# Optimizing mixing in the Rudner-Levitov lattice

I. Peshko1, M. Antsukh2, D. Novitsky1, D. Mogilevtsev1 B. I. Stepanov
Institute of Physics, NAS of Belarus, Nezavisimosti ave. 68, 220072 Minsk,
Belarus,
2Physics Department, Belarusian State University, Minsk, Belarus

(August 2023)

###### Abstract

Here we discuss optimization of mixing in finite linear and circular Rudner-
Levitov lattices (Su–Schrieffer–Heeger lattices with a dissipative
sublattice). We show that presence of exceptional points in the systems
spectra can lead to drastically different scaling of the mixing time with the
number of lattice nodes, varying from quadratic to the logarithmic one. When
operating in the region between the maximal and minimal exceptional points, it
is always possible to restore the logarithmic scaling by choosing the initial
state of the chain. Moreover, for the same localized initial state and values
of parameters, a longer lattice might mix much faster than the shorter one.
Also we demonstrate that an asymmetric circular Rudner-Levitov lattice can
preserve logarithmic scaling of the mixing time for an arbitrary large number
of lattice nodes.

##  I Introduction

Mixing in continuous quantum walks is a much discussed and exploited topic
[1]. It finds important applications in quantum computing [2]. It is also a
very important aspect of modeling transport phenomena in quantum systems [3].
Unitary quantum walks do not mix in a common classical sense of approaching an
asymptotic distribution, but they do mix on average: the long-time time-
averaged occupation probabilities of each lattice node can be considered as
the limiting distribution [4]. Decohering quantum walks were shown to mix in a
classical sense and to be able to do it faster than classical counterparts [5,
6, 7].

Here we address an aspect of mixing that was not much discussed before,
namely, optimizing mixing in practical devices exploiting designed loss for
producing delocalized stationary states from the localized input states. For
such devices (be it dissipative beam-splitters and ”equalizers” [8, 9, 10, 11,
12, 13, 14, 15], asymmetric distributors and propagators [16, 17, 18, 19],
non-classical states protectors and generators [8, 9, 20, 21, 10, 22])
minimization of the interaction time/length, i.e., mixing time, can be
crucially important for practical feasibility and integrability.

The systems with designed loss considered in the paper can be treated from the
standpoint of non-Hermitian physics [23]. The most intriguing aspect of non-
Hermitian systems is the possibility of the spectral degeneracies known as the
exceptional points much studied recently in classical and quantum optical
systems [24]. In particular, in the systems obeying parity-time
($\mathcal{PT}$) symmetry, an exceptional point has a clear physical meaning
as a point of spontaneous symmetry breaking resulting in transition between
$\mathcal{PT}$-symmetric and $\mathcal{PT}$-symmetry-broken phases [25, 26,
27]. Among the multitude of effects associated with the exceptional points, we
mention anisotropic transmission resonances [28, 29], locking of light
propagation direction [30], the effect of coherent perfect absorption and
lasing [31, 32, 33], loss compensation [34], sensing with enhanced response
[35, 36], resonant energy transfer enhancement [37], ”masking” of exceptional
points by quantum interference [38], and so on.

Here we show that in the non-Hermitian lattices considered in this work
exceptional points define the character of the mixing, or even its very
existence.

The main points of this work are as follows. We consider systems with the so-
called ”dark state”, i.e., the non-vacuum stationary state. This state is the
basic practically used feature of the considered systems. They are supposed to
function by projecting the initial state on this ”dark state”, thus generating
it. In difference with decohering walks [6], for generic lossy quantum walks
the normalized occupation probability distribution might not mix in a
classical sense. We connect character of mixing with the presence of
exceptional points in the systems spectrum that can drastically affect scaling
of the mixing time changing it from $O(N^{2})$ to $O(\log(N))$, $N$ being a
number of sub-systems (nodes) of the system. So, optimization of the mixing
might involve designing system parameters to be in a certain position with
respect to the exceptional points. We also show that choice of the initial
states can drastically fasten the mixing and considerably extend the region of
$O(\log(N))$ scaling, while still having the number of the initially excited
nodes much less than the total number of nodes. Thus, optimization should
include also a proper choice of the initial state.

The outline of this work is as follows. After a brief discussion of the mixing
concept in Section II, we illustrate in Section III the first point described
above with the example of the simplest three-mode dissipative system, namely,
a dissipative beam-splitter. This structure can be considered the shortest
kind of the finite Rudner-Levitov (RL) model [39] having a ”dark state” (or
the Su–Schrieffer–Heeger (SSH) model with the second sub-lattice being lossy
[40]). We also show how mixing and non-mixing regimes arise and how mixing
time behaves in dependence on the system parameters. In Section IV, we discuss
mixing time behaviour in a generic system with an exceptional point where all
the imaginary parts of the eigenvalues are coalesced. In Section V, we
consider a finite linear RL model and show how the mixing time depends on the
position of exceptional points. Here we show how one can initially excite only
few nodes to extend $O(\log(N))$ scaling. Also, we show how to exploit the
structure of eigenvectors to achieve the fastest localization with the minimal
number of the initially excited nodes, $M$. In Section VI, we demonstrate how
the found mixing optimization can be achieved for the circular RL lattice. In
particular, we show that for the same values of the interaction constants,
loss rates and initial states, the circlular lattice can exhibit much faster
mixing for the same number of nodes. Moreover, we demonstrate that the
asymmetric circular RL lattice can have log-like fast mixing scaling for
arbitrary $N$. Also, we show how for the circled lattice diabolic points [41]
can convalesce into an exceptional point.

##  II Mixing

Here we clarify the concept of mixing with respect to dissipative structures.
We consider systems described by the following generic equation for the vector
of amplitudes $\vec{\psi}$

$\frac{d}{dt}\vec{\psi}=-i{\mathbf{H}}\vec{\psi},$ (1)

where the elements of the vector $\vec{\psi}$, i.e., the amplitudes
$\psi_{j}$, correspond to the nodes of the lattice. This lattice is described
by the effective non-Hermitian Hamiltonian ${\mathbf{H}}$. Notice that
generally Eq.(1) describes a particular case of state dynamics resulting from
the master equation describing the lattice. For instance, the amplitudes
$\psi_{j}$ might be indeed the amplitudes of coherent states propagating
through the lattice of single-mode waveguides or the off-diagonal elements of
the single-particle density matrix [42].

As it was already mentioned in the Introduction, we consider practical systems
that function as passive state filters by projecting an initial state on the
particular ”dark” state (which can be a non-classical and even entangled one
[16, 17, 18, 19, 42]). Thus, the eigenvalues $\lambda_{k}$ of the Hamiltonian
${\mathbf{H}}$ satisfy $\mathrm{Im}\\{\lambda_{k}\\}\leq 0$, the index
$k=0,1\ldots\mathrm{dim}({\mathbf{H}})-1$. For simplicity sake, we assume that
the ”dark” state is unique, i.e., $\lambda_{0}=0$, and $|\lambda_{k}|>0$ for
$k>0$. Also, because of passivity, the system has the vacuum as the stationary
state, $\psi_{j}=0,\quad\forall j$.

Loss leads to the change of total probability
$P_{total}(t)=\sum\limits_{\forall j}|\psi_{j}(t)|^{2}$, which is no more
equal to $1$ (notice, that this can be accompanied by existence of other
integrals of motion; for examples, for coherent diffusive photonic systems
considered in Refs. [11, 42] one has preservation of the sum of coherencies,
$\sum\limits_{\forall j}\psi_{j}(t)=\mathrm{const}$). Thus, to describe mixing
one needs to introduce the normalized occupation probabilities

$p_{j}(t)={|\psi_{j}(t)|^{2}}/P_{total}(t)\underset{\scriptscriptstyle
t\rightarrow\infty}{\rightarrow}p_{j}^{(st)},$ (2)

with $p_{j}^{(st)}$ being the mixed time-independent occupation distribution.
Thus, the mixing time is defined in a standard classical way as [43]

$T_{mix}(\epsilon)=\min\\{t\geq 0:\sum\limits_{\forall
j}|p_{j}(t)-p^{st}_{j}|\leq\epsilon\\}$ (3)

for $\epsilon>0$. Notice that the normalization in Eq.( 2) actually makes
ambiguous the definition of the mixed distribution. If the initial state is
not orthogonal to the ”dark” state, the mixed distribution is unique and is
defined by this ”dark” state. This kind of mixing we call ”conventional”
throughout the paper. When the initial state is orthogonal to the ”dark”
state, the system eventually decays to the vacuum. However, the normalized
distribution (2) might also mix to some limiting distribution $p^{(st)}_{j}$.
This distribution might depend on both the initial state and the parameters of
the lattice. This kind of mixing we term as ”unconventional”. Finally, as we
will see below, for the initial state orthogonal to the ”dark” state, mixing
might not occur at all.

Figure 1: (a) A scheme of the dissipative beam-splitter (DBS) model. The nodes
$\alpha_{1,2}$ are unitary coupled with the node $\beta_{1}$ with the coupling
rates $v_{1,2}$, but not with each other; the node $\beta_{1}$ is subject to
the loss with the rate $\Gamma$. (b-c) Illustration of the mixing regimes of
the symmetric DBS described by the Hamiltonian (4) with $v_{1}=v_{2}$. (b)
Solid, dashed and dash-dotted lines correspond to the normalized probabilities
$p_{2,3,1}(t)$ given by Eq.(2) for the conventional mixing regime; initially
only the node $\alpha_{1}$ is excited, $v/\Gamma=0.4$. (c) Solid and dash-
dotted lines illustrate unconventional mixing for initial excitation of both
the nodes $\alpha_{1,2}$ with equal amplitudes; solid and dash-dotted lines
correspond to $p_{1,2}(t)$ for $v/\Gamma=0.4$. Dashed line illustrates absence
of mixing for $p_{1}(t)$ corresponding to $v/\Gamma=0.6$ for initial
excitation of both the nodes $\alpha_{1,2}$ with equal amplitudes.

##  III Dissipative beam-splitter

To demonstrate the essential points of our discussion, let us consider a
simple structure consisting of just three unitary coupled nodes with the
corresponding amplitudes $\alpha_{1,2}$ and $\beta_{1}$ with the middle one
corresponding to $\beta_{1}$ subject to loss (Fig.1(a)). This shortest case of
the RL model with a ”dark” state [39] can be realized, for example, with
single-mode waveguides [44, 11, 12, 13, 14, 15], with coherences of coupled
two-level systems [42] or three-site Bose-Hubbard model [45]. This dissipative
beam-splitter (DBS) is described by the following Hamiltonian

${\mathbf{H}}=\begin{pmatrix}0&v_{1}&0\\\ v_{1}&-i\Gamma&v_{2}\\\
0&v_{2}&0\end{pmatrix}$ (4)

where the real $\Gamma>0$ is the loss rate of the middle node denoted as
$\beta_{1}$ in Fig.1(a). For simplicity sake, we assume real unitary coupling
rates $v_{1,2}$.

Despite simplicity, the scheme described by Eq.(4) contains rich physics.
First of all, it can exhibit both $\mathcal{PT}$ and anti-$\mathcal{PT}$
symmetries. Indeed, for the limiting asymmetric case, when one of the coupling
rates vanishes, $v_{j}\rightarrow 0$, the remaining part of the system (two
coupled nodes) is $\mathcal{PT}$-symmetric [46, 47]. In that case it was also
observed how the strong designed loss in the node $\beta_{1}$ leads to the
effective decoupling of the remaining side node (termed ”loss-induced
transparency” [47]). In the other limit of
$\sqrt{v_{1}^{2}+v_{2}^{2}}/\Gamma\rightarrow 0$, the system becomes
anti-$\mathcal{PT}$-symmetric [48, 49], since one can adiabatically exclude
the middle node $\beta_{1}$ resulting in the so-called ”dissipative coupling”
[50, 45, 10].

Figure 2: Mixing time for the symmetric DBS in the units of $\Gamma^{-1}$ for
the initially excited node $\alpha_{1}$ as given by Eq.(3) with
$\epsilon=0.001$ in dependence on $v/\Gamma$; the vertical line denotes the
position of the exceptional point; in the inset imaginary parts of the
eigenvalues of the Hamiltonian (4) are shown in dependence on the ratio
$v/\Gamma$. The dashed line corresponds to $\lambda^{+}$, the dotted line
corresponds to $\lambda^{-}$ of Eq.(6) The vertical dashed line shows the
position of the exceptional point.

Here we show how another case of $\mathcal{PT}$-symmetry arises in this DBS
for arbitrary values of $v_{1,2}$ and $\Gamma$. This symmetry is connected
with the ”dark” state and the mixing rate. Indeed, let us introduce the
amplitude, $v$, and phase, $\phi$, parameters characterising the coupling such
that $v_{1}=v\cos{\phi},\quad v_{2}=v\sin{\phi}$,
$v=\sqrt{v_{1}^{2}+v_{2}^{2}}$, and collective variables $A$ and $\alpha$ are
corresponding to the amplitudes of the ”bright” and ”dark” states

$A=\alpha_{1}\cos{\phi}+\alpha_{2}\sin{\phi},\quad\alpha=\alpha_{2}\cos{\phi}-\alpha_{1}\sin{\phi}.$
(5)

It is easy to see from Eqs.(1) and (4) that $\alpha(t)=\mathrm{const}$.
Actually, it describes projection of the initial state on the ”dark”
eigenstate of the Hamiltonian (4) corresponding to the eigenvalue
$\lambda_{0}=0$. The two other eigenvalues are

$\lambda^{\pm}=-\frac{i}{2}(\Gamma\pm\sqrt{\Gamma^{2}-4v^{2}}).$ (6)

Dynamics of the variables $A,\beta_{1}$ are described by the effective
Hamiltonian

${\mathbf{H}}_{2}=\begin{pmatrix}0&v\\\ v&-i\Gamma\end{pmatrix},$ (7)

which is $\mathcal{PT}$-symmetric. As is typical for the
$\mathcal{PT}$-symmetric systems [46, 47, 38], the spectrum (6) shows an
exceptional point at $v/\Gamma=1/2$, where the eigenvalues $\lambda^{\pm}$
coalesce (see the inset in Fig.2). The position of this point with respect to
the system’s parameters, i.e., the ratio $v/\Gamma$, crucially affects both
mixing and the character of dynamics. If projection of the initial states on
the ”dark” state is finite, i.e., $\alpha(0)\neq 0$, for all the finite ratios
$v_{1,2}/\Gamma$ the system is always mixed conventionally with
$p_{2}^{(st)}=0$ and $p_{1,3}^{(st)}=v_{2,1}^{2}/v^{2}$. An illustration of
this behavior is given by Fig.1(b) at the point before the exceptional one,
$v/\Gamma=0.4$. This kind of dynamics underlies practical usability of DBSs.
By loosing part of input energy, a waveguide DBS robustly reproduces outputs
with needed amplitude ratios and fixed phase between them [12, 13].

However, nodal population dynamics becomes quite different for $\alpha(0)=0$.
This can be understood by considering the analytic solution of the system
given by Eq.(7):

$\displaystyle\beta_{1}(t)=c^{+}e^{-i\lambda^{+}t}-c^{-}e^{-i\lambda^{-}t},$
$\displaystyle\quad
c^{\pm}=\frac{-vA(0)+(i\Gamma+\lambda^{\mp})\beta_{1}(0)}{\lambda^{-}-\lambda^{+}},$
(8) $\displaystyle
A(t)=\frac{i}{v}(\frac{d}{dt}\beta_{1}(t)+\Gamma\beta_{1}(t)).$

This solution immediately shows that beyond the exceptional point, i.e., for
$v/\Gamma>1/2$, the system does not mix for $\alpha(0)=0$, so that Eqs.(8)
give non-decaying oscillations of the normalized occupation probabilities (see
the dashed line showing the first node normalized population in Fig.1(c)).
Before the exceptional point, unconventional mixing still occurs (see solid
and dash-dotted lines in Fig.1(c)), but the values of the stationary nodal
populations $p_{j}^{(st)}$ now become dependent on the initial excitation and
the decay rate $\Gamma$.

So, it is entirely unsurprising that conventional mixing time is very much
dependent on the position of the exceptional point with respect to the system
parameters. As one can see in the inset of Fig.2 showing imaginary parts of
the eigenvalues $\lambda^{\pm}$ in dependence on $v/\Gamma$, before the
exceptional point, i.e., for $v/\Gamma<1/2$, the mixing time (3) is defined by
$\lambda^{-}$ and quickly increases with $\Gamma$ (see the illustration in
Fig.2 for the symmetric structure with $v_{1}=v_{2}$ and $\epsilon=0.001$).
Beyond the exceptional point $T_{mix}(\epsilon)$ is defined mostly by the
required precision $\epsilon$. Curiously and importantly for the practical
design, the minimal mixing time is achieved in the vicinity of the exceptional
point.

Figure 3: (a) A scheme of the linear RL lattice. Each lossy node $\beta_{n}$
is unitarily coupled with the node $\alpha_{n,n+1}$ with the corresponding
coupling rates $v_{1,2}$, the node $\beta_{n}$ is subject to the loss with the
rate $\Gamma$. (b) Examples of stationary distributions corresponding to a
non-decaying eigenstate; thick, medium, and thin bars show $p_{k}^{st}$,
correspondingly, for $\phi=0.25\pi,0.21\pi,0.1\pi$. (c) $T_{mix}(\epsilon)$
dependence in the units of $\Gamma^{-1}$ on the ratio $v/\Gamma$ for different
$\phi$. The solid, dash-dotted and dashed curves correspond to
$\phi=0.25\pi,0.21\pi,0.1\pi$ and a single node initially excited,
$\alpha_{1}(0)\neq 0$; the dotted curve corresponds to $\phi=0.1\pi$ and only
$\alpha_{N_{L}+1}(0)\neq 0$. The solid lines in the inset show
$\mathrm{Im}(\lambda_{k})$ for the symmetric case, in dependence on the ratio
$v/\Gamma$, and dotted lines show $\mathrm{Im}(\lambda_{k})$ for the strongly
asymmetric case, $\phi=0.1\pi$. For all the panel (b), $N_{L}=9$.

##  IV General considerations

One can easily surmise that mixing features demonstrated by the DBS considered
in the previous Section are to be observed in more complicated systems with
similar spectral features. Indeed, let us have a single-state passive
filtering system described by Eq.(1) with the effective Hamiltonian
$\mathbf{H}$. The corresponding eigenequation is as follows

$\mathbf{H}\vec{\phi}_{n}=\lambda_{n}\vec{\phi}_{n},$ (9)

with eigenvalues satisfying $\lambda_{0}=0$, ${\rm Im}(\lambda_{n})<0,\forall
n\in[1,N-1]$ in all the considered range of the Hamiltonian parameters; $N$ is
the total number of nodes in the system; $\vec{\phi}_{n}$ are eigenvectors.
Now let us assume that in a certain region of the parameter space, for
$\forall n\in[1,N]$, one has the coalescence of all the imaginary parts, i.e.
$\lambda_{n}=-i\gamma+\omega_{n}$. Away from the exceptional spectral points,
one can write a solution

${\psi}_{k}=c_{k0}+e^{-\gamma
t}\sum\limits_{n=1}^{N-1}c_{kn}e^{-i\omega_{n}t}$ (10)

where the coefficients $c_{kn}$ are defined by the projection of the initial
state on the corresponding eigenvector $\vec{\phi}_{n}$. So, one has some
positive $C$, so that $|c_{kn}|\leq C,\forall k,n$. From Eq.(10) it is easy to
estimate the conventional mixing time for the normalized distribution (2). For
conventional mixing, when $c_{k0}\neq 0$ for some $k>0$, from Eqs. (2), (3),
and (10) it follows that

$\gamma T_{mix}(\epsilon)\leq O(\log\\{CN/\epsilon\\}).$ (11)

The estimate (11) is rather favorable for achieving mixing in the structures
with $N\gg 1$. On the other side, in the absence of eigenvalues coalescence,
i.e., for a set of different ${\rm Im}(\lambda_{n})<0$, the conventional
mixing time is defined in a standard way by a spectral gap ($\min|{\rm
Im}(\lambda_{n})|,n>0$). This scaling can be quite unfavorable. For example,
for a classical linear Markov chain $T_{mix}$ scales as $N^{2}$ [43].

For practical mixing structures, the question is when it is feasible to reach
the coalescence region of parameters and/or generally keep the scaling (11).
Also, the question is what can one do to minimize mixing time when it is not
feasible to reach the coalescence region, in particular, when only a part of
the eigenvalues coalesced.

Further we demonstrate that using symmetry of the eigenvectors for partially
coalesced spectrum, it is still possible to restore scaling (11) while keeping
the mixing device practically useful, i.e., keeping the number of the
initially excited nodes reasonably small and also keeping the set of these
nodes well localized. We demonstrate it on the example of two very common
workhorse models of dissipative quantum walks: linear and circular Rudner-
Levitov lattices [39].

##  V Linear RL lattice

Now let us consider how mixing occurs in a particular example of the generic
structures considered in the previous Section: the finite Rudner-Levitov (RL)
linear lattice [39] with $N_{L}$ lossy nodes and $N_{L}+1$ lossless ones. The
scheme of such lattice is shown in Fig.3(a), where the amplitudes
corresponding to the nodes of the lossless sublattice are termed as
$\alpha_{n}$ and the amplitudes corresponding to the nodes of the lossy
sublattice are termed as $\beta_{n}$. The effective Hamiltonian of the
considered structure is a tridiagonal matrix with the following non-zero
elements

$\displaystyle H_{2n,2n}=-i\Gamma,\quad H_{2n-1,2n}=H_{2n,2n-1}=v_{1},$
$\displaystyle H_{2n,2n+1}=H_{2n+1,2n}=v_{2},\quad n=1,2\ldots N_{L},$ (12)

where we similarly to the DBS case parameterize the unitary coupling rates
(taken for simplicity to be real) as $v_{1}=v\cos{\phi},\quad
v_{2}=v\sin{\phi}$, $v=\sqrt{v_{1}^{2}+v_{2}^{2}}$. The RL lattice described
by the Hamiltonian (12) is a simple extension of the DBS (4) retaining its
essential functional feature: a single non-vacuum stationary state allowing
for conventional mixing. Further, we will concentrate on this type of mixing.

###  V.1 Mixing and asymmetry

First of all, let us highlight the role of unequal coupling (i.e., $v_{1}\neq
v_{2}$) for the considered RL lattice devices (farther in the text we term
such unequal coupling as ”asymmetry” in coupling). From the Hamiltonian (12),
it is obvious that the ratio of the nodal amplitudes $\alpha_{n}/\alpha_{n+1}$
in the stationary ”dark” state is $v_{2}/v_{1}$. So, asymmetry of the coupling
leads to the asymmetry of the ”dark” state population distribution and might
severely limit functionality of the device aimed at the distribution of
population among the nodes. An example for the RL lattice with 19 nodes can be
seen in Fig.3(b): The thick, medium, and thin bars show $p_{k}^{st}$ for
symmetric ($\phi=\pi/4$), slightly asymmetric ($\phi=0.21\pi$), and strongly
asymmetric cases ($\phi=0.1\pi$), respectively. For $\phi=0.1\pi$ the ”dark”
state is strongly localized near the edge of the lattice, so that the initial
state would effectively mix only through a few edge nodes.

As we have shown in the previous Section, asymmetry does not affect the
spectrum of the DBS (6). It is not so for the RL lattice with larger number of
nodes. Qualitatively, the behaviour of the imaginary part of the eigenvalues,
$\mathrm{Im}(\lambda_{k})$, is similar for both the symmetric and asymmetric
cases as one can see for the lattice with $N=19$ nodes in the inset of
Fig.3(c): The solid lines show $\mathrm{Im}(\lambda_{k})$ as a function of the
ratio $v/\Gamma$ for the symmetric case, and the dotted lines show
$\mathrm{Im}(\lambda_{k})$ for the strongly asymmetric case, $\phi=0.1\pi$.
One can see in both cases that for sufficiently large ratio $v/\Gamma$, all
the imaginary parts of eigenvalues coalesce. With lowering $v/\Gamma$, the
system goes through a number of exceptional points (there are $N_{L}$ of
them). Eventually, for sufficiently small $v/\Gamma$, the system enters the
region of dissipative coupling. Both symmetric and asymmetric systems enter
this region for close values of $v/\Gamma$. However, the spread of exceptional
points is larger for the symmetric case. Coalescence is broken for larger
$v/\Gamma$. These features are obvious from the spectra of non-zero
eigenvalues easily obtained analytically due to a simple tridiagonal structure
of the matrix (12) [51]:

$\lambda^{\pm}_{k}(\phi)=-\frac{i}{2}\left(\Gamma\pm\sqrt{\Gamma^{2}-4v^{2}\mu_{k}(\phi)}\right),$
(13)

where $\mu_{k}(\phi)=1+\sin 2\phi\cos\left\\{\frac{k\pi}{N_{L}+1}\right\\}$
and $k=1,2\ldots N_{L}$.

Figure 4:  Dependence of the mixing time $T_{mix}(0.001)$ shown in the units
of $\Gamma^{-1}$ on the number of lossy nodes $N_{L}$ for the symmetric RL
lattice. Solid, dash-dotted, and dashed lines correspond to $v/\Gamma=3,5,7$,
respectively. For all the curves $\epsilon=0.001$. The inset shows $T_{mix}$
for smaller interval of $N_{L}$. Initially only the first losless node is
excited. Figure 5:  Main panel shows mixing time $T_{mix}(0.001)$ in the units
of $\Gamma^{-1}$ for different positions of the initially excited node in
dependence on $N_{L}$. Solid line corresponds to the initially excited first
node, circle marks show mixing time for the initially excited middle node
(i.e., $N_{L}$-th node for the odd $N_{L}$ and $N_{L}+1$-th node for the even
$N_{L}$ with total of $2N_{L}+1$ nodes). The lower inset shows absolute values
of the eigenvector elements for the eigenvector corresponding to the
eigenvalue with the second smallest imaginary part for an odd $N_{L}=19$, the
upper inset shows the same but for an even $N_{L}=20$. For all the panels,
$v/\Gamma=3$.

The spectral features mentioned above directly influence the mixing behavior.
Eq.(13) shows that similarly to the DBS case, the mixing time at $v/\Gamma$
larger than the position of all the exceptional points depends mainly on the
precision parameter $\epsilon$ (and, of course, the value of $\Gamma$). This
can be seen with the estimation of $T_{mix}(\epsilon)$ shown in the main panel
of Fig.3(c), where the dependence of $T_{mix}(\epsilon)$ on the ratio
$v/\Gamma$ is shown for different asymmetry angles $\phi$. The solid, dash-
dotted, and dashed curves correspond to $\phi=0.25\pi,0.23\pi,0.1\pi$ with a
single leftmost node initially excited, $\alpha_{1}(0)\neq 0$. The most
immediate observation is that for the same single-node initial excitation, the
symmetric structure mixes slower at $v/\Gamma$ larger than the exceptional
point corresponding to the largest value of this ratio for $\phi=\pi/4$ (we
call it ”LREP”). Indeed, it can be seen from Eq.(13) that before the LREP

$\mathrm{min}\left\\{{\rm
Im}(\lambda^{\pm}_{k}(\phi\neq\pi/4))\right\\}>\mathrm{min}\left\\{{\rm
Im}(\lambda^{\pm}_{k}(\pi/4))\right\\}$

for $\phi\in[0,\pi/2]$. As shown in Fig.3(c), for strongly asymmetric case
mixing might be orders of magnitude faster. However, in that case mixing
looses its practical meaning. As it was pointed above, the mixed state is
strongly localized.

Another curious albeit intuitive feature of asymmetric structures is the
rather strong quantitative difference in $T_{mix}$ dependence on $v/\Gamma$
for different initial excitations. In Fig.3(c), the dashed and dotted curves
differ only by the initial excitation; $\alpha_{1}(0)\neq 0$ for the dashed
curve and $\alpha_{N_{L}+1}(0)\neq 0$ for the dotted one with all the other
nodes being initially of zero amplitudes. When the initial excitation is far
from the region where the stationary distribution is localized, the mixing is
slower.

As we shall see below, the choice of initial excitation can strongly affect
mixing even in the symmetric case.

###  V.2 Mixing and chain length

As we have seen in the previous Subsection, asymmetry does not introduce any
qualitative changes in conventional mixing in RL lattices. So, for simplicity
sake, in this Subsection we discuss conventional mixing for symmetric RL
lattices in dependence on the number of nodes.

Generally, it conforms to the patterns discussed in the Section IV. As it is
obvious from the eigenvalues given by Eq.(13), for any given $v/\Gamma$ the
system will eventually cross the LREP as $N_{L}$ increases. Indeed, for the
position of LREP from Eq. (13), one has $v/\Gamma\approx N_{L}/\sqrt{2}\pi$
for $N_{L}\gg 1$. Thus, with increasing $N_{L}$, the system eventually becomes
susceptible to slowing of mixing (with $T_{mix}\propto N_{L}^{2}$, as for the
usual Markovian chain mixing [43]).

The examples of such behaviour are shown in Fig.4 for only the edge node being
initially excited. Here the dependence of $T_{mix}(0.001)$ on $N_{L}$ is shown
for different ratios $v/\Gamma$. The solid, dash-dotted, and dashed lines
correspond to $v/\Gamma=3,5,7$, respectively. It is seen that after crossing
LREP the mixing time grows like $N_{L}^{2}$. The inset shows that before
crossing the LREP, $T_{mix}$ grows much slower. Moreover, as discussed in
Section IV, before the LREP the mixing time is rather close for different
$v/\Gamma$.

Thus, we have the recipe for optimizing mixing time for the RL lattice of any
given length: one needs just to increase the interaction strength between the
nodes so that it crosses the LREP.

Figure 6: The scheme of the ring RL lattice. There are $N_{L}$ loss-free nodes
$\alpha_{k}$ and $N_{L}$ dissipative nodes $\beta_{k}$ with loss rate
$\Gamma$. The coupling constant of the nodes $\alpha_{k}$ and $\beta_{k}$ is
$v_{1}$; the coupling constant of the nodes $\beta_{k}$ and $\alpha_{k+1}$ is
$v_{2}$. The last dissipative node $\beta_{N_{L}}$ is coupled to the first
node $\alpha_{1}$ with the interaction constant $\delta v_{2}$; $\delta$ is a
balancing parameter. Figure 7: Mixing time as a function of $v/\Gamma$ for the
ring RL lattice with $N_{L}=3$ in the units of $\Gamma^{-1}$ for the initially
excited node $\alpha_{1}$ as given by Eq.(3) with $\epsilon=0.001$. Inset
shows the imaginary parts of the eigenvalues of the Hamiltonian (VI) with the
balancing parameter (15) in dependence on the ratio $v/\Gamma$. For all the
figure, the solid line corresponds to the symmetric structure, the dashed line
corresponds to $\phi=0.15\pi$.

###  V.3 Mixing and initial states

The recipe of increasing the mixing rate described in the previous Subsection
is not always available. One cannot increase the interaction strength
indefinitely. For example, in the system based on single-mode waveguides,
increasing interaction strength means decreasing the distance between the
waveguides. This eventually breaks the weak-coupling and single-mode
approximations [52].

However, it appears that one can avoid crossing the LREP and reach the fast
mixing regime by judicious choice of the initial state. Even for just a single
initially excited node, its choice might mean a lot. Fig.5 shows that exciting
initially a node in the middle of the lattice instead of the edge one, one can
make mixing much faster. Moreover, the region of log-like fast mixing is much
wider for even $N_{L}$ than for odd $N_{L}$ (the ratio is nearly two for the
example shown in Fig.5). The reason can be related to the structure of the
eigenvectors. For even $N_{L}$, the eigenvector corresponding to the second
smallest eigenvalue has zero element corresponding to the central node,
$k=N_{L}+1$. The initial excitation of this node effectively shifts the LREP
to the second largest exceptional point in Fig.3. Thus, having just one
initially excited node, one can drastically decrease the mixing time by making
the RL lattice longer.

Of course, it is possible to exploit the features of eigenstates in a similar
manner making the initial excitation non-local for effectively shifting the
LREP. Indeed, adjusting the amplitudes of initial excitation of $M+1$ nodes,
outside of exceptional points one can always make the initial state orthogonal
to any of $M$ eigenvectors extending the region of log-like mixing time
dependence with the system’s size.

Figure 8:  (a,b) Imaginary parts of the eigenvalues of the Hamiltonian (VI)
with the balancing parameter (15) in dependence on the ratio $v/\Gamma$ for
the RL ring with $N_{L}=10$. On both these panels the solid lines show
$\mathrm{Im}(\lambda_{k})$ for the symmetric case versus the ratio $v/\Gamma$.
The dashed line in the panel (a) shows $\mathrm{Im}(\lambda_{k})$ for the
slightly asymmetric case, $\phi=0.22\pi$. The dashed line in the panel (b)
shows $\mathrm{Im}(\lambda_{k})$ for the strongly asymmetric case,
$\phi=0.1\pi$. The panel (c) shows the position of the LREP in dependence on
the asymmetry angle $\phi$ for the linear structure (dash-dotted line) and the
ring structure (solid line). For the linear structure $N_{L}=9$, for the ring
structure $N_{L}=10$.

##  VI Ring RL lattice

It is easy to surmise that optimization of the mixing time can be achieved
also by changing geometry of the lattice. Here we demonstrate that a simple
modification of RL lattice, namely, closing it into a ring structure, can
bring considerable advantages in mixing. Such a ring structure can be readily
realized, for example, by laser waveguide writing in a bulk dielectric [11]. A
scheme of this modification is shown in Fig.6. It coincides with the scheme
shown in Fig.3(a) with just one difference: there is an additional lossy node
(say, $\beta_{N_{L}}$) coupling the nodes $\alpha_{1}$ and $\alpha_{N_{L}}$.
Also, the coupling of this node with the node $\alpha_{1}$ is weighted by a
parameter $\delta$. So, the Hamiltonian for the system shown in Fig.6 is given
by

$\displaystyle H_{2n,2n}=-i\Gamma,\quad H_{2n-1,2n}=H_{2n,2n-1}=v_{1},$
$\displaystyle H_{2n,2n+1}=H_{2n+1,2n}=v_{2},\quad n=1,2\ldots N_{L}-1,$ (14)
$\displaystyle H_{2N_{L}-1,2N_{L}}=H_{2N_{L},2N_{L}-1}=v_{1},$ $\displaystyle
H_{2N_{L},2N_{L}}=-i\Gamma,\quad H_{1,2N_{L}}=H_{2N_{L},1}=\delta v_{2}.$

The meaning of the balance parameter $\delta$ can be easily inferred from the
Hamiltonian (VI). To have a non-vacuum stationary state and a possibility of
conventional mixing, one needs to satisfy

$v_{1}^{N_{L}}+(-1)^{N_{L}+1}\delta v_{2}^{N_{L}}=0.$ (15)

In the case of a symmetric structure, it leads to $\delta=1$ for even $N_{L}$
and $\delta=-1$ for odd $N_{L}$. Further, we consider only the conventional
mixing with RL rings balanced as given by Eq.(15).

Figure 9: Dependence of the mixing time $T_{mix}$ on the number of lossy nodes
$N_{L}$ for the ring RL lattice. In the main panel, the solid and dash-dotted
lines correspond to $v/\Gamma=1,2$ to the symmetric structure. The dash-dotted
line corresponds to $v/\Gamma=2$, $\phi=0.22\pi$; the dotted line corresponds
to $v/\Gamma=2$, $\phi=0.2\pi$. The inset shows $T_{mix}$ for smaller vertical
scale; here solid, dash-dotted, dashed and dotted lines correspond to
$v/\Gamma=1,2,3,4$. For all the curves, $\epsilon=0.001$. Initially only the
first lossless node is excited.

###  VI.1 The simplest ring structure

To clarify how the mixing on a RL ring occurs, let us consider the simplest
ring structure with non-trivial non-vacuum stationary state. It is the ring
with three non-lossy nodes, $N_{L}=3$. As shown in the inset of Fig.7, for the
symmetric structure, the imaginary parts of the eigenvalues of the Hamiltonian
(VI) behave quite similarly to the ones for the DBS considered in Section III.
There is also just a single exceptional point. As the main panel of Fig.7
shows, the mixing time quickly grows for values of $v/\Gamma$ below this
point. However, there are two significant differences. Firstly, there is an
eigenvalue independent on $v$ and equal to $\Gamma$.

Secondly and rather curiously, here we have convalescence of two diabolical
points (more exactly, diabolical branches [41]) into the exceptional points.
The upper and lower solid curves in the inset in Fig.7 show the degenerate
eigenvalues corresponding to pairs of different eigenvectors. These features
are easily seen from the Hamiltonian (VI). Indeed, one can get from Eq.(VI)
the following equations for the compound amplitudes of dissipative nodes
$B_{1}=\beta_{1}-\beta_{3}$, $B_{2}=\beta_{1}+2\beta_{2}+\beta_{3}$,
$B_{3}=\beta_{2}-\beta_{1}-\beta_{3}$:

$\displaystyle\frac{d^{2}}{dt^{2}}B_{1,2}+\Gamma\frac{d}{dt}B_{1,2}+\frac{3}{2}v^{2}B_{1,2}=0,$
$\displaystyle\frac{d^{2}}{dt^{2}}B_{3}+\Gamma\frac{d}{dt}B_{3}=0.$ (16)

Curiously, as Eqs.(VI.1) show, the exceptional point retains its diabolic
character: it is actually a pair of exceptional points with different
eigenvectors.

Notice that the superposition $B_{3}$ corresponds to the unconventionally
mixed state. Also notice that the symmetric ring structure gives the
exceptional point at the interaction constant $\sqrt{3/2}$ times smaller that
in the case of the DBS (6). The ring structure provides faster mixing than the
linear DBS for the same $v$ values.

However, symmetry breaking in the case of the ring RL lattice leads to the
consequences opposite to those of the DBS. Asymmetric coupling leads to
lifting the degeneracy (see the dashed lines in the inset in Fig.7
corresponding to $\phi=0.15\pi$), so that the LREP shifts to the larger ratios
$v/\Gamma$. As expected, this results in slower mixing as seen in the main
panel of Fig.7 (see the dashed line corresponding to $\phi=0.15\pi$). Another
difference from the DBS is the independence of $T_{mix}$ of the initially
excited lossless node position in the asymmetric case.

Figure 10: (a) Dependence of the mixing time $T_{mix}$ on the number of lossy
nodes $N_{L}$ for the asymmetric ring RL lattice in comparison with the
symmetric RL structure; for all the curves $\epsilon=0.001$, $v/\Gamma=2$.
Solid, dashed and dash-dotted curves correspond to
$\phi=0.25\pi,0.24\pi,0.23\pi$ and initially excited ${N_{L}}$-th lossless
node; dotted curve corresponds to $\phi=0.23\pi$ and excitation of the
lossless node opposite to the ${N_{L}}$-th lossless node. (b) Dependence of
the second smallest $|Im(\lambda)|$ of the number of nodes $N_{L}$ for
$v/\Gamma=2$. Solid, dashed and dash-dotted curves correspond to
$\phi=0.25\pi,0.24\pi,0.23\pi$.

###  VI.2 Mixing and the ring perimeter

Now let us discuss how the mixing time depends on the ring perimeter which is
twice the number of dissipative or lossless nodes. Just like in the previous
subsection, we are discussing here only the balanced rings allowing for
conventional mixing. First of all, one needs to notice that the balanced ring
RL lattice shows dissipation rates spectra quite similar to those for the
linear RL lattice (see Fig.8(a,b)). One also has a set of exceptional points.
With increasing of the ratio $v/\Gamma$ the system eventually crosses the
LREP. However, there are also some rather remarkable differences apart from
the already noticed presence of the unconventionally mixed state corresponding
to $\mathrm{Im}(\lambda)=-\Gamma$. For the symmetric case and for odd $N_{L}$,
all the exceptional points are twice degenerate. For even $N_{L}$, the
exceptional point corresponding to the lowest ratio is not degenerate (for
that reason one sees just five exceptional points for solid lines in
Fig.8(a,b)). Making the lattice asymmetric (i.e., for $\phi\neq\pi/4$) removes
degeneracy. As one can see in Fig.8(c) (dash-dotted line), for the linear RL
lattice, increase in asymmetry resulted in monotonous shift of the LREP to the
region of lower $v/\Gamma$. This is not the case for the ring structure. If
the mixing angle decreases from the symmetric value $\pi/4$, the LREP shifts
to the larger values of $v/\Gamma$ (see the dashed line in Fig.8(a))
corresponding to $\phi=0.22\pi$). But further decrease of $\phi$ leads to the
reverse motion of the LREP which eventually shifts to lower $v/\Gamma$ than
for the symmetric RL ring (see the dashed line in Fig.8(b) corresponding to
$\phi=0.1\pi$ and the solid line in Fig.8(c) showing how the LREP moves with
the asymmetry angle $\phi$ for the ring structure).

Another important difference from the linear RL lattice is that for the
asymmetric ring structure, the exceptional point shifts to the lower ratios
$v/\Gamma$ with decreasing $\phi$ with no limit similar to those described by
Eq.(13). So, asymmetry can break the ”dissipative coupling” approximation
which is always valid for the linear RL structure for sufficiently low ratios
$v/\Gamma$ regardless of the asymmetry.

The specific features of the spectra for the ring structures significantly
affect the mixing. As one can see in Fig.9, for the symmetric structure the
scaling is just like as for the linear RL lattice. There is the log-like
scaling for the structure below the LREP and $N_{L}^{2}$ scaling above it.
Similar to the smaller ring structure discussed in the previous subsection,
the $N_{L}^{2}$ scaling starts at considerably larger values of $N_{L}$ for
the ring structure than for the linear structure with the same $v/\Gamma$ (for
example, one needs nearly twice as large $N_{L}$ for $v/\Gamma=3$, as follows
from the comparison between Figs.4 and 9). So, the symmetric balanced ring
structure is considerably better for a mixing device than a linear structure.

Much larger difference from the linear case appears for the balanced
asymmetric structure. One can see from Eq.(15) that the balancing parameter
$\delta$ for an asymmetric structure grows with the number of nodes. This
induces quite a remarkable spectral feature: stabilization of the second
smallest dissipation rate with the increasing number of nodes, $N_{L}$. An
example of such a spectral curiosity is shown in Fig.10(b) for $v/\Gamma=2$
and asymmetry angles $\phi=0.24\pi$ (dashed curve) and $\phi=0.23\pi$ (dash-
dotted curve). For comparison, the solid curve shows how the the second
smallest dissipation rate tends to zero with increasing $N_{L}$ for the
symmetric structure. Naturally, stabilization of $Im(\lambda)$ induces
returning to log-like scaling of the mixing time for asymmetric rings.
Examples of such fast mixing restoration are shown in Fig.10(a) for
$v/\Gamma=2$ and asymmetry angles $\phi=0.24\pi$ (dashed curve) and
$\phi=0.23\pi$ (dash-dotted and dotted curves). The dotted curve demonstrates
qualitative dependence of the mixing time on the choice of the initial state;
the initially excited lossless node is maximally distant from the $N_{L}$-th
lossless node.

Finally, it is worth reminding that asymmetry makes the mixed distribution
localized. In particular, for the balanced ring structure shown in Fig.6, the
mixed state localizes near $\alpha_{N_{L}}$ like $[\cot\phi]^{m}$, $m$ being a
number of lossless nodes counter-clockwise from the $N_{L}$-th lossless node.
So, even despite restoration of the fast scaling, asymmetric ring is hardly
feasible as a practical mixing device.

##  VII Conclusions

In this work, we have demonstrated how one can optimize mixing in linear and
circular Rudner-Levitov lattices [39], which are the Su–Schrieffer–Heeger
lattices with the second sub-lattice being lossy [40]). The RL lattices have a
number of important applications in photonics and quantum optics, so that the
mixing optimization in such structures is quite important for the practical
photonic devices realizing such structures. We have obtained a number of
important results. First of all, we have demonstrated that for the finite RL
lattice one can always reach a region of fast, log-like dependence of the
mixing time of the number of the lattice nodes just by engineering the
coupling between nodes and loss rates of the lossy nodes. The key to such a
design is the position of the exceptional points of the lattice. For any fixed
ratio of the interaction constant and the loss rate, as the number of nodes is
increased, the system eventually crosses the last exceptional point and the
mixing time acquires square dependence on the number of nodes. However, one
can extend the region of log-like dependence by judicious choice of the
initial state keeping only a few nodes initially excited. We have also shown
that the ring RL lattice provides the wider region of log-like dependence than
the linear RL lattice for the same ratios of interaction constant and loss
rate and the same initial states. Also, we have shown that asymmetry of the
balanced ring structure can restore log-like scaling for an arbitrary number
of nodes.

## Acknowledgments

I. P. and D. M acknowledge financial support from the The Belarusian
Republican Foundation for Fundamental Research, grant F22B-008.

## References

  * [1] S. Venegas-Andraca, “Quantum walks: A comprehensive review,” Quantum Information Processing 11 (2012).
  * [2] R. Portugal, _Quantum Walks and Search Algorithms_ (Springer Publishing Company, Incorporated, 2018), 2nd ed.
  * [3] O. Mülken and A. Blumen, “Continuous-time quantum walks: Models for coherent transport on complex networks,” Physics Reports 502, 37–87 (2011).
  * [4] D. Aharonov, A. Ambainis, J. Kempe, and U. Vazirani, “Quantum walks on graphs,” in _Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing,_ (Association for Computing Machinery, New York, NY, USA, 2001), STOC ’01, p. 50–59.
  * [5] V. Kendon and B. Tregenna, “Decoherence can be useful in quantum walks,” Phys. Rev. A 67, 042315 (2003).
  * [6] V. Kendon, “Decoherence in quantum walks – a review,” Mathematical Structures in Computer Science 17, 1169–1220 (2007).
  * [7] D. N. Biggerstaff, R. Heilmann, A. A. Zecevik, M. Gräfe, M. A. Broome, A. Fedrizzi, S. Nolte, A. Szameit, A. G. White, and I. Kassal, “Enhancing coherent transport in a photonic network using controllable decoherence,” Nature Communications 7, 11282 (2016).
  * [8] D. Braun, “Creation of entanglement by interaction with a common heat bath,” Phys. Rev. Lett. 89, 277901 (2002).
  * [9] F. Benatti, R. Floreanini, and M. Piani, “Environment induced entanglement in markovian dissipative dynamics,” Phys. Rev. Lett. 91, 070402 (2003).
  * [10] D. Mogilevtsev and V. S. Shchesnovich, “Single-photon generation by correlated loss in a three-core optical fiber,” Opt. Lett. 35, 3375–3377 (2010).
  * [11] S. Mukherjee, D. Mogilevtsev, G. Slepyan, T. H. Doherty, R. Thomson, and N. Korolkova, “Dissipatively coupled waveguide networks for coherent diffusive photonics,” Nature Communications 8, 1909 (2017).
  * [12] S. Ke, D. Zhao, Q. Liu, S. Wu, B. Wang, and P. Lu, “Optical imaginary directional couplers,” J. Lightwave Technol. 36, 2510–2516 (2018).
  * [13] R. Alrifai, V. Coda, J. Peltier, A. A. Rangelov, and G. Montemezzani, “Ultrabroadband beam splitting in a dissipative system of three waveguides,” Phys. Rev. A 103, 023527 (2021).
  * [14] F.-Q. Dou, Y.-T. Wei, M.-P. Han, and J.-A. Sun, “Controllable multiple beam splitting in hermitian and non-hermitian symmetric coupled waveguide systems,” Journal of Optics 24, 065801 (2022).
  * [15] H. Fan, J. Chen, Z. Zhao, J. Wen, and Y.-P. Huang, “Antiparity-time symmetry in passive nanophotonics,” ACS Photonics7, 3035–3041 (2020).
  * [16] A. Metelmann and A. A. Clerk, “Nonreciprocal photon transmission and amplification via reservoir engineering,” Phys. Rev. X 5, 021025 (2015).
  * [17] Y. Matsuzaki, V. M. Bastidas, Y. Takeuchi, W. J. Munro, and S. Saito, “One-way transfer of quantum states via decoherence,” Journal of the Physical Society of Japan 89, 044003 (2020).
  * [18] X. Huang, C. Lu, C. Liang, H. Tao, and Y. Liu, “Loss-induced nonreciprocity,” Light, Science & Applications 10, 30 (2021).
  * [19] I. Peshko, D. Pustakhod, and D. Mogilevtsev, “Breaking reciprocity by designed loss,” J. Opt. Soc. Am. B 39, 1926–1935 (2022).
  * [20] H. Ezaki, E. Hanamura, and Y. Yamamoto, “Generation of phase states by two-photon absorption,” Phys. Rev. Lett. 83, 3558–3561 (1999).
  * [21] A. R. R. Carvalho, P. Milman, R. L. de Matos Filho, and L. Davidovich, “Decoherence, pointer engineering, and quantum state protection,” Phys. Rev. Lett. 86, 4988–4991 (2001).
  * [22] F. Verstraete, M. M. Wolf, and J. Ignacio Cirac, “Quantum computation and quantum-state engineering driven by dissipation,” Nature Physics 5, 633–636 (2009).
  * [23] Y. Ashida, Z. Gong, and M. Ueda, “Non-hermitian physics,” Advances in Physics 69, 249–435 (2020).
  * [24] M.-A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363, eaar7709 (2019).
  * [25] L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity–time symmetry,” Nat. Photon. 11, 752 (2017).
  * [26] R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11 (2018).
  * [27] Ş. K. Özdemir, S. Rotter, F. Nori, and L. Yang, “Parity-time symmetry and exceptional points in photonics,” Nat. Mater. 18, 783–798 (2019).
  * [28] L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one-dimensional $\mathcal{PT}$-symmetric photonic heterostructures,” Phys. Rev. A 85, 023802 (2012).
  * [29] A. Novitsky, D. Lyakhov, D. Michels, A. A. Pavlov, A. S. Shalin, and D. V. Novitsky, “Unambiguous scattering matrix for non-hermitian systems,” Phys. Rev. A 101, 043834 (2020).
  * [30] D. V. Novitsky, A. Karabchevsky, A. V. Lavrinenko, A. S. Shalin, and A. V. Novitsky, “$\mathcal{PT}$ symmetry breaking in multilayers with resonant loss and gain locks light propagation direction,” Phys. Rev. B 98, 125102 (2018).
  * [31] Z. J. Wong, Y.-L. Xu, J. Kim, K. O’Brien, Y. Wang, L. Feng, and X. Zhang, “Lasing and anti-lasing in a single cavity,” Nat. Photon. 10, 796–801 (2016).
  * [32] D. V. Novitsky, “Cpa-laser effect and exceptional points in pt-symmetric multilayer structures,” Journal of Optics 21, 085101 (2019).
  * [33] D. V. Novitsky, A. C. Valero, A. Krotov, T. Salgals, A. S. Shalin, and A. V. Novitsky, “Cpa-lasing associated with the quasibound states in the continuum in asymmetric non-hermitian structures,” ACS Photonics 9, 3035–3042 (2022).
  * [34] A. V. Hlushchenko, V. I. Shcherbinin, D. V. Novitsky, and V. R. Tuz, “Multimode parity-time symmetry and loss compensation in coupled waveguides with loss and gain,” Phys. Rev. A 104, 013507 (2021).
  * [35] J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: Application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112, 203901 (2014).
  * [36] W. Chen, S. Ozdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
  * [37] A. Novitsky, F. Morozko, D. Gao, L. Gao, A. Karabchevsky, and D. V. Novitsky, “Resonance energy transfer near higher-order exceptional points of non-hermitian hamiltonians,” Phys. Rev. B 106, 195410 (2022).
  * [38] S. Longhi, “Quantum interference and exceptional points,” Opt. Lett. 43, 5371–5374 (2018).
  * [39] M. S. Rudner and L. S. Levitov, “Topological transition in a non-hermitian quantum walk,” Phys. Rev. Lett. 102, 065703 (2009).
  * [40] W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42, 1698–1701 (1979).
  * [41] M. V. Berry and M. Wilkinson, “Diabolical points in the spectra of triangles,” Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 392, 15–43 (1984).
  * [42] D. Mogilevtsev, G. Y. Slepyan, E. Garusov, S. Y. Kilin, and N. Korolkova, “Quantum tight-binding chains with dissipative coupling,” New Journal of Physics 17, 043065 (2015).
  * [43] D. J. Aldous, “Some inequalities for reversible markov chains,” Journal of the London Mathematical Society s2-25, 564–576 (1982).
  * [44] T. Eichelkraut, S. Weimann, S. Stützer, S. Nolte, and A. Szameit, “Radiation-loss management in modulated waveguides,” Opt. Lett. 39, 6831–6834 (2014).
  * [45] V. S. Shchesnovich and D. S. Mogilevtsev, “Three-site bose-hubbard model subject to atom losses: Boson-pair dissipation channel and failure of the mean-field approach,” Phys. Rev. A 82, 043621 (2010).
  * [46] A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of $\mathcal{P}\mathcal{T}$-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
  * [47] S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of branch points in $\mathcal{P}\mathcal{T}$-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
  * [48] F. Yang, Y.-C. Liu, and L. You, “Anti-$\mathcal{PT}$ symmetry in dissipatively coupled optical systems,” Phys. Rev. A 96, 053845 (2017).
  * [49] Y. Qin, H. Chen, D. Luo, C. Pan, H. Hu, Y. Zhang, and D. Wei, “Quantum interference in anti-parity-time symmetric coupled waveguide system,” Opt. Express 29, 29175–29185 (2021).
  * [50] S. Longhi, “Optical analog of population trapping in the continuum: Classical and quantum interference effects,” Phys. Rev. A 79, 023811 (2009).
  * [51] S. Noschese, L. Pasquini, and L. Reichel, “Tridiagonal toeplitz matrices: properties and novel applications,” Numerical Linear Algebra with Applications 20, 302–326 (2013).
  * [52] Y. Suematsu and K. Kishino, “Coupling coefficient in strongly coupled dielectric waveguides,” Radio Science 12, 587–592 (1977).