q_{2}$. On the other hand, the signal $P_{2}$ in the
second case interacts with two rings ($\text{R}_{2}$ and $\text{R}_{3}$)
before it is coupled to the output channel $Q_{1}$. Finally, the trajectory of
the input signal $P_{1}$ in the third case involves three resonators and the
output port $Q_{1}$. The results discussed above can be understood by noting
that each ring contributes a Lorentzian response and that total response is
the product of the individual responses (due to the series connection of the
rings). In more complex systems where such a simple trajectory picture does
not exist, it is necessary to develop a general theoretical framework to
analyze and understand the system’s response.

###  .2 Completeness and biorthogonality of the eigenvectors associated with
defective matrices

The eigenvectors of non-Hermitian matrices are in general non-orthogonal.
Instead, they obey a modified bi-orthogonality relation, i.e. orthogonality
between the right eigenvector and its corresponding left eigenvectors. The
spectral analysis of a non-Hermitian matrix becomes rather complicated when it
is defective, i.e. exhibiting EPs in its spectrum. In what follows, we provide
a concise discussion that elaborates on this situation. Let us assume an
$N\times N$ matrix $\hat{H}_{EP}$ that has one EP of order $M$ (generalization
to multiple EPs is straightforward). The right eigenvectors
($\ket{\psi_{n}^{r}}$) of $\hat{H}_{EP}$ are defined by

$(\hat{H}_{EP}-\Omega_{n}\hat{I})\ket{\psi_{n}^{r}}=0,$ (14)

where $n=1,2,\ldots,N-M$. Evidently, these vectors do not span the underlying
vector space. It is well-known however that the bases can be completed by
using the Jordan canonical chain procedure, i.e. by defining the generalized
right eigenvectors ($\ket{J^{r}_{m}}$) of $\hat{H}_{EP}$ by the Jordan chain

$(\hat{H}_{EP}-\Omega_{EP}\hat{I})\ket{J_{m}^{r}}=\chi_{m}\ket{J_{m-1}^{r}},$
(15)

where $m=1,2,\ldots,M$ and $\chi_{1}=0$. The eigenvectors $\ket{J_{m}^{r}}$
are linearly independent. This can be shown as follows. Consider the relation
$\sum_{m=1}^{M}z_{m}\ket{J_{m}^{r}}=0$. We now apply the operator
$(H-\Omega_{EP}\hat{I})^{M-1}$ to both sides. All the terms on the left-hand-
side will vanish except the last term, which gives $z_{M}=0$. By repeating
this procedure on the remaining terms, we find that all the coefficients
$z_{m}$ have to be zero. Similarly, one can show that any vector
$\ket{J_{m}^{r}}$ is linearly independent from all the vectors
$\ket{\psi_{n}^{r}}$. In other words, the set $\\{\ket{\psi_{n}^{r}}\\}$ spans
an $N-M$ dimensional space (call it $\mathcal{D}_{\phi}$) while the set
$\\{\ket{J_{n}^{r}}\\}$ spans the complementary $M$ dimensional space
$\mathcal{D}_{EP}$. Taken together, the set
$\\{\ket{\psi_{n}^{r}},\ket{J_{n}^{r}}\\}$ forms a complete basis that spans
$\mathcal{D}_{\phi}\cup\mathcal{D}_{EP}$. Thus any vector in that space can be
decomposed in terms of these bases. However, as we mentioned, in general
$\braket{\psi_{m}^{r}}{\psi_{n}^{r}}\neq\delta_{m,n}$. This, in turn
complicates the procedure for finding the projection of any general vector
onto the bases vectors. This difficulty can be overcome by using the concept
of left eigenvectors:

$\bra{\psi_{n}^{l}}(\hat{H}_{EP}-\Omega_{n}\hat{I})=0.$ (16)

The set $\\{\bra{\psi_{m}^{l}}\\}$ defines a dual space of
$\mathcal{D}_{\phi}$ (see Fig. 5a). From Eq. (14), we can write
$\braket{\psi_{n_{2}}^{l}}{\hat{H}_{EP}}{\psi_{n_{1}}^{r}}=\Omega_{n_{1}}\braket{\psi_{n_{2}}^{l}}{\psi_{n_{1}}^{r}}$.
Similarly, from Eq. (16) we obtain
$\braket{\psi_{n_{2}}^{l}}{\hat{H}_{EP}}{\psi_{n_{1}}^{r}}=\Omega_{n_{2}}\braket{\psi_{n_{2}}^{l}}{\psi_{n_{1}}^{r}}$.
For these two relations to be consistent for
$\Omega_{n_{1}}\neq\Omega_{n_{2}}$, we must have
$\braket{\psi_{n_{2}}^{l}}{\psi_{n_{1}}^{r}}=0$ for $n_{1}\neq n_{2}$. This is
known as biorthogonality. By using this last relation iteratively in the
recursive system of Eqs. (15), we also find that
$\braket{\psi_{m}^{l}}{J_{n}^{r}}=0$ for any $m$ and $n$. For instance, by
using $(\hat{H}_{EP}-\Omega_{EP}\hat{I})^{2}\ket{J_{2}^{r}}=0$, we find that
$\braket{\psi_{m}^{l}}{(\hat{H}_{EP}-\Omega_{EP}\hat{I})^{2}}{J_{2}^{r}}=0$.
On the other hand, by using
$\bra{\psi_{m}^{l}}(\hat{H}_{EP}-\Omega_{EP}\hat{I})^{2}=\bra{\psi_{m}^{l}}(\Omega_{m}-\Omega_{EP})^{2}$,
the same expression can be evaluated to be
$\braket{\psi_{m}^{l}}{(\hat{H}_{EP}-\Omega_{EP}\hat{I})^{2}}{J_{2}^{r}}=(\Omega_{m}-\Omega_{EP})^{2}\braket{\psi_{m}^{l}}{J_{2}^{r}}$,
which gives $\braket{\psi_{m}^{l}}{J_{2}^{r}}=0$.

Similarly, one can define the left generalized eigenvectors of $\hat{H}_{EP}$
by following the same procedure, namely:

$\bra{J_{m}^{l}}(\hat{H}_{EP}-\Omega_{EP}\hat{I})={\chi}_{m}\bra{J^{l}_{m-1}}.$
(17)

Following the same procedure as before, one can show that the vectors
$\bra{J_{m}^{l}}$ are linearly independent from each other and from
$\bra{\psi_{n}^{l}}$, i.e. they define a dual space of $\mathcal{D}_{EP}$ (see
Fig. 5a in the main text). Also, it is straightforward to show that
$\braket{J_{m}^{l}}{\psi_{n}^{r}}=0$ for any $m$ and $n$. On the other hand,
the relation between the vectors $\bra{J^{l}}$ and $\ket{J^{r}}$ is subtle due
to the self-orthogonality: $\braket{J_{m}^{l}}{J^{r}_{m}}=0$ for
$m\leq\frac{M}{2}$. This can be proven by considering the expression
$\braket{J^{l}_{m}}{(\hat{H}_{EP}-\Omega_{EP}\hat{I})^{m}}{J^{r}_{2m}}$. When
evaluated by first calculating the term
$\bra{J^{l}_{m}}(\hat{H}_{EP}-\Omega_{EP}\hat{I})^{m}$, we find that the
result is zero. On the other hand, when we first evaluate the term
$(\hat{H}_{EP}-\Omega_{EP}\hat{I})^{m}\ket{J^{r}_{2m}}$, we obtain
$\chi_{2m}\chi_{2m-1}\ldots\chi_{m+1}\braket{J^{l}_{m}}{J^{r}_{m}}$, which
proves the self-orthogonality relation.

In order to overcome the above problem, we first note that the set of the left
eigenvectors $\\{\bra{J_{m}^{l}}\\}$ spans the dual space of
$\mathcal{D}_{EP}$. Thus any vector formed by an arbitrary superposition of
$\\{\bra{J_{m}^{l}}\\}$ also lies in the dual space of $\mathcal{D}_{EP}$. Let
us now define a new set of vectors $\bra{\tilde{J}_{m}^{l}}$, each of which
lies in the dual space of $\mathcal{D}_{EP}$, i.e.
$\bra{\tilde{J}_{m}^{l}}=\sum_{k=1}^{M}t_{m,k}\bra{J_{k}^{l}}$. In order to
find the $M^{2}$ coefficients $t_{m,k}$, we need to impose $M^{2}$
normalization conditions. These can be chosen to satisfy
$\braket{\tilde{J}_{m}^{l}}{J_{k}^{r}}=\delta_{m,k}$. Note that the new
vectors $\bra{\tilde{J}_{m}^{l}}$ are not generalized eigenvectors of the
Hamiltonian $\hat{H}_{EP}$.

###  .3 Simultaneous normalization of right and left eigenvectors

The simultaneous normalization of the right and left eigenvectors associated
with a non-Hermitian Hamiltonian was discussed in Siegman (1989). For
completeness, we reiterate some of these results here from a more general
point of view. The main result of this section is that for non-normal matrices
(matrices that do not commute with their Hermitian conjugates), the three
different normalization conditions $\braket{\psi_{n}^{l}}{\psi_{n}^{r}}=1$,
$\braket{\psi^{r}_{n}}{\psi^{r}_{n}}=1$, and
$\braket{\psi_{n}^{l}}{\psi_{n}^{l}}=1$ cannot be satisfied simultaneously.
For non-defective Hamiltonians, this is a direct consequence of Cauchy-Schwarz
inequality:
$|\braket{\psi_{n}^{l}}{\psi_{n}^{r}}|^{2}\leq\braket{\psi_{n}^{r}}{\psi_{n}^{r}}\braket{\psi_{n}^{l}}{\psi_{n}^{l}}$.
The equality holds if and only if the vectors $\ket{\psi_{n}^{r}}$ and
$\ket{\psi_{n}^{l}}$ are the same up to a constant. This occurs when $\hat{H}$
is normal, i.e. $[\hat{H},\hat{H}^{\dagger}]=0$. To prove this, we note that
the above relation implies that both $\hat{H}$ and $\hat{H}^{\dagger}$ share
the same right and left eigenvectors. It follows that $\ket{\psi_{n}^{r}}$ and
$\bra{\psi_{n}^{l}}$ are also right and left eigenvectors of the operator
$\hat{H}\hat{H}^{\dagger}$. This last operator is Hermitian and thus, up to a
constant, the right and left eigenvectors are identical.

###  .4 Series expansion of the resolvent

In this section, we present a detailed derivation for the series expansion of
the resolvent in Eq. (6). By considering an arbitrary input signal
$\ket{u}=\sum_{n=1}^{N-M}c_{n}\ket{\psi_{n}^{r}}+\sum_{m=1}^{M}d_{m}\ket{J_{m}^{r}}$,
where the coefficients $c_{n}$ and $d_{m}$ are known (or can be calculated
using the projection rules for the right/left eigenvectors as discussed in the
previous section), and by assuming a similar series representation of the
response signal with known coefficients $\tilde{c}_{n}$ and $\tilde{d}_{m}$,
we can write:

$\displaystyle(\omega\hat{I}-\hat{H}_{EP})\left(\sum_{n=1}^{N-M}\tilde{c}_{n}\ket{\psi_{n}^{r}}+\sum_{m=1}^{M}\tilde{d}_{m}\ket{J_{m}^{r}}\right)=\sum_{n=1}^{N-M}c_{n}\ket{\psi_{n}^{r}}+\sum_{m=1}^{M}d_{m}\ket{J_{m}^{r}}.$
(18)

By using Eqs. (14) and (15), we can express the left-hand-side of Eq. (18) as
$\sum_{n=1}^{N-M}\tilde{c}_{n}(\omega-\Omega_{n})\ket{\psi_{n}^{r}}+\sum_{m=1}^{M}\left[\tilde{d}_{m}(\omega-\Omega_{EP})-\tilde{d}_{m+1}\chi_{m+1}\right]\ket{J_{m}^{r}}$.
By recalling that the vectors $\ket{\psi_{n}^{r}}$ and $\ket{J_{m}^{r}}$ are
all linearly independent, we arrive at:

$\displaystyle\tilde{c}_{n}$ $\displaystyle=\frac{c_{n}}{\omega-\Omega_{n}}$
(19) $\displaystyle\tilde{d}_{m}$
$\displaystyle=\frac{d_{m}}{\omega-\Omega_{EP}}+\frac{\chi_{m+1}\tilde{d}_{m+1}}{\omega-\Omega_{EP}}.$
(20)

The recursive relation in Eq. (20) can be further simplified by noting that
$\chi_{M+1}=0$, which gives $\tilde{d}_{M}=\frac{d_{M}}{\omega-\Omega_{EP}}$.
By inserting this result back in Eq. (20), we obtain the general expression:

$\displaystyle\tilde{d}_{m}=$
$\displaystyle\frac{d_{m}}{\omega-\Omega_{EP}}+\frac{\chi_{m+1}d_{m+1}}{(\omega-\Omega_{EP})^{2}}+\frac{\chi_{m+1}\chi_{m+2}d_{m+2}}{(\omega-\Omega_{EP})^{3}}+\cdots+\frac{\chi_{m+1}\chi_{m+2}\cdots\chi_{M}d_{M}}{(\omega-\Omega_{EP})^{M-m+1}}$
$\displaystyle=$
$\displaystyle\sum_{k=m}^{M}\alpha_{k}^{(m)}\frac{d_{k}}{(\omega-\Omega_{EP})^{k-m+1}},$
(21)

where we defined $\alpha_{m}^{(m)}=1$ and
$\alpha_{k}^{(m)}=\alpha_{k-1}^{(m)}\chi_{k}=\chi_{k}\chi_{k-1}\ldots\chi_{m+1}$
for $k=m+1,\ldots,M$. Finally, by using $c_{n}=\braket{\psi_{n}^{l}}{u},\
d_{k}=\braket{\tilde{J}_{k}^{l}}{u}$ (see supplementary note .2) we obtain:

$\displaystyle\hat{G}_{EP}\ket{u}$
$\displaystyle=\biggr{(}\sum_{n=1}^{N-M}\frac{\ket{\psi^{r}_{n}}\bra{\psi_{n}^{l}}}{\omega-\Omega_{n}}+\sum_{m=1}^{M}\sum_{k=m}^{M}\alpha_{k}^{(m)}\frac{\ket{J^{r}_{m}}\bra{\tilde{J}^{l}_{k}}}{(\omega-\Omega_{EP})^{k-m+1}}\biggr{)}\ket{u}.$
(22)

Since $\ket{u}$ is an arbitrary vector, it follows that the series summation
on the right-hand-side represents the resolvent which proves Eq. (6).

###  .5 Non-interfering excitation channels

Here we discuss the classification of the excitation channels based on their
interference properties. Consider an input $\ket{u}=\ket{u_{1}}+\ket{u_{2}}$,
with $s(t)=e^{-i\omega_{e}t}$. This rather unphysical assumption of an input
signal that extends over the time axis is well suited for treating long time
limits after the transient response has died out, which is the case of
interest here (alternatively, one can, of course, use the Laplace transform
and take the long-time limit $t\rightarrow\infty$ explicitly). The normalized
energy (with respect to some reference energy value), stored in the system
after the transient response fades away, is given by
$\braket{E}\equiv\lim_{t\rightarrow\infty}\overline{\braket{a(t)}{a(t)}}$,
with the overline indicating time average. Equivalently, the normalized stored
energy can be also expressed as
$\braket{E}=\braket{\tilde{A}(\omega_{e})}{\tilde{A}(\omega_{e})}$ where
$\ket{A(\omega)}=\ket{\tilde{A}(\omega_{e})}\delta(\omega-\omega_{e})$, or
$\ket{\tilde{A}(\omega_{e})}=\sum_{n=1}^{N}\frac{c_{n}}{\omega_{e}-\Omega_{n}}\ket{\psi_{n}^{r}}$.
In general, the above expression will contain contributions from (1)
$\ket{u_{1}}$ only in the absence of $\ket{u_{2}}$; (2) $\ket{u_{2}}$ only in
the absence of $\ket{u_{1}}$; (3) interference component due to the non-
orthogonality of $\ket{u_{1,2}}$. To illustrate this with a concrete example,
consider an input
$\ket{f}=\left[\alpha_{m}\ket{\psi_{m}^{r}}+\alpha_{n}\ket{\psi_{n}^{r}}\right]e^{-i\omega_{e}t}$.
The normalized stored energy in this case is given by
$\braket{E}=|\frac{\alpha_{m}}{\omega_{e}-\Omega_{m}}|^{2}\braket{\psi_{m}^{r}}{\psi_{m}^{r}}+|\frac{\alpha_{n}}{\omega_{e}-\Omega_{n}}|^{2}\braket{\psi_{n}^{r}}{\psi_{n}^{r}}+2\text{Re}\\{\frac{\alpha_{m}^{*}}{\omega_{e}-\Omega_{m}^{*}}\frac{\alpha_{n}}{\omega_{e}-\Omega_{n}}\braket{\psi_{m}^{r}}{\psi_{n}^{r}}\\})$,
with the last term representing the interference term. This raises the
question of whether this feature is pertinent to any input profile or if it is
possible to construct some excitation channels whose energy contribution
inside the resonators do not interfere. The main result of this section indeed
affirms the latter possibility. To demonstrate this, we first note that the
resolvent $\hat{G}_{EP}(\omega)$ has the same spectral structure of the
Hamiltonian $\hat{H}_{EP}$, i.e. they have the same eigenvectors and their
eigenvalues are given by $\frac{1}{\omega-\Omega_{n}}$ and $\Omega_{n}$
respectively. Consequently, they also share the same EP (see supplementary
note 5). As a result, one can in principle complete the basis by constructing
the Jordan vectors associated with $\hat{G}_{EP}(\omega)$ instead of
$\hat{H}_{EP}$ as we have done before:

$\displaystyle(\hat{G}_{EP}-\mu_{EP}\hat{I})\ket{K_{1}^{r}}$ $\displaystyle=0$
(23) $\displaystyle(\hat{G}_{EP}-\mu_{EP}\hat{I})\ket{K_{2}^{r}}$
$\displaystyle=\xi\ket{K_{1}^{r}}$ $\displaystyle\vdots$
$\displaystyle(\hat{G}_{EP}-\mu_{EP}\hat{I})\ket{K_{M}^{r}}$
$\displaystyle=\xi\ket{K_{M-1}^{r}}.$

In the above, $\mu_{EP}=\frac{1}{\omega-\Omega_{EP}}$ and the coefficient
$\xi$ is taken to be of unit value and of dimensions similar to $\mu_{EP}$. We
emphasize that both $\hat{G}_{EP}$ and $\mu_{EP}$ are functions of $\omega$.
It is straightforward to show that $\ket{K_{1}^{r}}=\ket{J_{1}^{r}}$. A more
general Jordan vector $\ket{K_{m}^{r}}$ with $m>1$, can be expressed as a
linear superposition of the vectors $\ket{J^{r}_{m}}$ (see supplementary note
6). We now consider the case when the matrix representation of $\hat{G}_{EP}$
is given by the Jordan canonical form. In this case, the generalized
eigenvectors are orthogonal. This is of course a feature of the geometry of
the problem. For instance, if the effective Hamiltonian of the optical
structure under study is given by $\hat{H}=$
$\begin{pmatrix}\Omega&J&\kappa\\\ 0&\Omega&J\\\ 0&0&\Omega\end{pmatrix}$ ,
then we find
$\hat{G}(\omega=\Omega-\frac{J^{2}}{\kappa})=\frac{\kappa^{2}}{J^{3}}$
$\begin{pmatrix}-\frac{J}{\kappa}&1&0\\\ 0&-\frac{J}{\kappa}&1\\\
0&0&-\frac{J}{\kappa}\end{pmatrix}$ with $\Ket{K^{r}_{1}}=[1,0,0]^{T}$,
$\Ket{K^{r}_{2}}=[0,1,0]^{T}$, and $\Ket{K^{r}_{3}}=[0,0,1]^{T}$. By assuming
that the above condition is satisfied, we now consider an input of the form
$\ket{f}=\left[\alpha_{m}\ket{K_{m}^{r}}+\alpha_{n}\ket{K_{n}^{r}}\right]e^{-i\omega_{e}t}$.
Clearly, if $|n-m|=1$, interference terms will appear in the expression for
$\braket{E}$. On the other hand, when $|n-m|>1$, the energy expression will
not contain any interference terms. For instance, for $m=2$ and $n=4$, we will
obtain $\braket{E}=\braket{E_{2}}+\braket{E_{4}}$, where
$\braket{E_{2}}=|\alpha_{2}|^{2}\left[|\xi|^{2}\braket{K_{1}^{r}}{K_{1}^{r}}+|\mu_{EP}(\omega_{e})|^{2}\braket{K_{2}^{r}}{K_{2}^{r}}\right]$
and
$\braket{E_{4}}=|\alpha_{4}|^{2}\left[|\xi|^{2}\braket{K_{3}^{r}}{K_{3}^{r}}+|\mu_{EP}(\omega_{e})|^{2}\braket{K_{4}^{r}}{K_{4}^{r}}\right]$.
If, on the other hand, the natural matrix representation (i.e. the
representation that arises from the geometry of the problem without performing
any linear mapping) of $\hat{G}$ is not in the Jordan canonical form, the
generalized eigenvectors do not have to be orthogonal. However, by using the
similarity transformation that relates the natural and canonical bases one can
find the non-interfering channels in the natural bases. These in general will
be a superposition between the excitation channels associated with the natural
bases.
The above analysis in terms of the Jordan vectors associated with the
resolvent illustrates another interesting effect. Consider the two different
inputs $\ket{f_{1,2}}=\alpha_{1,2}\ket{K_{1,2}^{r}}e^{-i\omega_{e}t}$, with
the constants $\alpha_{1,2}$ chosen to satisfy the conditions
$|\alpha_{1,2}|^{2}\braket{K_{1,2}^{r}}{K_{1,2}^{r}}=1$. In this case, the
expressions for the corresponding energy stored in the structure are given by:
$\braket{E_{1}}=|\mu_{EP}(\omega_{e})|^{2}$, and
$\braket{E_{2}}=|\mu_{EP}(\omega_{e}))|^{2}+|\xi|^{2}|\alpha_{2}/\alpha_{1}|^{2}$.
In other words,
$\braket{E_{2}}/\braket{E_{1}}=1+\frac{\xi^{2}}{|\mu_{EP}|^{2}}|\alpha_{2}/\alpha_{1}|^{2}>1$.
This result indicates that in this case, mode matching does not lead to the
most efficient excitation scheme- a feature that resembles the notion of
adjoint coupling in the context of free space unstable laser resonators
Siegman (1995).

###  .6 Generalized eigenvectors of the resolvent

A direct consequence of resolvent definition is that it shares the same
eigenvectors with its corresponding Hamiltonian, including the exceptional
vector, i.e. $\ket{K_{1}^{r}}=\ket{J_{1}^{r}}$. However, the generalized
eigenvectors of the Hamiltonian and those of the corresponding resolvent are
not identical. Additionally, as we have mentioned before, $\ket{K_{n}^{r}}$
and $\ket{J_{n}^{r}}$ are not unique. However, once the two sets of vectors
are chosen according to a specific criterion, one can express the former in
terms of the latter. In particular, one can write
$\ket{K_{m}^{r}}=\sum_{i=1}^{M}C_{i}^{m}\ket{J_{i}^{r}}$, where the
coefficients $C_{i}^{m}$ are given by
$C_{i}^{m}=\braket{\tilde{J}_{i}^{l}}{K_{m}^{r}}$ (see supplementary note .2
for the discussion of the left vectors $\bra{\tilde{J}_{i}^{l}}$).
In general, the above series expansion should be computed numerically.
However, one particular solution can be expressed in closed form. In the
following, we derive this result. Consider a choice of the vectors
$\ket{K_{m}^{r}}$ as $\ket{K^{r}_{m}}=\sum_{i=2}^{m}C_{i}^{m}\ket{J^{r}_{i}}$,
with $m=2,3,\ldots,M$. Note that the upper limit of the summation is $m$, not
$M$, i.e. it varies from one vector to another. The strategy is to substitute
this ansatz into the formula
$(\hat{G}_{EP}-\mu_{EP}\hat{I})\ket{K^{r}_{m}}=\xi\ket{K^{r}_{m-1}}$ and solve
for the vectors $\ket{K_{m}^{r}}$. Before we proceed, we recall
$\mu_{EP}\equiv(\omega-\Omega_{EP})^{-1}$. Let us now consider the case $m=2$.
It follows that
$(\hat{G}_{EP}-\mu_{EP}\hat{I})C_{2}^{2}\ket{J^{r}_{2}}=\xi\ket{J^{r}_{1}}$.
By multiplying both sides by
$1/\hat{G}_{EP}\equiv(\omega\hat{I}-\hat{H}_{EP})$, we obtain
$\mu_{EP}C_{2}^{2}\chi_{2}\ket{J_{1}^{r}}=\xi(\omega-\Omega_{EP})\ket{J_{1}^{r}}$,
which in turn leads to:

$C_{2}^{2}=\frac{\xi}{\mu_{EP}^{2}\chi_{2}}.$ (24)

By following similar steps for $m>2$, we arrive at the relation
$\sum_{i=2}^{m}C_{i}^{m}\mu_{EP}\chi_{i}\ket{J^{r}_{i-1}}=\xi\sum_{i=2}^{m-1}\frac{C_{i}^{m-1}}{\mu_{EP}}\ket{J^{r}_{i}}-\xi\sum_{i=2}^{m-1}C_{i}^{m-1}\chi_{i}\ket{J^{r}_{i-1}}$.
By changing the summation index $i\rightarrow i+1$ in the first and third
terms, the above formula reads
$\sum_{i=1}^{m-1}C_{i+1}^{m}\mu_{EP}\chi_{i+1}\ket{J^{r}_{i}}=\xi\sum_{i=2}^{m-1}\frac{C_{i}^{m-1}}{\mu_{EP}}\ket{J^{r}_{i}}-\xi\sum_{i=1}^{m-2}C_{i+1}^{m-1}\chi_{i+1}\ket{J^{r}_{i}}$.
By rearranging the terms, we obtain $\left(C_{2}^{m}\mu_{EP}\chi_{2}+\xi
C_{2}^{m-1}\chi_{2}\right)\ket{J^{r}_{1}}+\sum_{i=2}^{m-2}\left(C_{i+1}^{m}\mu_{EP}\chi_{i+1}-\xi\frac{C_{i}^{m-1}}{\mu_{EP}}+\xi
C_{i+1}^{m-1}\chi_{i+1}\right)\ket{J^{r}_{i}}+\left(C_{m}^{m}\mu_{EP}\chi_{m}-\xi\frac{C_{m-1}^{m-1}}{\mu_{EP}}\right)\ket{J^{r}_{m-1}}=0$.
By invoking the linear independence of vectors $\ket{J_{i}^{r}}$ and using the
above expression for $C_{2}^{2}$ , we arrive at the following expressions for
the coefficients $C_{2}^{m}$ and $C_{m}^{m}$:

$\displaystyle C_{2}^{m}$
$\displaystyle=\left(-\frac{1}{\mu_{EP}}\right)^{m}\frac{\xi^{m-1}}{\chi_{2}}$
$\displaystyle C_{m}^{m}$
$\displaystyle=\left(\frac{\xi}{\mu_{EP}^{2}}\right)^{m-1}\frac{1}{\chi_{2}\chi_{3}\ldots\chi_{m}}$
(25)

together with the recursive relation:

$\displaystyle
C^{m}_{i}=\left(\frac{1}{\mu_{EP}}\right)^{2}\frac{\xi}{\chi_{i}}C_{i-1}^{m-1}-\frac{\xi}{\mu_{EP}}C_{i}^{m-1},$
(26)

where $i=3,4,\ldots,m-1$. Relation (26) is a recursive equation with variable
coefficients. The unknown expansion constants $C_{i}^{m}$ in Eq. (26) can be
obtained by the aid of the known coefficients in Eq. (.6). For instance, the
value of $C_{3}^{4}$ depends on those of $C_{2}^{3}$ and $C_{3}^{3}$ which are
obtained from the first and second lines of Eq. (.6). In general, one can
confirm by direct substitution that the solution of Eq. (26) can be written
as:

$\displaystyle C_{i}^{m}=(-1)^{m+i}\begin{pmatrix}m-2\\\
i-2\end{pmatrix}\frac{\mu_{EP}^{2-m-i}}{\chi_{2}\chi_{3}\ldots\chi_{i}}\xi^{m-1},$
(27)

where $\begin{pmatrix}m\\\ n\end{pmatrix}\equiv\frac{m!}{n!(m-n)!}$ is the
binomial coefficient. Thus we finally arrive at:

$\displaystyle\ket{K^{r}_{m}}=\sum_{i=2}^{m}(-1)^{m+i}\begin{pmatrix}m-2\\\
i-2\end{pmatrix}\frac{\mu_{EP}^{2-m-i}\xi^{m-1}}{\chi_{2}\chi_{3}\ldots\chi_{i}}\ket{J^{r}_{i}}.$
(28)

It is worth noting that the expansion coefficients obtained above are
functions of the frequency $\omega$ of the input drive signal.

###  .7 Systems with multiple exceptional points

Let us assume a Hamiltonian with two exceptional points of order, say, $M_{1}$
and $M_{2}$ and energy eigenvalues $\Omega_{EP}^{(1)}$, $\Omega_{EP}^{(2)}$
respectively (the case with several exceptional points is a straightforward
generalization). The generalized eigenvalue equations for the left and right
generalized eigenvectors associated with the two EPs are given by:

$\displaystyle\left(\hat{H}_{EP}-\Omega_{EP}^{(1)}\right)\ket{J_{m}^{r}}$
$\displaystyle=\chi_{m}^{(1)}\ket{J_{m-1}^{r}}$ (29)
$\displaystyle\bra{J_{m}^{l}}\left(\hat{H}_{EP}-\Omega_{EP}^{(1)}\right)$
$\displaystyle=\chi_{m}^{(1)}\bra{J_{m-1}^{l}}$ (30)

and

$\displaystyle\left(\hat{H}_{EP}-\Omega_{EP}^{(2)}\right)\ket{I_{k}^{r}}$
$\displaystyle=\chi_{k}^{(2)}\ket{I_{k-1}^{r}}$ (31)
$\displaystyle\bra{I_{k}^{l}}\left(\hat{H}_{EP}-\Omega_{EP}^{(2)}\right)$
$\displaystyle=\chi_{k}^{(2)}\bra{I_{k-1}^{l}},$ (32)

where $m=1,2,\ldots,M_{1}$ and $k=1,2,\ldots,M_{2}$ and $\chi_{1}^{(1,2)}=0$.
In what follows, we show that the generalized eigenvectors associated with the
two different EPs are biorthogonal. Let us start by inspecting the expression
$\braket{J_{m}^{l}}{\left(\hat{H}_{EP}-\Omega_{EP}^{(1)}\right)^{m}}{I_{1}^{r}}$.
It can be evaluated by first applying the operator
$\left(\hat{H}_{EP}-\Omega_{EP}^{(1)}\right)^{m}$ to the vector
$\ket{I_{1}^{r}}$ from the left or by acting on the vector $\bra{J_{m}^{l}}$
from the right. By doing so, we obtain $\braket{J_{m}^{l}}{I_{1}^{r}}=0$.
Similarly, one can show that $\braket{J_{1}^{l}}{I_{k}^{r}}=0$.

Figure 11: General structure of the linear response associated with a non-
Hermitian system that exhibits two different EPs. As can be seen, each domain
is coupled only to itself. In other words, there is no coupling between the
two sub-spaces spanned by the generalized eigenvectors associated with the two
EPs. Thick arrows indicate collective coupling between the domains. The
details of the coupling (not shown here) follow a scheme identical to that
depicted in Fig. 5b.

Next, we evaluate the expression
$\braket{J_{m}^{l}}{\left(\hat{H}_{EP}-\Omega_{EP}^{(1)}\right)}{I_{k}^{r}}$
for $k,m>1$. By following the procedure described above and by using Eqs. (30)
and (31), we find
$\left(\Omega_{EP}^{(2)}-\Omega_{EP}^{(1)}\right)\braket{J_{m}^{l}}{I_{k}^{r}}=\chi_{m}^{(1)}\braket{J_{m-1}^{l}}{I_{k}^{r}}-\chi_{k}^{(2)}\braket{J_{m}^{l}}{I_{k-1}^{r}}$.
The above formula can be used recursively to replace
$\braket{J_{m-1}^{l}}{I_{k}^{r}}$ and $\braket{J_{m}^{l}}{I_{k-1}^{r}}$ by
generalized eigenvectors having lower order indices to arrive at
$\braket{J_{m}^{l}}{I^{r}_{k}}=\sum_{m^{\prime}=2}^{m}c_{m^{\prime}}^{(1)}\braket{J_{m^{\prime}}^{l}}{I_{1}^{r}}+\sum_{k^{\prime}=1}^{k}c_{k^{\prime}}^{(2)}\braket{J_{1}^{l}}{I_{k^{\prime}}^{r}}$.
But the right hand side of this last expression is zero, and hence
$\braket{J_{m}^{l}}{I^{r}_{k}}=0$. Similarly,
$\braket{I_{k}^{l}}{J_{m}^{r}}=0$. In other words, the generalized
eigenvectors of the two different EPs form a biorthogonal set of vectors.

Equipped with this information, we can consider any arbitrary input signal
$\sum_{n=1}^{N-M_{1}-M_{2}}c_{n}\ket{\psi_{n}^{r}}+\sum_{m=1}^{M_{1}}d_{m}\ket{J_{m}^{r}}+\sum_{k=1}^{M_{2}}f_{k}\ket{I_{k}^{r}}$,
and repeat the same procedure as in supplementary note .4, to obtain:

$\displaystyle\hat{G}_{EP}\ket{u}=$
$\displaystyle\left(\sum\limits_{n=1}^{N-M_{1}-M_{2}}\frac{\ket{\psi_{n}^{r}}\bra{\psi_{n}^{l}}}{\omega-\Omega_{n}}+\sum\limits_{m=1}^{M_{1}}\sum\limits_{m^{\prime}=m}^{M_{1}}\alpha_{m^{\prime}}^{(m)}\frac{\ket{J_{m}^{r}}\bra{\tilde{J}_{m^{\prime}}^{l}}}{\left(\omega-\Omega_{EP}^{(1)}\right)^{m^{\prime}-m+1}}+\right.$
$\displaystyle\qquad\left.\sum\limits_{k=1}^{M_{2}}\sum\limits_{k^{\prime}=k}^{M_{2}}\beta_{k^{\prime}}^{(k)}\frac{\ket{I_{k}^{r}}\bra{\tilde{I}_{k^{\prime}}^{l}}}{\left(\omega-\Omega_{EP}^{(2)}\right)^{k^{\prime}-k+1}}\right)\ket{u}$
(33)

where $\alpha_{m}^{(m)}=1,\
\alpha_{m^{\prime}}^{(m)}=\alpha_{m^{\prime}-1}^{(m)}\chi_{m^{\prime}}^{(1)}$
and $\beta_{k}^{(k)}=1,\
\beta_{k^{\prime}}^{(k)}=\beta_{k^{\prime}-1}^{(k)}\chi_{k^{\prime}}^{(2)}$.
As discussed in supplementary note .2 for the case of only one EP, the vectors
$\bra{\tilde{J}_{m}^{l}}$ and $\bra{\tilde{I}_{k}^{l}}$ span the dual domains
associated with $\mathcal{D}_{EP}^{(1,2)}$ and satisfy the relations
$\braket{\tilde{J}_{m}^{l}}{J_{n}^{r}}=\delta_{n,m}$ and
$\braket{\tilde{I}_{m}^{l}}{I_{n}^{r}}=\delta_{n,m}$. The general structure
described by the above expansion is shown schematically in supplementary Fig.
11.

###  .8 Photonic simulations

Here we explain the details of the simulations associated with Fig. 6a in the
main text. The outer radius and width of the microring resonator was chosen to
be $R=5\>\upmu$m and $w=0.25\>\upmu$m, respectively. The material refractive
index is assumed to be $n_{r}=3.47$ and the background index was taken as
$n_{b}=1.44$ (relevant to silicon photonics). The straight waveguides are of
the same material and width as the ring. The edge-to-edge separation between
the ring and each of the waveguides is $d=0.2\>\upmu$m. By first using an add-
drop configuration (in the absence of the mirror) and plotting the drop signal
as a function of frequency (see supplementary Fig. 12), we evaluate the value
of the decay rate: $\gamma=123\>$GHz. From the same figure, we also find that
$\omega_{o}=1217\times 10^{12}\>\text{sec}^{-1}$, corresponding to
$f_{o}=193.7\>\text{THZ}$ or equivalently $\lambda_{o}=1548\>$nm for the TE
optical mode. The mirror is implemented by using a 100-nm-thick silver layer.
The absolute value of the amplitude reflection coefficient of the mirror is
obtained by simulating the waveguide-mirror system in the absence of the ring
resonator, and is found to be $|r|\approx 0.98$. The stored energy enhancement
$\eta$ is calculated by integrating the time averaged energy density over the
ring resonator volume for resonant excitation from ports $\text{P}_{2}$ and
$\text{P}_{1}$ under steady-state conditions (see Fig. 6b in the main text).
All simulations were performed by using 2D finite element method available
from COMSOL software package.

Figure 12: (a) Add-drop filter configuration for evaluating the resonant
frequency and decay rate of a microring resonator. The design parameters of
the ring resonator and the waveguides are listed in the text. (b) Optical
power transmission as a function of frequency detuning from the resonant
frequency (estimated from the simulation data to be $f_{o}=193.7\>\text{THZ}$)
as obtained by frequency domain finite element numerical analysis. From the
plot, we estimate that $\gamma=123\>$GHZ.

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