#  Eigenfunction non-orthogonality factors and the shape of CPA-like dips in a
single-channel reflection from lossy chaotic cavities.

Yan V. Fyodorov  Department of Mathematics, King’s College London, London WC26
2LS, United Kingdom  L.D.Landau Institute for Theoretical Physics, Semenova
1a, 142432 Chernogolovka, Russia  Mohammed Osman  Department of Mathematics,
King’s College London, London WC26 2LS, United Kingdom

###### Abstract

Motivated by the phenomenon of Coherent Perfect Absorption, we study the shape
of the deepest minima in the frequency-dependent single-channel reflection of
waves from a cavity with spatially uniform losses. We show that it is largely
determined by non-orthogonality factors $O_{nn}$ of the eigenmodes associated
with the non-selfadjoint effective Hamiltonian. For cavities supporting
chaotic ray dynamics we then use random matrix theory to derive, fully non-
perturbatively, the explicit probability density ${\cal P}(O_{nn})$ of the
non-orthogonality factors for systems with both broken and preserved time
reversal symmetry. The results imply that $O_{nn}$ are heavy-tail distributed,
with the universal tail ${\cal P}(O_{nn}\gg 1)\sim O_{nn}^{-3}$.

A surge of interest related to the proposal of constructing the so-called
Coherent Perfect Absorbers (CPA)[1] has been evident in recent years. The
simplest CPA’s can be looked at as a scattering system ( e.g. a cavity) with a
small amount of loss which completely absorbs a monochromatic wave incident at
a particular frequency [2]. Nonlinear effects may further help to make CPA’s a
broad-band phenomenon[3] increasing prospects of its practical implementation,
including optical filters and switches or logic gates for use in optical
computers. Recently, a CPA in a rectangular cavity with randomly positioned
scatterers and absorption due to a single antenna has been realised
experimentally [4], paving a way to the construction of CPAs based on
disordered cavities. In another recent experiment, a CPA has been realised
with a two-port microwave graph system, both with and without time-reversal
symmetry [5], and yet in another way with chaotic cavity with programmable
meta-atom inclusions [6]. In those settings the relevant framework is that of
chaotic wave scattering, and the CPA state corresponds to an eigenstate of the
scattering matrix $S(\omega)$ with zero eigenvalue at a real frequency. This
fact naturally motivates rising interest in a more general question of
characterizing S-matrix complex zeros, as well as their manifestation in
physical observables, which has not been systematically studied for wave
chaotic systems with losses until very recently [7, 8, 9, 10, 11, 12].

Although CPA’s are clearly a new exciting development, a somewhat related
phenomena in lossy chaotic scattering have been discussed actually long ago.
To this end one may invoke an early influential experiment by Doron, Smilansky
and Frenkel who studied the frequency-dependent reflection coefficient
$R(\omega)=|S(\omega)|^{2}$ of an electromagnetic wave sent via a single-mode
waveguide to a cavity shaped in the form of a chaotic billiard. This line of
experimental activity maintained its strength over several decades [14, 15,
16, 17, 18] providing important insights and stimulating theoretical research.

Returning to the setting of [13] one should recall that the scattering matrix
of a single-channel system without gain or loss must be unimodular due to flux
conservation: $S(\omega)=e^{i\delta(\omega)}$, where real $\delta(\omega)$ is
known as the scattering phaseshift. The frequency derivative of the scattering
phaseshift is an important scattering characteristics called Wigner time delay
[19] and is well-described by the sum of Lorentzians of widths $\Gamma_{n}$
centered at positions $E_{n}$,

$\tau_{W}(\omega):=\frac{d}{d\omega}\delta(\omega)=2\sum_{n=1}^{N}\frac{\Gamma_{n}}{(\omega-
E_{n})^{2}+\Gamma_{n}^{2}}$ (1)

One of the main experimental observations made in [13] was that the reflection
coefficient $R(\omega)$ showed considerable variations with frequency
$\omega$, with many pronounced dips to low values $R(\omega)\lesssim 0.1$ at
some frequencies, reminiscent of an ”imperfect” version of modern CPA. Such a
behaviour is clearly incompatible with flux conservation and has been
reasonably attributed to presence of uniform losses in resonator walls. Such
losses had been then taken into account phenomenologically by adding a small
imaginary increment to the real frequency: $\omega\to\omega+i\epsilon$ with
$\epsilon>0$. Assuming that absorption is weak, i.e. $\epsilon\ll\Delta$,
where $\Delta$ stands for the mean spacing between eigenfrequencies in the
closed cavity in a given frequency range, one may expand in $\epsilon$
yielding a relation between $R(\omega)$ and the Wigner time-delay:
$|S(\omega+i\epsilon)|^{2}\approx e^{-2\epsilon\,\tau_{W}(\omega)}$. The paper
thus was among the first promoting interest in statistics of Wigner time
delays in wave-chaotic scattering, which after three decades still remains an
active research topic, see e.g. [20, 21, 22, 23, 24, 25, 26, 27, 28] as well
as [29, 30, 31, 32, 33, 34] and references therein. Measuring the phaseshifts
$\delta(\omega)$ independently allowed to test experimentally the relation
between $R(\omega)$ and the Wigner time-delay, and overall good agreement has
been reported in [13], with discrepancies close to the deepest minima
attributed to inacuracies in numerical differentiation.

Our goal in this paper is to have a closer look at $R(\omega)$ in the above
setting and to demonstrate that its behaviour at the deepest CPA-like dips
bears important physical information which seems to have been not discussed
before when addressing the shape of CPA minima, as e.g. in [6]. The proper
framework for such an analysis is the so called effective Hamiltonian
formalism for wave-chaotic scattering [35, 36, 37, 38, 39, 40, 41]. In such a
formalism the parameters $E_{n}$ and $\Gamma_{n}$ featuring in (1) describe
the positions $z_{n}=E_{n}-i\Gamma_{n},\,n=1,\ldots,N$ of $S-$ matrix poles in
the lower half of the complex frequency plane, which in turn are considered to
be complex eigenvalues of an $N\times N$ non-selfadjoint matrix ${\cal
H}_{eff}=\mathbf{H}_{N}-i{\bf w}\otimes{\bf w}^{\dagger}$ known as the
effective Hamiltonian. In this setting the eigenfrequencies $\omega_{n}$ of
the closed resonator cavity are associated with the real eigenvalues of the
self-adjoint part $\mathbf{H}_{N}=\mathbf{H}_{N}^{\dagger}$, with
corresponding eigenmodes forming an orthonormal basis. Those eigenfrequencies
are converted to the complex $S-$ matrix poles due to coupling of the cavity
to continuum via a single open channel supporting waves coming from (and
escaping to) infinity. The coupling between the cavity and the channel is then
characterised by the vector ${\bf w}=(w_{1},\ldots,w_{N})$ of coupling
amplitudes. The single-channel scattering matrix $S(\omega+i\epsilon)$ of a
system with spatially-uniform losses $\epsilon>0$ in this approach can be
represented as

$\vspace{-0.2cm}S(\omega+i\epsilon)=\prod_{n=1}^{N}\frac{\omega+i\epsilon-z^{*}_{n}}{\omega+i\epsilon-
z_{n}},$ (2)

where $z_{n}^{*}$ stands for the complex conjugate of $z_{n}$. Such relation
immediately implies that in the absence of losses
$S(\omega)=e^{i\delta(\omega)}$ as dictated by flux conservation, and Eq.(1)
then easily follows. In presence of a uniform absorption $\epsilon>0$
unimodularity is lost, and it is completely clear that the deepest dips in
$|S(\omega+i\epsilon)|^{2}$ happen when the condition
$\omega+i\epsilon=E_{n}+i\Gamma_{n}$ is approximately satisfied.

Figure 1: A typical profile of the reflection coefficient at the centre of the
spectrum for $N=200,\,\gamma=0.5$ and two different absorptions:
$\epsilon=10^{-3}\ll\Delta$ (top) and $\epsilon=0.01\sim\Delta$ (bottom)

It is important to note that for a single-channel wave-chaotic system the set
of parameters $\Gamma_{n}$ (known as the resonance widths) is generically
random. Note that the value of the uniform loss $\epsilon$ can be reliably
extracted from scattering data, as discussed, e.g. in [42]. By changing the
frequency $\omega$ one always can ensure satisfying $\omega=E_{n}$ for a given
$n$, but if choosing uniform absorption rate $\epsilon$ is not fully under
experimental control the condition $\epsilon=\Gamma_{n}$ can be satisfied only
purely accidentally. If such coincidence however does happen, one comes
exactly to the CPA situation of vanishing reflection, see Fig.(1). In what
follows we assume that the reflection dip is located at the frequency
$\omega=E_{n}$ but may be not perfect, though deep enough, i.e the value at
the minimum $0\leq R_{min}\ll 1$. This is ensured by imposing the relations
$\omega=E_{n}+\delta\omega$ and $\Gamma_{n}=\epsilon+\delta_{n}$ assuming the
range of frequencies around the dip $\delta\omega\ll\Delta$ and the resonance
mismatch parameter $|\delta_{n}|\ll\epsilon$. Under these conditions using
Eq.(2) we immediately see that the shape of the reflection dip centered at the
frequency $\omega=E_{n}$ should be well described by the following profile:

$R(\omega)=K_{n}\,\frac{(\omega-E_{n})^{2}+\delta^{2}_{n}}{(\omega-
E_{n})^{2}+4\epsilon^{2}},\quad K_{n}=\prod_{k\neq
n}^{N}\frac{|z_{k}-z_{n}|^{2}}{|z^{*}_{k}-z_{n}|^{2}},$ (3)

Such a profile is then uniquely characterized by its depth $R_{min}$ and the
curvature at the minimum
$C:=\frac{d^{2}}{d\omega^{2}}R(\omega)|_{\omega=E_{n}}$ which are given by

$R_{min}=K_{n}\,\frac{\delta^{2}_{n}}{4\epsilon^{2}},\quad
C=\frac{K_{n}}{2\epsilon^{2}}.$ (4)

The fluctuations of the dip shapes over different reflection minima are thus
controlled by the statistics of resonance widths $\Gamma_{n}$ and that of the
factors $K_{n}$. Whereas the statistical information about $\Gamma_{n}$ in a
few-channel system is available theoretically [43, 44, 45, 46] and is in a
good agreement with experiments [47], the factor $K_{n}$ looks less familiar,
and its meaning is not immediately obvious.

To this end we observe below that such a factor can be actually given a direct
interpretation, by relating it to nonorthogonal eigenfunctions of the
effective Hamiltonian ${\cal H}_{eff}$. The latter Hamiltonian is not only
non-selfadjoint, but also non-normal, i.e. does not commute with its adjoint,
hence is characterized by the pair of left ${\bf l}_{n}$ and right ${\bf
r}_{n}$ eigenvectors corresponding to the same complex eigenvalue $z_{n}$, or
equivalently ${\cal H}_{eff}{\bf r}_{n}=z_{n}{\bf r}_{n}$ and ${\cal
H}^{\dagger}_{eff}{\bf l}_{n}=z^{*}_{n}{\bf l}_{n}$. One can always choose the
sets of right- and left eigenvectors to be bi-orthogonal, with scalar products
satisfying ${\bf l}^{\dagger}_{n}{\bf r}_{m}=\delta_{nm}$. The overlap non-
orthogonality matrix is then defined as $O_{mn}=({\bf r}^{\dagger}_{m}{\bf
r}_{n})({\bf l}^{\dagger}_{n}{\bf l}_{m})$ and its diagonal entries
$O_{nn}=({\bf r}^{\dagger}_{n}{\bf r}_{n})({\bf l}^{\dagger}_{n}{\bf l}_{n})$
are frequently referred to as self-overlaps. For normal matrices ${\bf
l}_{n}={\bf r}_{n}$ implying $O_{nn}=1,\forall n$, whereas for a generic non-
normal matrix $O_{nn}>1$ so that the difference $t_{n}=O_{nn}-1$ can be used
as a measure of non-normality. It is therefore intriguing to find that the
factor $K_{n}$ featuring in Eq.(3) turns out to be reciprocal to this simplest
characteristics of non-normality: $K_{n}=O^{-1}_{nn}$.

Statistics of non-orthogonal eigenvectors in non-normal random matrices is a
topic of active research interest in Mathematical and Theoretical Physics
starting from the seminal works [48, 49]. Various aspects of non-orthogonality
received a lot of attention recently [50, 51, 52, 53, 54, 55, 56, 57, 58, 59,
60, 61, 62, 63]. In particular, self-overlaps are controlling sensitivity of
the complex eigenvalues to perturbation of matrix entries, and
$\kappa_{n}:=\sqrt{O_{nn}}$ are frequently called in the mathematical
literature the eigenvalue condition numbers.

To this end, the phenomenon of wave-chaotic scattering provides quite a unique
framework for addressing eigenfunction non-orthogonality and its
manifestations not only theoretically [64, 65, 66, 67, 68, 69, 70, 71] but
also experimentally [72, 73, 74]. In the chaotic scattering context self-
overlaps are known as Petermann factors, and being responsible for the excess
noise in lasing resonators have been addressed in an early paper [65]. Whereas
the mean value for self-overlaps for wave-chaotic scattering in systems with
broken time-reversal invariance has been calculated nonperturbatively using
random Matrix Theory (RMT) techniques in [65, 68, 69], any other statistical
characteristics of self-overlaps were only so far addressed in the framework
of second-order perturbation theory in the limit of weak channel coupling:
$\gamma:=|{\bf w}|^{2}\ll 1$.

In this paper we develop a method of non-perturbative evaluation of the
probability density for the variable $t_{n}=O_{nn}-1$ valid for the single-
channel chaotic scattering at any coupling $\gamma>0$. For generic non-normal
matrices the relations between overlaps $O_{nn}$ and complex eigenvalues
$z_{n}$ are very involved and hard to utilize. However in the special case of
${\cal H}_{eff}$ given by a rank-one perturbation of the self-adjoint matrix
$\mathbf{H}_{N}$ such relation is very explicit [68], and shows that
$O_{nn}=K_{n}^{-1}$, with $K_{n}$ exactly as defined in Eq.(3). For
completeness we provide a direct verification of this relation in the
Supplementary Materials section [75].

We expose main features of our method on a somewhat simpler example of systems
with broken time-reversal invariance, the case of preserved invariance being
discussed at the end, with details given in [75]. We rely on the standard
assumption that the $N\times N$ Hermitian matrix ${\bf H}_{N}$ featuring in
the definition of effective Hamiltonian belongs to the Gaussian Unitary
ensemble of RMT. In the physically relevant limit $N\gg 1$ the corresponding
eigenvalue density is then given by $\rho=\frac{1}{2\pi}\sqrt{4-X^{2}}$, hence
the mean eigenvalue spacing $\Delta=1/(N\rho)\sim N^{-1}$. The coupling
amplitudes $w_{i},i=1,\ldots,N$ are normalized in such a way that
$\gamma=\sum_{i}|w_{i}|^{2}$ stays finite as $N\to\infty$. With this choice as
$N\gg 1$ statistics of all scattering characteristics varying at frequency
scales of the order of $\Delta$ around a spectral point $|X|<2$ can be shown
to be universal and depending only on the “renormalized” coupling strength
$g=\frac{1}{2\pi\rho}\left(\gamma+\frac{1}{\gamma}\right)\geq 1$ but not on
the particular choice of $w_{i}$, see e.g.[35, 76]. E.g. we can choose ${\bf
w}=\sqrt{\gamma}\tilde{{\bf w}}$, with components $\tilde{w}_{i}$ independent,
mean zero complex Gaussian, with covariance
$\left\langle\tilde{w}^{*}_{i}\tilde{w}_{j}\right\rangle=N^{-1}\delta_{ij}$,
where the angular brackets always denote averaging over relevant
distributions. Equivalently, one may choose $\tilde{{\bf w}}$ to be any fixed
of unit length, in particular $\tilde{{\bf w}}={\bf e}$ with ${\bf
e}^{T}=(1,0,\ldots,0)$. Under these assumptions typically
$\Gamma_{n}\sim\Delta$, and we introduce $y_{n}=\pi\Gamma_{n}/\Delta$. This is
the main parameter controlling both non-Hermiticity and non-normality of the
effective Hamiltonian ${\cal H}_{eff}$. As long as $g\gg 1$ the departure from
Hermiticity/normality is small and to a large extent can be studied
perturbatively, whereas in the so-called perfect coupling case $g=1$ those
effects are the strongest.

With those preparations we can formulate our main result. Define, using Dirac
$\delta(x)$ and $\delta^{(2)}(z):=\delta(Re\,z)\delta(Im\,z)$ the following
object

${\cal
P}(t;z)=\left\langle\frac{1}{N}\sum_{i=1}^{N}\delta(O_{nn}-1-t)\delta^{(2)}(z-z_{n})\right\rangle$
(5)

interpreted as the (conditional) probability density of the non-orthogonality
factor $t=O_{nn}-1$ corresponding to eigenvalues in the vicinity of a point
$z=X-iY,\,Y>0$ in the complex plane. Then the limiting density ${\cal
P}^{(2)}_{y}(t):=\lim_{N\to\infty}\frac{1}{\pi\rho N}{\cal
P}(t;z=X-i\frac{y}{\pi\rho N})$ takes the following form for the present
symmetry class:

${\cal
P}^{(2)}_{y}(t)=\frac{16}{t^{3}}\,e^{-2gy}\,\mathbb{L}_{2}\,e^{-2gy\left(1+\frac{2}{t}\right)}I_{0}\left(\frac{4y}{t}\sqrt{(g^{2}-1)(1+t)}\right)$
(6)

where $I_{\nu}(x)$ stands for the modified Bessel function and
$\mathbb{L}_{2}$ is a differential operator acting on functions $f(y)$ as

$\mathbb{L}_{2}\,f(y)=\left\\{1+\left(\frac{\sinh{2y}}{2y}\right)^{2}+\frac{1}{2y}\left(1-\frac{\sinh{4y}}{4y}\right)\frac{d}{dy}\right.$
(7)

$\left.+\frac{1}{4}\left(\left(\frac{\sinh{2y}}{2y}\right)^{2}-1\right)\frac{d^{2}}{dy^{2}}\right\\}\,y^{2}f(y).$

The following remarks on the above formula are due. First, the most prominent
feature in (6) is the heavy-tail behaviour ${\cal P}^{(2)}_{y}(t)\sim t^{-3}$
for $t\gg 1$ rendering all moments $\left\langle
O^{l}_{nn}\right\rangle,\,l\geq 2$ divergent. This tail behaviour is exactly
the same as found earlier in other complex-valued non-normal random matrices
[53, 54, 58] and seems to be the most universal feature of random diagonal
overlaps. Second, the integrals $\int{\cal P}^{(2)}_{y}(t)t^{l}\,dt$ can be
explicitly performed for $l=0$ and $l=1$, reproducing the known mean density
of the resonance widths [43] and the mean diagonal overlap [68]. Finally, with
suitable adjustments, the method should be applicable to the rank-one family
of subunitary deformations of the Haar-distributed unitary random CUE
matrices. The corresponding model introduced in [77, 78] naturally appears in
the context of time-periodic scattering. The statistics of non-orthogonality
factors are then expected to be exactly the same as given by (6), with the
perfect coupling case $g=1$ corresponding in that context to the so-called
truncated CUE [79].

Our starting point is to consider (negative) moments of the diagonal overlap
defined for $z=X-iY$ and real $p$ as ${\cal
O}_{p}=\left\langle\frac{1}{N}\sum_{n=1}^{N}\delta^{(2)}(z-z_{n})O_{nn}^{-p}\right\rangle$.
Fixing the coupling parameter $\gamma>0$ it is technically convenient to
consider first the fixed coupling amplitudes ${\bf w}=\sqrt{\gamma}{\bf e}$.
We note that as $\sum_{n}Imz_{n}=-\gamma$ and $Imz_{n}\leq 0,\,\forall n$ it
is enough to consider $0\leq Y<\gamma$. Following the procedure explained for
$p=-1$ in [68], we denote $\gamma_{*}=\gamma-Y>0$, use the known joint
probability density of all $z_{n},\,n=1,\ldots,N$ [44, 80, 81], and rely on
GUE invariance. After defining $\tilde{z}=\sqrt{\frac{N}{N-1}}z$, considering
$(N-1)\times(N-1)$ matrices $\tilde{{\cal
H}}_{eff}:=\mathbf{H}_{N-1}-i\sqrt{\frac{N}{N-1}}\gamma_{*}{\bf
e}_{N-1}\otimes{\bf e}_{N-1}^{T}$ and disregarding the explicit
proportionality constants we arrive at the following representation

${\cal
O}_{p}(z)\propto\frac{\gamma_{*}^{N-2}}{\gamma^{N-1}}\,e^{-\frac{N}{2}(X^{2}-Y^{2}+\gamma^{2})+\frac{N-1}{2}\gamma_{*}^{2}}{\cal
M}^{(\beta=2)}_{p}(z)$ (8)

where

${\cal
M}^{(\beta=2)}_{p}(z)=\left\langle\frac{\left[\det(\tilde{z}-\tilde{{\cal
H}}_{eff})\det(\tilde{z}^{*}-\tilde{{\cal
H}}^{\dagger}_{eff})\right]^{p+1}}{\left[\det(\tilde{z}-\tilde{{\cal
H}}^{\dagger}_{eff})\det(\tilde{z}^{*}-\tilde{{\cal
H}}_{eff})\right]^{p}}\right\rangle$ (9)

with the average now going over the reduced GUE matrices $\mathbf{H}_{N-1}$ of
the size $(N-1)\times(N-1)$. The major task is to evaluate ${\cal
M}^{(\beta=2)}_{p}(z)$ as $N\to\infty$ for a given $\gamma>0$ and real $p$
assuming the scaling $Imz=Y\sim N^{-1}$. It is easy to check that in such a
limit one may safely replace $\tilde{{\cal H}}_{eff}$ with the original ${\cal
H}_{eff}=\mathbf{H}_{N}-i\gamma\,\tilde{{\bf w}}\otimes\tilde{{\bf
w}}^{\dagger}$, without changing the result. When evaluating ${\cal
M}^{(\beta=2)}_{p}(z)$ we also find it more technically convenient to consider
the random complex gaussian channel vector ${\bf\tilde{w}}$ rather than
keeping it of unit length, and can show that both choices lead to the same
result.

The crucial trick which allows to evaluate ${\cal M}^{(\beta=2)}_{p}(z)$
efficiently is to define a version of the matrix element of the GUE resolvent
(Green’s function), also known in scattering theory as the $K-$matrix, via

$G(z):={\bf\tilde{w}}^{\dagger}\frac{1}{z-\mathbf{H}_{N}}{\bf\tilde{w}}=u+iv,\,v>0$
(10)

and to use the identity $\det(z-{\cal
H}_{eff})=\det(z-\mathbf{H}_{N})(1+i\gamma G(z))$ to rewrite our main object
of interest as

${\cal
M}^{(\beta=2)}_{p}(z)=\left\langle|\det(z-\mathbf{H}_{N})|^{2}\frac{\left|1+i\gamma
G(z)\right|^{2(p+1)}}{\left|1+i\gamma G(z^{*})\right|^{2p}}\right\rangle$ (11)
$=\int_{-\infty}^{\infty}du\int_{0}^{\infty}dv\frac{\left[(1-\gamma
v)^{2}+\gamma^{2}u^{2}\right]^{p+1}}{\left[(1+\gamma
v)^{2}+\gamma^{2}u^{2}\right]^{p}}{\cal P}^{(\beta=2)}_{z}(u,v)$ (12)

in terms of the function

${\cal
P}^{(\beta=2)}_{z}(u,v)=\left\langle|\det(z-\mathbf{H}_{N})|^{2}\right.$ (13)
$\left.\times\delta\left(u-\mbox{Re}G(z)\right)\,\delta\left(v-\mbox{Im}G(z)\right)\right\rangle$

where now averaging includes one over $\tilde{{\bf w}}$ as well. Such function
represents a “deformed” version of the probability density for the Green’s
function entry, Eq.(10). Without deformation it enjoyed thorough attention,
starting from [82], and methods to evaluate it within RMT context (and beyond)
has been discussed e.g. in [83, 84]. Following one of those methods, we
consider the Fourier-Laplace transform ${\cal F}(s,p)=\int
du\frac{e^{isu}}{2\pi}\int_{0}^{\infty}dve^{-pv}{\cal P}_{z}(u,v)$, and
perform the average over $\tilde{{\bf w}}$ first, before that over the GUE
matrices $\mathbf{H}_{N}$. Along those lines we notice that

$is\mbox{Re}\,G(z)-p\mbox{Im}\,G(z)={\bf\tilde{w}}^{\dagger}\frac{is(X-\mathbf{H}_{N})-pY}{(z-\mathbf{H}_{N})(\overline{z}-\mathbf{H}_{N})}{\bf\tilde{w}}$

so that the average over ${\bf\tilde{w}}$ amounts to performing a simple
Gaussian integral. For simplicity of the presentation we set $X=0$ below, the
results for any $|X|<2$ recovered by a simple rescaling, and scale $Y=iy/N$,
Doing this we arrive after simple algebraic manipulations to the following
identity

${\cal
F}^{(\beta=2)}(s,p)=\left\langle\frac{|\det(\frac{iy}{N}-\mathbf{H}_{N})|^{4}}{\det\left(-i\frac{\tau_{+}}{2N}-\mathbf{H}_{N}\right)\det\left(-i\frac{\tau_{-}}{2N}-\mathbf{H}_{N}\right)}\right\rangle$
(14)

where we defined $\tau_{+}=s+\sqrt{s^{2}+4y(y+p)}>0$ and
$\tau_{-}=s-\sqrt{s^{2}+4y(y+p)}<0$. The objects like those featuring in the
right-hand side of (14) have a long history of studies, starting from [85],
and then in increasing generality in [86, 87] culminating in [88]. Using the
latter results an explicit expression for ${\cal F}(s,p)$ as $N\to\infty$ can
be found as explained in [75]. Then on performing the Laplace-Fourier
inversion of ${\cal F}(s,p)$ [75] one arrives at a quite compact result:

${\cal
P}^{(\beta=2)}_{z}(u,v)\propto\mathbb{L}_{2}\frac{1}{v^{3}}e^{-y\frac{u^{2}+v^{2}+1}{v}}$
(15)

with the differential operator $\mathbb{L}_{2}$ defined in (7). Now we can
substitute (15) back to (12) and after introducing a new variable
$x=\frac{u^{2}+v^{2}+1}{2v}>1$ arrive to the formula for
$M^{(\beta=2)}_{p}(y)=\lim_{N\to\infty}{\cal
M}^{(\beta=2)}_{p}(z=-\frac{iy}{N})$ [75]:

$M^{(\beta=2)}_{p}(y)\propto\gamma\mathbb{L}_{2}\int_{1}^{\infty}da\frac{(a-1)^{p+1}}{(a+1)^{p}}$
(16) $\times e^{-2yga}I_{0}(2y\sqrt{(a^{2}-1)(g^{2}-1)})$

The above formula is well-defined for any $p>-2$. Evaluating straightforwardly
the limit $N\to\infty$ in (8) while keeping $X=0,Y=y/N$ gives us the moments
of the self-overlap as ${\cal O}_{p}\propto e^{-2gy}M_{p}(y)$ and the main
result (6) then follows by a straightforward moment inversion after setting
$a=2t^{-1}+1$.

The same method can be adapted mutatis mutandis for deriving a similar result
for systems with preserved time-reversal symmetry, with the matrix
$\mathbf{H}_{N}$ belonging to the Gaussian Orthogonal Ensemble of RMT and the
channel coupling vector being real-valued[90]. In that setting the central
result needed for our goal is a correlation function involving half-integer
powers of characteristic polynomials evaluated in [89]. Necessary steps of the
derivation are presented in [75], with the final result being now

${\cal
P}^{(1)}_{y}(t)=\frac{1}{2}\frac{\,e^{-gy}}{\sqrt{t^{5}(1+t)}}\,\,\mathbb{L}_{1}\,e^{-gy\left(1+\frac{2}{t}\right)}I_{0}\left(\frac{2y}{t}\sqrt{(g^{2}-1)(1+t)}\right)$
(17)

where $\mathbb{L}_{1}$ is the following differential operator

$\mathbb{L}_{1}=2\sinh{2y}-\left(\cosh{2y}-\frac{\sinh{2y}}{2y}\right)\left(\frac{3}{y}+2\frac{d}{dy}\right)$
(18)

We see that the heavy-tail behaviour ${\cal P}^{(1)}_{y}(t)\sim t^{-3}$ for
$t\gg 1$ is also manifest in such a case. One can further integrate over $t$
to obtain the mean density of the scaled resonance widths
$y=\pi\Gamma_{n}/\Delta$ in the form:

$\rho^{(1)}(y)=\frac{1}{4\sqrt{2}}e^{-gy}\,\,\mathbb{L}_{1}\Phi(y)$ (19)

where

$\Phi(y)=\int_{1}^{\infty}da\,e^{-gay}\frac{(a-1)}{\sqrt{a+1}}I_{0}\left(y\sqrt{(g^{2}-1)(a^{2}-1)}\right)$
(20)

This expression looks considerably simpler than the 3-fold integral
representation derived earlier for the same density in [45], though one can
check numerically that they are fully equivalent [75].

In conclusion, we related the shape of deep, CPA-like dips in a single-channel
reflection from lossy cavities with the diagonal non-orthogonality factor of
eigenfunctions of the underlying effective Hamiltionian. Using RMT framework
we derived, fully non-perturbatively, the explicit distribution of these
factors for wave-chaotic scattering in systems with both broken and preserved
time reversal symmetry. The results imply that $O_{nn}$ are heavy-tail
distributed, sharing this feature with other instances of non-orthogonality
factors of non-Hermitian ensembles [53, 54, 58, 63] and supporting the claim
of the universality of such a behaviour. Experimentally, statistics of
$O_{nn}$ should be accessible within the framework of ”harmonic inversion”
method [47], or via accurate study of the shape of reflection dips.

Y.V.F. acknowledges financial support from EPSRC Grant EP/V002473/1 ”Random
Hessians and Jacobians: theory and applications”.

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##  I SUPPLEMENTARY MATERIALS

###  I.1 Non-orthogonality factors in the single-channel effective Hamiltonian

Recall the definitions of the effective Hamiltonian ${\cal
H}_{eff}=\mathbf{H}_{N}-i{\bf w}\otimes{\bf w}^{\dagger}$ and its left-right
eigenvectors and eigenvalues:

${\cal H}_{eff}{\bf r}_{n}=z_{n}{\bf r}_{n},\,\,{\bf l}^{\dagger}_{n}{\cal
H}_{eff}=z_{n}{\bf l}^{\dagger}_{n}\quad\mbox{ and }{\bf r}^{\dagger}_{n}{\cal
H}^{\dagger}_{eff}=z^{*}_{n}{\bf r}^{\dagger}_{n}\,\,\quad{\cal
H}^{\dagger}_{eff}{\bf l}_{n}=z^{*}_{n}{\bf l}_{n}\,.$ (21)

We choose the sets of right- and left eigenvectors to be bi-orthogonal, with
scalar products satisfying ${\bf l}^{\dagger}_{n}{\bf r}_{m}=\delta_{nm}$.

In these terms the single-channel scattering matrix at the complex plane
$z=\omega+i\epsilon$ can be rewritten, after a simple algebra, in several
equivalent forms

$S(z)=\prod_{n=1}^{N}\frac{z-z^{*}_{n}}{z-z_{n}}=\frac{\det{\left(z-{\cal
H}_{eff}^{\dagger}\right)}}{\det{\left(z-{\cal H}_{eff}\right)}}$ (22)
$=1-2i{\bf w}^{\dagger}\frac{1}{z-{\cal H}_{eff}}{\bf
w}=1-2i\sum_{n=1}^{N}\frac{\left({\bf w}^{\dagger}{\bf r}_{n}\right)\left({\bf
l}^{\dagger}_{n}{\bf w}\right)}{z-z_{n}}$ (23)

Comparing the residues at the poles $z_{n}$ in (22) and (23) one finds the
relation:

$-2i\left({\bf w}^{\dagger}{\bf r}_{n}\right)\left({\bf l}^{\dagger}_{n}{\bf
w}\right)=(z_{n}-z_{n}^{*})\prod_{k\neq n}\frac{z_{k}-z_{n}^{*}}{z_{k}-z_{n}}$
(24)

Conjugation gives also:

$2i\left({\bf w}^{\dagger}{\bf l}_{n}\right)\left({\bf r}^{\dagger}_{n}{\bf
w}\right)=(z^{*}_{n}-z_{n})\prod_{k\neq
n}\frac{z^{*}_{k}-z_{n}}{z^{*}_{k}-z^{*}_{n}}$ (25)

Multiplying (24) and (25) and rearranging gives:

$\left[2i\left({\bf l}^{\dagger}_{n}{\bf w}\right)\left({\bf w}^{\dagger}{\bf
l}_{n}\right)\right]\left[-2i\left({\bf r}^{\dagger}_{n}{\bf
w}\right)\left({\bf w}^{\dagger}{\bf
r}_{n}\right)\right]=-(z_{n}-z_{n}^{*})^{2}\prod_{k\neq
n}\frac{|z^{*}_{k}-z_{n}|^{2}}{|z_{k}-z_{n}|^{2}}$ (26)

On the other hand one may notice from the definition that $2i{\bf
w}\otimes{\bf w}^{\dagger}={\cal H}_{eff}^{\dagger}-{\cal H}_{eff}$, hence

$2i\left({\bf l}^{\dagger}_{n}{\bf w}\right)\left({\bf w}^{\dagger}{\bf
l}_{n}\right)={\bf l}^{\dagger}_{n}\,\left[2i{\bf w}\otimes{\bf
w}^{\dagger}\right]\,{\bf l}_{n}={\bf l}^{\dagger}_{n}\,\left[{\cal
H}_{eff}^{\dagger}-{\cal H}_{eff}\right]\,{\bf
l}_{n}=(z_{n}^{*}-z_{n})\left({\bf l}^{\dagger}_{n}{\bf l}_{n}\right)$ (27)

and similarly:

$-2i\left({\bf r}^{\dagger}_{n}{\bf w}\right)\left({\bf w}^{\dagger}{\bf
r}_{n}\right)={\bf r}^{\dagger}_{n}\,\left[-2i{\bf w}\otimes{\bf
w}^{\dagger}\right]\,{\bf r}_{n}={\bf r}^{\dagger}_{n}\,\left[{\cal
H}_{eff}-{\cal H}_{eff}^{\dagger}\right]\,{\bf
r}_{n}=(z_{n}-z_{n}^{*})\left({\bf r}^{\dagger}_{n}{\bf r}_{n}\right)$ (28)

Substituting back to (25) shows that:

$\left({\bf l}^{\dagger}_{n}{\bf l}_{n}\right)\left({\bf r}^{\dagger}_{n}{\bf
r}_{n}\right)=\prod_{k\neq n}\frac{|z^{*}_{k}-z_{n}|^{2}}{|z_{k}-z_{n}|^{2}}$
(29)

Finally recalling the definition of overlap non-orthogonality matrix as
$O_{mn}=({\bf r}^{\dagger}_{m}{\bf r}_{n})({\bf l}^{\dagger}_{n}{\bf l}_{m})$
we see that the right-hand side of (29) provides the expression for diagonal
entries $O_{nn}$, and comparing with the definition Eq.(3) of the factor
$K_{n}$ we see that the two are reciprocal.

Finally note that as residues of the scattering matrix at resonance poles can
be extracted from the same harmonic inversion procedure used to get
experimental access to widths $\Gamma_{n}$, the relation (24) could provide a
basis for experimentally studying the statistics of the non-orthogonality
factors.

###  I.2 Correlation function of characteristic polynomials for GUE matrices,
eq.(14)

For convenience of the reader we collect the formulas from Theorem 1.3.2 in
[1] below most useful for our purposes. Denoting the limiting mean eigenvalue
density of $N\times N$ GUE matrices $\mathbf{H}_{N}$ to be
$\rho=\frac{1}{2\pi}\sqrt{4-X^{2}}$ we define
$D(\alpha)=\det{\left(X+\frac{\alpha}{N\rho(X)}-\mathbf{H}_{N}\right)}$ and
keep $\alpha$ fixed as $N\to\infty$. Then in such a limit

$\left\langle\frac{D\left(\alpha^{-}_{1}\right)D\left(\alpha^{-}_{2}\right)D\left(\beta^{-}_{1}\right)D\left(\beta^{-}_{2}\right)}{D(\alpha_{+})D(\beta_{+})}\right\rangle\propto
e^{N\frac{X^{2}}{2}+\frac{X}{\rho(x)}}\frac{\left(\alpha_{1}^{-}-\alpha_{+}\right)\left(\alpha_{2}^{-}-\alpha_{+}\right)\left(\beta_{1}^{-}-\beta_{+}\right)\left(\beta_{2}^{-}-\beta_{+}\right)}{\left(\alpha_{1}^{-}-\alpha_{2}^{-}\right)\left(\beta_{1}^{-}-\beta_{2}^{-}\right)}\,\det{{\cal
D}}$ (30)

where ${\cal D}$ is the following $3\times 3$ matrix:

${\cal
D}=\left(\begin{array}[]{ccc}S(\alpha_{1}^{-},\beta_{1}^{-})&S(\alpha_{1}^{-},\beta_{2}^{-})&S(\alpha_{1}^{-},\alpha_{+})\\\
S(\alpha_{2}^{-},\beta_{1}^{-})&S(\alpha_{2}^{-},\beta_{2}^{-})&S(\alpha_{2}^{-},\alpha_{+})\\\
S(\beta_{+},\beta_{1}^{-})&S(\beta_{+},\beta_{2}^{-})&S(\beta{+},\alpha_{+})\end{array}\right)$
(31)

where in our particular case:

$\alpha_{1,2}^{-}=i\pi\rho(X)y_{1,2},\quad\beta_{1,2}^{-}=-i\pi\rho(X)y_{1,2},\quad\alpha_{+}=-i\frac{\pi\rho(X)}{2}\tau_{-},\quad\beta_{+}=-i\frac{\pi\rho(X)}{2}\tau_{+}$

As $\mbox{Im}\alpha_{+}>0,\,\mbox{Im}\beta_{+}<0$ we have:

$S(\alpha_{1}^{-},\beta_{1}^{-})=\frac{1}{\pi}\frac{\sin{\left(\alpha_{1}^{-}-\beta_{1}^{-}\right)}}{\alpha_{1}^{-}-\beta_{1}^{-}}=\frac{1}{\pi}\frac{\sin{2\pi\rho(X)y_{1}}}{2\pi\rho(X)y_{1}}$
(32)

and similarly

$S(\alpha_{2}^{-},\beta_{2}^{-})=\frac{1}{\pi}\frac{\sin{2\pi\rho(X)y_{2}}}{2\pi\rho(X)y_{2}},\quad
S(\alpha_{1}^{-},\beta_{2}^{-})=S(\alpha_{2}^{-},\beta_{1}^{-})=\frac{1}{\pi}\frac{\sin{2\pi\rho(X)(y_{1}+y_{2})}}{2\pi\rho(X)(y_{1}+y_{2})}.$
(33)

Further

$S(\beta_{+},\beta_{1,2}^{-})=\frac{e^{\pi\rho(X)\left(y_{1,2}-\frac{\tau_{+}}{2}\right)}}{i\pi\rho(X)\left(y_{1,2}-\frac{\tau_{+}}{2}\right)},\quad
S(\alpha_{1,2}^{-},\alpha_{+})=\frac{e^{\pi\rho(X)\left(y_{1,2}+\frac{\tau_{-}}{2}\right)}}{i\pi\rho(X)\left(y_{1,2}+\frac{\tau_{-}}{2}\right)}$
(34)

and finally

$S(\beta_{+},\alpha_{+})=-2\pi
i\frac{e^{\alpha_{+}-\beta_{+}}}{\alpha_{+}-\beta_{+}}=2\pi\frac{e^{\frac{\pi\rho(X)}{2}\left(\tau_{-}-\tau_{+}\right)}}{\frac{\pi\rho(X)}{2}\left(\tau_{-}-\tau_{+}\right)},\quad$
(35)

Eventually we need to consider the limit $y_{1}\to y_{2}=y$ in (30) which
after straightforward manipulations produces (after setting $X=0$, hence
$\pi\rho(x)=1$) the following expression for the Laplace-Fourier transform
defined in Eq.(14) of the main paper:

${\cal F}^{(\beta=2)}(s,p)=A(s,p)+B(s,p)+C(s,p)$ (36)

where

$A(s,p)=2\frac{e^{\frac{1}{2}(\tau_{-}-\tau_{+})}}{\frac{1}{2}(\tau_{-}-\tau_{+})}\left[\left(y+\frac{\tau_{-}}{2}\right)\left(y-\frac{\tau_{+}}{2}\right)\right]^{2}\,\frac{1}{4y^{2}}\left(\frac{\sinh^{2}{2y}}{4y^{2}}-1\right),$
(37)

$B(s,p)=\frac{1}{2}e^{\frac{1}{2}(\tau_{-}-\tau_{+})}\left(y+\frac{\tau_{-}}{2}\right)\left(y-\frac{\tau_{+}}{2}\right)\,\frac{1}{4y^{3}}\left(e^{4y}-1-4y-8y^{2}\right),$
(38)

and

$C(s,p)=\frac{1}{2}e^{\frac{1}{2}(\tau_{-}-\tau_{+})}\left(2y+\frac{\tau_{-}-\tau_{+}}{2}\right)\frac{1}{4y^{2}}\left(1+4y-e^{4y}\right)+\frac{1}{2}e^{\frac{1}{2}(\tau_{-}-\tau_{+})}\frac{1}{2y}\left(e^{4y}-1\right).$
(39)

####  I.2.1 Inverting the Laplace-Fourier transform

Substituting the definitions

$\tau_{+}=s+\sqrt{s^{2}+4y(y+p)}\quad\mbox{and}\quad\tau_{-}=s-\sqrt{s^{2}+4y(y+p)}$
(40)

to the expression for the Fourier-Laplace transform $F^{(\beta=2)}(s,p)$
presented in Eqs.(37)-(39) above, we notice that it can be represented as

$A(s,p)=2\frac{e^{-\sqrt{s^{2}+4y(y+p)}}}{\sqrt{s^{2}+4y(y+p)}}\left(2y+p-\sqrt{s^{2}+4y(y+p)}\right)^{2}\frac{1}{4}\left(\frac{\sinh^{2}{2y}}{4y^{2}}-1\right),$
(41)
$B(s,p)=\frac{1}{2}e^{-\sqrt{s^{2}+4y(y+p)}}\left(2y+p-\sqrt{s^{2}+4y(y+p)}\right)\frac{1}{4y^{2}}\left(e^{4y}-1-4y-8y^{2}\right),$
(42)

and

$C(s,p)=\frac{1}{2}e^{-\sqrt{s^{2}+4y(y+p)}}\left\\{\left(2y-\sqrt{s^{2}+4y(y+p)}\right)\frac{1}{4y^{2}}\left(1+4y-e^{4y}\right)+\frac{1}{2y}\left(e^{4y}-1\right)\right\\}.$
(43)

To represent above as an explicit Laplace-Fourier transform we define for
$p>0$, any real $s$ and $\alpha$ the functions:

$\phi_{\alpha}(s,p;y)=\int_{0}^{\infty}dve^{-pv}\int_{-\infty}^{\infty}\frac{du}{\sqrt{2\pi}}e^{isu}\frac{1}{v^{\alpha+2}}e^{-y\frac{u^{2}+v^{2}+1}{v}}=\frac{1}{2^{\alpha}\,y^{\alpha+1}}\left(s^{2}+4y(y+p)\right)^{\frac{1}{2}\left(\alpha+\frac{1}{2}\right)}\,K_{\alpha+\frac{1}{2}}\left(\sqrt{s^{2}+4y(y+p)}\right)$
(44)

where we performed first the Gaussian integral over $u$ and then used the
formula

$\int_{0}^{\infty}\,dv\,v^{\nu-1}\,e^{-\frac{\beta}{v}-\alpha
v}=2\left(\frac{\beta}{\alpha}\right)^{\nu/2}K_{\nu}(2\sqrt{\beta\alpha}),\quad\beta>0,\alpha>0$

remembering $K_{\nu}(x)=K_{-\nu}(x)$.

In particular:

$\phi_{1}(s,p;y)=\sqrt{\frac{\pi}{2}}\frac{e^{-\sqrt{s^{2}+4y(y+p)}}}{2y^{2}}\left(1+\sqrt{s^{2}+4y(y+p)}\right),$
(45)

and using this we can easily check that

$A(s,p)=\frac{1}{2}\left(\frac{\sinh^{2}{2y}}{4y^{2}}-1\right)\left(\frac{1}{2}\frac{d^{2}}{dy^{2}}+2\frac{d}{dy}+2\right)\sqrt{\frac{2}{\pi}}y^{2}\phi_{1}(s,p;y),$
(46)

Further, after rewriting

$B(s,p)+C(s,p)=\frac{1}{2}e^{-\sqrt{s^{2}+4y(y+p)}}\,(2y+p)\frac{1}{4y^{2}}\left(e^{4y}-1-4y-8y^{2}\right)+e^{-\sqrt{s^{2}+4y(y+p)}}\left(1+\sqrt{s^{2}+4y(y+p)}\right),$
(47)

it turns out to be possible to represent it in the form

$B(s,p)+C(s,p)=\left(2-\frac{1}{8y^{2}}\left(e^{4y}-1-4y-8y^{2}\right)\frac{d}{dy}\right)\sqrt{\frac{2}{\pi}}y^{2}\phi_{1}(s,p;y),$
(48)

Finally, one may check that adding (46) and (48) produces, after simple
algebra, eq.(15) of the main text.

###  I.3 Derivation of eq.(16) of the main text.

According to Eqs.(12) and (15) of the main text we need to deal with the
following integral:

$F^{(2)}_{p}(y)=\int_{-\infty}^{\infty}du\int_{0}^{\infty}dv\frac{\left[(1-\gamma
v)^{2}+\gamma^{2}u^{2}\right]^{p+1}}{\left[(1+\gamma
v)^{2}+\gamma^{2}u^{2}\right]^{p}}\frac{1}{v^{3}}e^{-y\frac{u^{2}+v^{2}+1}{v}}$
(49)

which after introducing the combination
$x=\frac{1}{2}\left(v+\frac{1}{v}\right)+\frac{u^{2}}{2v}>1$ as the
integration variable instead of $u^{2}$ allows to rewrite the above as

$F_{p}(y)\propto\int_{1}^{\infty}dxe^{-2x}\int_{v_{1}}^{v_{2}}\frac{dv}{v}\frac{1}{\sqrt{(v-v_{1})(v_{2}-v)}}\frac{\left[\frac{1-\gamma^{2}}{2v}-\gamma+\gamma^{2}x\right]^{p+1}}{\left[\frac{1-\gamma^{2}}{2v}+\gamma+\gamma^{2}x\right]^{p}}$
(50)

where $v_{1,2}=x\mp\sqrt{x^{2}-1}$, with $v_{1}v_{2}=1$. Changing $v\to 1/v$
and then substituting
$v=\frac{1}{2}\left(v_{1}+v_{2}\right)+\frac{1}{2}\left(v_{2}-v_{1}\right)\cos{\theta}$,
where $\theta\in[0,\pi]$ brings the right-hand side to

$F_{p}(y)\propto\int_{1}^{\infty}dxe^{-2x}\int_{0}^{\pi}\frac{\left[x(1+\gamma^{2})-2\gamma+(1-\gamma^{2})\sqrt{x^{2}-1}\cos{\theta}\right]^{p+1}}{\left[x(1+\gamma^{2})+2\gamma+(1-\gamma^{2})\sqrt{x^{2}-1}\cos{\theta}\right]^{p}}\,d\theta$
(51)

After introducing $g=\frac{1}{2}\left(\gamma+\frac{1}{\gamma}\right)$ and
using that $\frac{1}{2}\left(\frac{1}{\gamma}-\gamma\right)=\pm\sqrt{g^{2}-1}$
with sign dependent on whether $\gamma<1$ or $\gamma>1$ (this sign is
immaterial for the value of the integral and can be omitted) the above is
brought to the following form:

$F_{p}(y)\propto\gamma\int_{1}^{\infty}dxe^{-2x}\int_{0}^{\pi}d\theta\frac{[gx-1+\cos{\theta}\sqrt{(g^{2}-1)(x^{2}-1)}]^{p+1}}{[gx+1+\cos{\theta}\sqrt{(g^{2}-1)(x^{2}-1)}]^{p}}$
(52)

Further introducing $a=gx+\cos{\theta}\sqrt{(g^{2}-1)(x^{2}-1)}>1$ as a new
variable one can transform (52) to

$F_{p}(y)\propto\gamma\int_{1}^{\infty}da\frac{(a-1)^{p+1}}{(a+1)^{p}}\int_{x_{1}(a)}^{x_{2}(a)}\,e^{-2yx}\frac{dx}{\sqrt{(x-x_{1}(a))(x_{2}(a)-x)}}$
(53)

$x_{1,2}(a)=ga\mp\sqrt{(a^{2}-1)(g^{2}-1)}$. Substituting
$x=\frac{1}{2}\left(x_{1}(a)+x_{2}(a)\right)+\frac{1}{2}\left(x_{2}(a)-x_{1}(a)\right)\cos{\phi}$,
where $\phi\in[0,\pi]$ brings the right-hand side to

$F_{p}(y)\propto\gamma\int_{1}^{\infty}da\frac{(a-1)^{p+1}}{(a+1)^{p}}e^{-2gay}\int_{0}^{\pi}\,e^{-2y\sqrt{(a^{2}-1)(g^{2}-1)}\cos{\phi}}\,d\phi$
(54)

which is equivalent to (16) of the main text.

Numerical simulations provide a reasonably good support for the final result,
Eq.(6) of the main text, already for relatively small matrices.

Figure 2:  Eq.6 compared with the empirical density of overlaps $O_{nn}$ for
$n=10^{6}$ realizations of $32\times 32$ GUE matrices, with the choice
$y=0.05$ and $\gamma=0.5$.

###  I.4 Non-orthogonality factors in systems with preserved time-reversal
invariance

Our starting point is the well-known joint probability density (JPD) of
eigenvalues for $N\times N$ non-selfadjoint matrix ${\cal
H}_{eff}=\mathbf{H}_{N}-i\gamma{\bf e}\otimes{\bf e}^{T}$, with
$\mathbf{H}_{N}$ taken from the Gaussian Orthogonal Ensemble[2,3]:

${\cal
P}_{N}(z_{1},\ldots,z_{N})\propto\frac{e^{-\frac{N}{4}\left(\gamma^{2}+\sum_{j=1}^{N}(\Re
z_{j}^{2})\right)}}{\gamma^{\frac{N}{2}-1}}\prod_{j,k=1}^{N}\frac{1}{\sqrt{|z_{j}-z^{*}_{k}|}}\prod_{j<k}^{N}|z_{j}-z_{k}|^{2}\delta(\sum_{j=1}^{N}\Im
z_{j}+\gamma)$ (55)

Using it we follow essentially the same procedure of reduction to averaging
over matrices of size $(N-1)\times(N-1)$ as in the GUE case (cf. also [4]) and
come to the following relation:

${\cal
O}^{(\beta=1)}_{p}(z)\propto\frac{1}{(\gamma_{*}Y)^{1/2}}\left(\frac{\gamma_{*}}{\gamma}\right)^{\frac{N}{2}-1}\,e^{-\frac{N}{4}(X^{2}-Y^{2}+\gamma^{2})+\frac{N-1}{4}\gamma_{*}^{2}}{\cal
M}^{(\beta=1)}_{p}(z)$ (56)

where

${\cal
M}^{(\beta=1)}_{p}(z)=\left\langle\frac{\left[\det(\tilde{z}-\tilde{{\cal
H}}_{eff})\det(\tilde{z}^{*}-\tilde{{\cal
H}}^{\dagger}_{eff})\right]^{p+1}}{\left[\det(\tilde{z}-\tilde{{\cal
H}}^{\dagger}_{eff})\det(\tilde{z}^{*}-\tilde{{\cal
H}}_{eff})\right]^{p+\frac{1}{2}}}\right\rangle$ (57)

which, assuming for simplicity $X=0$ and the scaling $Y=y/N$, asymptotically
in the limit $N\to\infty$ is equivalent to

${\cal
O}^{(\beta=1)}_{p}(z)\propto\frac{e^{-\frac{y}{2}\left(\gamma+\frac{1}{\gamma}\right)}}{y^{1/2}}\lim_{N\to\infty}{\cal
M}^{(\beta=1)}_{p}(z)$ (58)

where one replaces $\tilde{{\cal H}}_{eff}$ with the original ${\cal
H}_{eff}=\mathbf{H}_{N}-i\gamma\,\tilde{{\bf w}}\otimes\tilde{{\bf
w}}^{\dagger}$ without changing the result and may consider the random real
gaussian channel vector ${\bf\tilde{w}}$ rather than keeping it of unit
length. The same steps as in the GUE case then give:

${\cal
M}^{(\beta=1)}_{p}(z)=\left\langle|\det(z-\mathbf{H}_{N})|\frac{\left|1+i\gamma
G(z)\right|^{2(p+1)}}{\left|1+i\gamma G(z^{*})\right|^{2p+1}}\right\rangle$
(59) $=\int_{-\infty}^{\infty}du\int_{0}^{\infty}dv\frac{\left[(1-\gamma
v)^{2}+\gamma^{2}u^{2}\right]^{p+1}}{\left[(1+\gamma
v)^{2}+\gamma^{2}u^{2}\right]^{p+\frac{1}{2}}}{\cal P}_{z}(u,v)$ (60) ${\cal
P}^{(\beta=1)}_{z}(u,v)=\left\langle|\det(z-\mathbf{H}_{N})|\delta\left(u-\mbox{Re}G(z)\right)\,\delta\left(v-\mbox{Im}G(z)\right)\right\rangle$

After performing the Fourier-Laplace transform of the above function and
averaging over random Gaussian real vectors $\tilde{\bf w}$ we reduce the
problem to evaluating of the following function

${\cal
F}^{(\beta=1)}(s,p)=\left\langle\frac{|\det(\frac{iy}{N}-\mathbf{H}_{N})|^{2}}{\left[\det\left(-i\frac{\tau_{+}}{2N}-\mathbf{H}_{N}\right)\det\left(-i\frac{\tau_{-}}{2N}-\mathbf{H}_{N}\right)\right]^{\frac{1}{2}}}\right\rangle$
(61)

where in the present case $\frac{1}{2}\tau_{+}=s+\sqrt{s^{2}+y(y+2p)}>0$ and
$\frac{1}{2}\tau_{-}=s-\sqrt{s^{2}+4y(y+p)}<0$ and the averaging is to be
performed over GOE matrices $\mathbf{H}_{N}$ as $N\to\infty$. Such a
correlation function has been evaluated in [5], with the result being given by

${\cal
F}^{(\beta=1)}(s,p)\propto\frac{\sinh{2y}}{2y}\,\frac{|\tau_{+}-\tau_{-}|}{2}\,K_{1}\left(\frac{|\tau_{+}-\tau_{-}|}{4}\right)+\beta(p,q),$
(62)

where

$\beta(p,q)=\frac{2}{(2y)^{3}}\left[2y\cosh{2y}-\sinh{2y}\right]\left(y^{2}-\frac{1}{4}\tau_{+}\tau_{-}\right)K_{0}\left(\frac{|\tau_{+}-\tau_{-}|}{4}\right)$

Substituting here the values of $\tau_{\pm}$ one then arrives at

${\cal F}^{(\beta=1)}(s,p)\propto
2\frac{\sinh{2y}}{2y}\,\sqrt{s^{2}+y(y+2p)}\,K_{1}\left(\sqrt{s^{2}+y(y+2p)}\right)+\beta(p,q),$
(63)

with

$\beta(p,q)=\frac{2}{(2y)^{3}}\left[2y\cosh{2y}-\sinh{2y}\right]\left(y+p\right)K_{0}\left(\sqrt{s^{2}+y(y+2p)}\right)$

The Fourier-Laplace inversion is performed using the functions defined in
(44), namely

$\phi_{-\frac{1}{2}}(s,p;y/2)=\frac{2}{\sqrt{y}}\,K_{0}(\sqrt{s^{2}+y^{2}+2py}),\quad\phi_{\frac{1}{2}}(s,p;y/2)=\frac{2}{y^{3/2}}\sqrt{s^{2}+y^{2}+2py}\,K_{1}(\sqrt{s^{2}+y^{2}+2py})$
(64)

which also can be used to prove the following identity:

$2p\,K_{0}(\sqrt{s^{2}+y^{2}+2py})=-\sqrt{y}\int_{0}^{\infty}e^{-pv}\,\int_{-\infty}^{\infty}\frac{du}{\sqrt{2\pi}}e^{isu}\frac{e^{-\frac{y}{2}\frac{u^{2}+v^{2}+1}{v}}}{v^{\frac{3}{2}}}\left[\frac{3}{2v}+\frac{y}{2}\left(2-\frac{u^{2}+v^{2}+1}{v^{2}}\right)\right]$

These identities when applied to the inversion of (62) yield

${\cal
P}^{(\beta=1)}_{z}(u,v)\propto\left\\{\left(\cosh{2y}-\frac{\sinh{2y}}{2y}\right)\left(-\frac{3}{y}+\frac{u^{2}+v^{2}+1}{v^{2}}\right)+2\sinh{2y}\right\\}\frac{\sqrt{y}}{4}\frac{1}{v^{\frac{5}{2}}}\,e^{-\frac{y}{2}\frac{u^{2}+v^{2}+1}{v}}$
(65)

and the final expression (17) for ${\cal P}^{(\beta=1)}_{y}(t)$ is
straightforwardly obtained following the same manipulations as described for
the GUE-based case.

Numerical simulations provide a reasonably good support for the final result,
Eq.(18) of the main text, already for relatively small matrices.

Figure 3:  Eq.18 compared with the empirical density of overlaps $O_{nn}$ for
$n=10^{6}$ realizations of $32\times 32$ matrices, with $y=0.05.$ and
$\gamma=0.5$.

###  I.5 Comparison of the formula Eq.(19) for the mean resonance density with
the form in [6]

We start with recalling Eq.(4) from [6]:

$\rho^{(\beta=1)}(y)=\frac{1}{4\pi}\frac{d^{2}}{dy^{2}}\int_{-1}^{1}(1-\lambda^{2})e^{2\lambda
y}(g-\lambda){\cal F}(\lambda,y)\,d\lambda$ (66)

where

${\cal F}(\lambda,y)=\int_{g}^{\infty}dp_{1}\frac{e^{-yp_{1}}}{(\lambda-
p_{1})^{2}\sqrt{(p_{1}^{2}-1)(p_{1}-g)}}\int_{1}^{g}dp_{2}\frac{e^{-yp_{2}}(p_{1}-p_{2})}{(\lambda-
p_{2})^{2}\sqrt{(p_{2}^{2}-1)(g-p_{2}})}$

Below we consider only the perfect coupling case $g\to 1$. For this first
change the variable: $p_{2}=1+(g-1)t,\,t\in[0,1]$, implying
$\frac{dp_{2}}{\sqrt{(p_{2}-1)(g-p_{2})}}=\frac{dt}{\sqrt{t(1-t)}}$. This
allows to evaluate the integral over $t$ in the limit $g\to 1$, the result
being equal to $\frac{\pi}{\sqrt{2}}e^{-y}\frac{(p_{1}-1)}{(\lambda-1)^{2}}$,
yielding the formula:

$\rho^{(\beta=1)}_{g=1}(y)=\frac{1}{4\sqrt{2}}\int_{1}^{\infty}\frac{dp}{\sqrt{p+1}}e^{-y(p+1)}\int_{1}^{1}e^{2\lambda
y}\frac{(1+\lambda)(2\lambda-p-1)^{2}}{(\lambda-p)^{2}}$ (67)

which should be compared to the $g=1$ limit of Eq.(19)-(20) in the paper.
Although we were not yet able to show analytically the equivalence of the two
formulas, the numerical comparison is flawless:

Figure 4:  Eq.(19) of the main text compared with Eq.(67)

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