# Non-unitary transformation approach to $\mathcal{PT}$ dynamics

Braulio M. Villegas-Martínez  Instituto Nacional de Astrofísica Óptica y
Electrónica. Calle Luis Enrique Erro No. 1, Santa María Tonantzintla, Pue.,
72840, Mexico  Francisco Soto-Eguibar Corresponding author<EMAIL_ADDRESS>Instituto Nacional de Astrofísica Óptica y Electrónica. Calle Luis Enrique
Erro No. 1, Santa María Tonantzintla, Pue., 72840, Mexico  Sergio A. Hojman
Departamento de Ciencias, Facultad de Artes Liberales, Universidad Adolfo
Ibáñez, Santiago, 7491169, Chile Departamento de Física, Facultad de Ciencias,
Universidad de Chile, Santiago, 7800003, Chile Centro de Recursos Educativos
Avanzados, CREA, Santiago, 7500018, Chile.  Felipe A. Asenjo  3Facultad de
Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Santiago, 7491169, Chile.
Héctor M. Moya-Cessa  Instituto Nacional de Astrofísica Óptica y Electrónica.
Calle Luis Enrique Erro No. 1, Santa María Tonantzintla, Pue., 72840, Mexico

###### Abstract

We show that several Hamiltonians that are $\mathcal{PT}$ symmetric may be
taken to Hermitian Hamiltonians via a non-unitary transformation and vice
versa. We also show that for some specific Hamiltonians such non-unitary
transformations may be associated, via a fractional-Wick rotation, to complex
time.

##  1 Introduction

Non-Hermitian Hamiltonian systems have become a multifaceted research frontier
which encompasses a wide range of theoretical as well as experimental
interdisciplinary fields [1, 2, 3, 4, 5, 6, 7]. The choice of the proper
framework to investigate the non-Hermitian Hamiltonians depends on the nature
of their eigenvalues. A particular emphasis of interest centers on a class of
these systems obeying the combined parity-time ($\mathcal{PT}$) symmetry. The
main reason is that this generalization of the conventional quantum mechanics
to the complex domain has opened an alternative condition (but not sufficient)
to guarantee the reality of the spectrum for non-Hermitian Hamiltonians [8, 9,
10, 11]. This field of research traces back to Bender’s original conjecture
[8, 9, 10, 12, 13, 14] where the hermiticity condition can be relaxed under
the assumption of space-time reflection symmetry; soon after, a more general
theoretical proposal than the $\mathcal{PT}$ concept was given in [15, 16,
17]. Naturally, Bender’s pioneering work has marked the grounds for these non-
Hermitian Hamiltonians to acquire a new meaning with $\mathcal{PT}$-symmetry
in non-Hermitian quantum mechanics [18, 19, 20, 21]. These concepts indeed
have already led to compelling applications in compound photonic structures,
for example $\mathcal{PT}$-symmetric Hamiltonians found an ideal paragon on
coupled waveguides differentiated by absorption and amplification into their
refractive index profiles [22, 23, 24, 25, 26, 27, 28]. On the other hand, in
the light of elementary quantum theory, the acceptance of
$\mathcal{PT}$-symmetric Hamiltonians have been flourishing for a new
framework where can be mapped into a Hermitian ones by under specific
transformations that underlie the reality of their spectrum. So far most
effort has gone the grounds for treating time-independent and the time-
dependent scenarios of such classes of Hamiltonians through Dyson maps, gauge-
like transformations, Bogoliubov, Darboux or non-unitary transformations [29,
30, 31, 32, 33, 34, 35]; most of them, however, involve cumbersome procedures
and complicated requirements, which currently limits the scope of their
practical applications. Following the seminal insight of these developments,
we hence proceed in this work to look for non-unitary transformations that
applied to Hermitian Hamiltonians, i.e. Hamiltonians that have real
eigenvalues, produce $\mathcal{PT}$-symmetric Hamiltonians, and vice versa.
Broadly speaking, our purpose is not to engage in a comprehensive review;
rather, we want to show that under an adequate choice of the arguments of the
non-unitary transformations, it is possible to produce some renown
$\mathcal{PT}$-symmetric Hamiltonians. Resulting non-unitary transformations
offer new perspectives, and are substantially simpler than those reported
above in literature. Particularly, we show that one of those transformations
may be associated with a fractional-Wick rotation to complex time.

##  2 Non-conservative binary oscillator

We consider first the $\mathcal{PT}$ symmetric Hamiltonian used in [36],

$\hat{H}^{(\mathcal{PT})}=\epsilon 1_{2\times
2}+k\sigma_{x}+i\gamma\sigma_{z},$ (1)

that describes two equally tuned oscillators at energy level $\epsilon$ and
with their attenuation rates differing by $2\gamma$. In this Hamiltonian
$\sigma_{x},\,\sigma_{y},\,\sigma_{z}$ are the Pauli spin matrices,
$1_{2\times 2}$ is the unit $2\times 2$ matrix, and $\epsilon,\,k,\,\gamma$
are arbitrary real constants; we will also assume, for simplicity and without
loss of generality, that $k$ and $\gamma$ are strictly positive. Notice that
the Hamiltonian (1) is $\mathcal{PT}$ symmetric but it is not Hermitian.
The Hamiltonian (1) may be transformed via $\exp(-\theta\sigma_{y})$ to

$\hat{H}^{(\mathcal{PT})}=\exp(-\theta\sigma_{y})\hat{H}\exp(\theta\sigma_{y})$
(2)

with

$\hat{H}=\epsilon 1_{2\times 2}+\alpha\sigma_{x}.$ (3)

We may obtain the angle of rotation from

$\displaystyle\exp(-\theta\sigma_{y})$
$\displaystyle\hat{H}\exp(\theta\sigma_{y})=\epsilon 1_{2\times
2}+\alpha[\sigma_{x}\cosh(2\theta)+i\sigma_{z}\sinh(2\theta)],$ (4)

which establishes the conditions $\alpha\cosh(2\theta)=k$ and
$\alpha\sinh(2\theta)=\gamma$; among the infinite number of solutions of these
equations, for simplicity we choose for $k\neq\gamma$,

$\theta=\frac{1}{2}\ln\left(\frac{k+\gamma}{\sqrt{k^{2}-\gamma^{2}}}\right),\qquad\alpha=\sqrt{k^{2}-\gamma^{2}},\qquad|k|\neq\gamma$
(5)

for $k=\gamma$, we get $\tanh(2\theta)=1$ that does not have a solution and
the transformation $\exp(-\theta\sigma_{y})$ breaks down. As we will see
below, this corresponds to the exceptional points of the $\mathcal{PT}$
symmetric Hamiltonian.
The two eigenvalues of $\hat{H}$, (3), are $\epsilon\pm\alpha$. These two
eigenvalues are real as long as $\alpha$ is real; which, of course, also
establishes the region for which the Hamiltonian is Hermitian. For the
$\mathcal{PT}$ symmetric Hamiltonian $\hat{H}^{(\mathcal{PT})}$, (1), the
eigenvalues are $\epsilon\pm\sqrt{k^{2}-\gamma^{2}}$.
We then have three regions:

  1. 1.

When $k>\gamma$, the difference between the attenuation rates of the two
equally tuned oscillators must be smaller than the natural frequency of the
oscillators. In this case, $\theta$ and $\alpha$ are real, the Hamiltonian
$\hat{H}$, (3), is Hermitian, and so its two eigenvalues are real. The
transformation $\exp(-\theta\sigma_{y})$ is not unitary. The two eigenvalues
of the $\mathcal{PT}$ symmetric Hamiltonian $\hat{H}^{(\mathcal{PT})}$, (1),
are also real, and we are in the unbroken symmetry region.

  2. 2.

When $k<\gamma$, the angle $\theta$ becomes complex and the transformation
$\exp(-\theta\sigma_{y})$ consists of the composition of a unitary
transformation and a non unitary one. The parameter $\alpha$ also becomes
complex, and the eigenvalues of both Hamiltonians, $\hat{H}^{(\mathcal{PT})}$
and $\hat{H}$ are complex numbers. In this case, we are in the broken symmetry
region.

  3. 3.

When $k=\gamma$, the corresponding matrices to $\hat{H}^{(\mathcal{PT})}$ and
$\hat{H}$ are defective; we are, in this case, in the exceptional points. At
this value of the parameters, the eigenvalues coalesce and there is crossing
between the energy levels. At these points there is only one eigenvalue,
$\epsilon$, and then the corresponding Hamiltonian matrix becomes defective,
which means that it is non diagonalizable.

The behavior of the eigenvalues as function of the real parameter $k$,
described in the previous points is depicted in Fig 1.

Figure 1: The real and imaginary parts of the energy eigenvalues as function
of the parameter $k$ for $\epsilon=2$ and $\gamma=1$.

From the mathematical point of view, the relative complexity of the behavior
of the eigenvalues comes from the fact that the function
$\sqrt{k^{2}-\gamma^{2}}$ is multi-valued and therefore has branch points; as
explained by Bender [21], we get a complete picture of the behavior of the
eigenvalues when we make the complex deformation of the problem. As $\epsilon$
is just an additive constant, it will be enough to analyze the behavior of the
multivalued function $\sqrt{k^{2}-\gamma^{2}}$, when we extend the interaction
parameter $k$ to become complex.
In Figure 2, we present the “positive” branch of $\sqrt{k^{2}-\gamma^{2}}$ as
function of $k$, extending this variable to the complex domain. We observe the
branch points, that in the left subfigure are marked in black; in the right
subfigure we can view the discontinuity in the argument of the function.

(a) Complex plot of $\sqrt{k^{2}-\gamma^{2}}$

(b) $\arg\left(\sqrt{k^{2}-\gamma^{2}}\right)$

Figure 2: The positive branch of $\sqrt{k^{2}-\gamma^{2}}$ as function of $k$
complex for $\gamma=1$.

In Figure 3, we present the “negative” branch of $\sqrt{k^{2}-\gamma^{2}}$ as
function of $k$, as a complex variable. Exactly as in the other branch, we
have marked the branch points in black in the left subfigure; also the
discontinuity of the argument of the function can be seen in the right
subfigure.

(a) Complex plot of $-\sqrt{k^{2}-\gamma^{2}}$

(b) $\arg\left(-\sqrt{k^{2}-\gamma^{2}}\right)$

Figure 3: The negative branch of $\sqrt{k^{2}-\gamma^{2}}$ as function of $k$
complex for $\gamma=1$.

##  3 $\hat{J}_{x}$ lattice

Let us consider the angular momentum operators
$\left\\{\hat{J}_{x},\hat{J}_{y},\hat{J}_{z}\right\\}$, defined by the usual
commutations rules [38, 39, 37]

$\left[\hat{J}_{x},\hat{J}_{y}\right]=i\hat{J}_{z},\qquad\left[\hat{J}_{y},\hat{J}_{z}\right]=i\hat{J}_{x},\qquad\left[\hat{J}_{z},\hat{J}_{x}\right]=i\hat{J}_{y}.$
(6)

We introduce the Hamiltonian [40]

$\hat{H}^{\left(\mathcal{PT}\right)}=\epsilon\hat{I}+i\gamma\hat{J}_{z}+k\hat{J}_{x},$
(7)

with $\epsilon,\;\gamma$ and $k$ reals.
Note that:

  1. 1.

The Hamiltonian $\hat{H}^{\left(\mathcal{PT}\right)}$, Eq. (7), is not
Hermitian (symmetric).

  2. 2.

The Hamiltonian $\hat{H}^{\left(\mathcal{PT}\right)}$, Eq. (7), is
$\mathcal{PT}$ symmetric when the parity transformation $\mathcal{P}$ is the
conjugated transposition.

  3. 3.

The first part of the Hamiltonian $\hat{H}^{\left(\mathcal{PT}\right)}$,
$\epsilon\hat{I}+i\gamma\hat{J}_{z}$ , is diagonal and can be considered as an
ensemble of systems without interaction or as one system with several
energetic levels that don’t interact either. The second part of the
Hamiltonian, $k\hat{J}_{x}$, introduces an interaction between the elements
that compose it or between the levels that it has. Thus, we can consider $k$
as the parameter of interaction [21].

We make now the transformation

$\hat{H}=\exp\left(\theta\hat{J}_{y}\right)\hat{H}^{(\mathcal{PT})}\exp\left(-\theta\hat{J}_{y}\right),$
(8)

where the parameter $\theta$ can be a complex number, and applying the
Hadamard’s lemma [41, 42, 43], we obtain

$\hat{H}=\epsilon\hat{I}+i\left[\gamma\cosh\left(\theta\right)-k\sinh\left(\theta\right)\right]\hat{J}_{z}+\left[k\cosh\left(\theta\right)-\gamma\sinh\left(\theta\right)\right]\hat{J}_{x}.$
(9)

We choose the parameter $\theta$ in such a way that the coefficient of
$\hat{J}_{z}$ above be zero; thus,

$\theta_{0}=\ln\left(\sqrt{\frac{k+\gamma}{k-\gamma}}\right),$ (10)

and

$\hat{H}_{0}=\epsilon\hat{I}+\sqrt{k^{2}-\gamma^{2}}\hat{J}_{x}.$ (11)

Note that there are an infinite number of ways to choose the $\theta$
parameter so as to make the coefficient of $\hat{J}_{z}$ zero. We have
arbitrarily chosen the one given by Eq. (10) because it is the one that best
suits us for simplicity; however, we could have chosen any other of the
possible ones and the essence of the result would be exactly the same.
Remark also that:

  1. 1.

The Hamiltonian (11) is Hermitian if and only if $|k|\geq|\gamma|$; note that
in that case, $\theta_{0}$ will be real.

  2. 2.

If $|k|<|\gamma|$, $\sqrt{k^{2}-\gamma^{2}}$ will be a complex number and the
Hamiltonian (11) is not Hermitian; however, it is $\mathcal{PT}$ symmetric
when the parity transformation $\mathcal{P}$ is defined as the conjugated
transposition.

  3. 3.

When $|k|=|\gamma|$, the Hamiltonian (11) becomes trivial and the $\theta$
parameter is undefined. As we will see below, this case corresponds to the
exceptional points.

The spectrum of the Hamiltonian (7) is

$E_{l}=\epsilon+l\sqrt{k^{2}-\gamma^{2}},\quad l=-j,...,+j;\qquad
j=\frac{1}{2},1,\frac{3}{2},2,\dots,\quad N=2j+1,$ (12)

where $N=2j+1$ is the dimension of the space. Note that the case
$j=\frac{1}{2}$ reduces to the non-conservative binary oscillator of the
previous section.
As expected, if $|k|\geq|\gamma|$ the spectrum is real as such as the
Hamiltonian is Hermitian; however, when $|k|=|\gamma|$ the parameter $\theta$
is undefined, the spectrum consists of only one value, $\epsilon$, and the
Hamiltonian becomes defective, i.e., non-diagonalizable. As we will see below,
this case corresponds to the exceptional points.
If we remove the restriction that $|k|\geq|\gamma|$, things get more
interesting because we will have three regions. First, when $|k|>|\gamma|$ we
will have the unbroken symmetry region; second, when $|k|<|\gamma|$ we will
have the broken symmetry region; and finally and third, when $|k|=|\gamma|$ we
will have the exceptional points.
In the unbroken symmetry region, i.e., when $|k|>|\gamma|$, the spectrum is
real, the eigenstates oscillates and the systems, or levels of the system,
remains in equilibrium between them.
In the broken symmetry region, i.e., when $|k|<|\gamma|$, the spectrum is
complex, eigenvalues appear in pairs, being complex conjugates of each other.
Some of the eigenstates grow and others decay; the systems, or levels of the
system, are not in equilibrium.
As we already mentioned, when $|k|=|\gamma|$, we get the exceptional points.
At this value of the parameters, the eigenvalues coalesce and there is
crossing between the energy levels. At these points there is only one
eigenvalue, $\epsilon$, and then the corresponding Hamiltonian matrix becomes
defective, which means that it is non diagonalizable.
Exactly in the same way that the case of the previous section, the complicated
behavior comes from the fact that the function $\sqrt{k^{2}-\gamma^{2}}$ is
multi-valued and therefore has branch points. The arguments presented in
Figure 1 are the same for all dimensions.
In Figure 4, we plot the real and imaginary parts of the energy eigenvalues
for $j=1$ (dimension 3) as function of the parameter $k$ for $\epsilon=2$ and
$\gamma=1$. In this case we have three eigenvalues,
$\left\\{\epsilon-\sqrt{k^{2}-\gamma^{2}},\epsilon,\epsilon+\sqrt{k^{2}-\gamma^{2}}\right\\}$.
In this case again the non-broken symmetry region is the region where $|k|>1$,
the broken symmetry region is the one where $|k|<1$ and the exceptional points
are $k=-1$ and $k=1$.It is worth mentioning that we have three eigenvalues
coalesce at $k=1,-1$ and the system exhibits a third-order exceptional point.
In the unbroken symmetry region, all the eigenvalues are real. In the broken
symmetry region, two of the eigenvalues are complex, and one is the complex
conjugate of the other; the other eigenvalue, $\epsilon$ is always real.

Figure 4: The real and imaginary parts of the energy eigenvalues for $j=1$ as
function of the parameter $k$ for $\epsilon=2$ and $\gamma=1$.

In Figure 5, we plot the real and imaginary parts of the energy eigenvalues
for $j=3/2$ (dimension 4) as function of the parameter $k$ for $\epsilon=2$
and $\gamma=1$. In this case we have four eigenvalues,

$\left\\{\epsilon-\frac{3}{2}\sqrt{k^{2}-\gamma^{2}},\epsilon-\frac{1}{2}\sqrt{k^{2}-\gamma^{2}},\epsilon+\frac{1}{2}\sqrt{k^{2}-\gamma^{2}},\epsilon+\frac{3}{2}\sqrt{k^{2}-\gamma^{2}}\right\\}.$

We observe that the eigenenergies are real for $|k|>1$ as corresponds to the
unbroken symmetry region; for $|k|<1$, we are in the broken symmetry region
and the eigenergies are complex and become in pairs; at the exceptional
points, again $k=-1$ and $k=1$, the eigenenergies merge. In this case, four
eigenvalues coalesce sharing a single eigenstate, this is a simple example of
a system which exhibits a four-order exceptional point; such behavior it can
be observed from the energy bifurcation diagram of Fig.5. Therefore, the
nature of the dimension $N=2j+1$ of the Hamiltonian (7) leads to the existence
of higher order exceptional points.

Figure 5: The real and imaginary parts of the energy eigenvalues for $j=3/2$
as function of the parameter $k$ for $\epsilon=2$ and $\gamma=1$.

##  4 The complex deformed harmonic oscillator

A very well known $\mathcal{PT}$ symmetric Hamiltonian, but not Hermitian, is
the complex deformed harmonic oscillator Hamiltonian given by

$\hat{H}=\hat{p}^{2}+\hat{x}^{2}(i\hat{x})^{\epsilon},$ (13)

where $\epsilon$ is a non-negative real number [12, 13, 21].
We make the transformation $\ket{\psi}=\hat{S}_{r}^{-1}\ket{\psi}$, with
$\hat{S}_{r}=\exp\left[\frac{r}{2}\left(\hat{x}\hat{p}+\hat{p}\hat{x}\right)\right]$,
such that the transformed Hamiltonian is

$\hat{{\mathcal{H}}}=\hat{S}_{r}\hat{H}\hat{S}_{r}^{-1}.$ (14)

We can write explicitly the previous Hamiltonian using the expressions

$\displaystyle\hat{S}_{r}\hat{x}\hat{S}_{r}^{-1}$
$\displaystyle=\exp\left[\frac{r}{2}(\hat{x}\hat{p}+\hat{p}\hat{x})\right]\hat{x}\exp\left[-\frac{r}{2}(\hat{x}\hat{p}+\hat{p}\hat{x})\right]$
$\displaystyle=\exp\left(-ir\right)\hat{x},$ (15a)
$\displaystyle\hat{S}_{r}\hat{p}\hat{S}_{r}^{-1}$
$\displaystyle=\exp\left[\frac{r}{2}(\hat{x}\hat{p}+\hat{p}\hat{x})\right]\hat{p}\exp\left[-\frac{r}{2}(\hat{x}\hat{p}+\hat{p}\hat{x})\right]$
$\displaystyle=\exp\left(ir\right)\hat{p},$ (15b)

which are derived from the commutation relation $[\hat{x},\hat{p}]=i$ and the
Hadamard´s lemma [43, 41, 42]
$e^{g\hat{A}}\hat{B}e^{-g\hat{A}}=\hat{B}+g[\hat{A},\hat{B}]+\frac{g^{2}}{2!}[\hat{A},[\hat{A},\hat{B}]]+\dots$
when $g=\frac{r}{2}$; we obtain

$\hat{\mathcal{H}}=\exp\left(2ir\right)\hat{p}^{2}+\exp\left[-ir\left(2+\epsilon\right)+i\frac{\pi}{2}\epsilon\right]\hat{x}^{2+\epsilon}.$
(16)

As $r$ is an arbitrary real constant, we choose it such that

$\exp\left(2ir\right)=\exp\left[-ir\left(2+\epsilon\right)+i\frac{\pi}{2}\epsilon\right],$
(17)

and we get

$r=\frac{\pi(\epsilon+4k)}{2(\epsilon+4)},$ (18)

with $k$ an integer. Thus, the transformed Hamiltonian reads as

$\hat{\mathcal{H}}_{k}=\exp\left(i\pi\frac{\epsilon+4k}{4+\epsilon}\right)\left(\hat{p}^{2}+\hat{x}^{2+\epsilon}\right),\quad
k\in\mathbb{Z}.$ (19)

If we consider now a continuous time Wick rotation, defining the imaginary
time

$\tau=t\exp\left(i\pi\frac{\epsilon+4k}{4+\epsilon}\right)\,,$ (20)

we get finally the Schrödinger equation

$i\frac{\partial\psi(x,\tau)}{\partial\tau}=\left(\hat{p}^{2}+\hat{x}^{2+\epsilon}\right)\psi(x,\tau)$
(21)

with $\hat{p}^{2}+\hat{x}^{2+\epsilon}$ a Hermitian Hamiltonian.
The Wick rotation (20) belongs to a family of general continuous rotations
with the form $t\rightarrow\exp(i\theta)t$, with $0\leq\theta\leq\pi/2$;
therefore, Wick rotations may take us from Schrödinger-like equations to
diffusion-like equations. They have also been proposed to develop a
supersymmetric Euclidean theory from any supersymmetric Minkowski theory for
Dirac spinors, thus producing an equivalence between bosons and fermions [44,
45].

##  5 Complex deformation of a general Hamiltonian

We may show that the very general $\mathcal{PT}$ symmetric, but non-Hermitian,
Hamiltonian

$\hat{H}=\hat{p}^{2}+V(i\hat{x})$ (22)

may be transformed with the non-unitary squeezing transformation
$\hat{S}=\exp\left[\frac{\pi}{4}\left(\hat{x}\hat{p}+\hat{p}\hat{x}\right)\right]$
(which is the same used in the case of the complex deformed harmonic
oscillator with $r=\pi/2$) to

$\hat{\mathcal{H}}=\exp\left(i\pi\right)\hat{p}^{2}+V(i\hat{x}e^{-i\frac{\pi}{2}}),$
(23)

which becomes the Hermitian Hamiltonian

$\hat{\mathcal{H}}=-\hat{p}^{2}+V(\hat{x}).$ (24)

As this Hamiltonian is Hermitian its evolution will be unitary and therefore
the usual techniques used in quantum mechanics may be applied.

##  6 Generalized Swanson oscillator

Let us now address another type of non-Hermitian system, a generalized version
of the quadratic Swanson model
$H_{SW}=\lambda\left(\hat{n}+1/2\right)+\beta_{1}\hat{a}^{\dagger
2}+\beta_{2}\hat{a}^{2}$ [46, 47]; in effect, we are going to consider the
Hamiltonian defined as

$\hat{H}_{GSW}=\lambda\left(\hat{n}+\frac{1}{2}\right)+\alpha_{1}\hat{a}^{\dagger}+\alpha_{2}\hat{a}+\beta_{1}\hat{a}^{\dagger
2}+\beta_{2}\hat{a}^{2},\quad\lambda,\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}\in\mathbb{R},$
(25)

where $\hat{a}^{\dagger}$ and $\hat{a}$ are the creation and annihilation
operators of the standard harmonic oscillator and
$\hat{n}=\hat{a}^{\dagger}\hat{a}$ is the number operator [48, 49, 50]; it is
worthwhile to note that this Hamiltonian is a non-Hermitian version of a
single-mode squeezed coherent harmonic oscillator. The presence of the linear
terms makes the Hamiltonian in Eq. (25) non-$\mathcal{PT}$-symmetric; this can
be readily demonstrated applying the parity operator
$\hat{P}:\hat{a}\rightarrow-\hat{a}$ and the time-reversal operator
$\hat{T}:\hat{a}\rightarrow\hat{a}$, with similar transformations for
$\hat{a}^{\dagger}$. For the particular case $\alpha_{1}=\alpha_{2}=0$, we get
back to the original Swanson’s quadratic model, which has a $\mathcal{PT}$
symmetric non-Hermitian Hamiltonian [51, 52]. Conversely, when $\alpha_{1}\neq
0$, $\alpha_{2}\neq 0$ but $\beta_{1}=\beta_{2}=0$, the system becomes a non-
Hermitian forced harmonic oscillator.
In a sharp contrast with our recent examples which present a $\mathcal{PT}$
symmetry, here we are interested in translating the
non-$\mathcal{PT}$-symmetric Hamiltonian Eq. (26) to a
$\mathcal{PT}$-symmetric one and show that it admits an equivalent Hermitian
representation to the harmonic oscillator. Hence, we directly consider to
solve the corresponding Schrödinger equation:
$i\frac{d}{dt}\ket{\psi\left(t\right)}=\hat{H}\ket{\psi\left(t\right)}$; we
will do that by means of three non-unitary transformations. We begin by
introducing the transformation
$\hat{\eta}_{1}=\exp\left[\frac{\hat{n}}{2}\ln\left(\frac{\alpha_{2}}{\alpha_{1}}\right)\right]$
with $\alpha_{1}\neq 0$ and $\alpha_{2}\neq 0$; the transformed Hamiltonian
becomes

$\displaystyle\hat{H}_{1}=\hat{\eta}_{1}\hat{H}_{GSW}\hat{\eta}_{1}^{-1}=\lambda\left(\hat{n}+\frac{1}{2}\right)+\sqrt{\alpha_{1}\alpha_{2}}\left(\hat{a}^{\dagger}+\hat{a}\right)+\left(\frac{\alpha_{2}\beta_{1}}{\alpha_{1}}\hat{a}^{\dagger
2}+\frac{\alpha_{1}\beta_{2}}{\alpha_{2}}\hat{a}^{2}\right).$ (26)

Note that this transformed Hamiltonian is non-Hermitian and
non-$\mathcal{PT}$-symmetric. Remark also that the application of the operator
$\hat{\eta}_{1}$ makes that the linear terms in $\hat{a}$ and
$\hat{a}^{\dagger}$ can be expressed as a sum of both operators multiplied by
the factor $\sqrt{\alpha_{1}\alpha_{2}}$.
We get rid of the linear terms by means of the simple non-unitary
transformation
$\hat{\eta}_{2}=\exp\left(\gamma_{1}\hat{a}^{\dagger}-\gamma_{2}\hat{a}\right)$,
where $\gamma_{1}$ and $\gamma_{2}$ are two unknown parameters to be
determined. Remark that $\hat{\eta}_{2}$ translates into the usual
displacement operator once $\gamma_{1}=\gamma_{2}$; then, the application of
$\hat{\eta}_{2}$ to the Hamiltonian $\hat{H}_{1}$, Eq. (26), yields to

$\displaystyle\hat{H}_{2}=\hat{\eta}_{2}\hat{H}_{1}\hat{\eta}_{2}^{-1}=$
$\displaystyle\lambda\left(\hat{n}+\frac{1}{2}\right)+\frac{\alpha_{2}\beta_{1}}{\alpha_{1}}\hat{a}^{\dagger
2}+\frac{\alpha_{1}\beta_{2}}{\alpha_{2}}\hat{a}^{2}+\left(\sqrt{\alpha_{1}\alpha_{2}}-\gamma_{1}\lambda-\frac{2\gamma_{2}\alpha_{2}\beta_{1}}{\alpha_{1}}\right)\hat{a}^{\dagger}$
$\displaystyle+\left(\sqrt{\alpha_{1}\alpha_{2}}-\gamma_{2}\lambda-\frac{2\gamma_{1}\alpha_{1}\beta_{2}}{\alpha_{2}}\right)\hat{a}+\lambda\gamma_{1}\gamma_{2}-\sqrt{\alpha_{1}\alpha_{2}}\left(\gamma_{1}+\gamma_{2}\right)+\frac{\gamma_{1}^{2}\alpha_{1}\beta_{2}}{\alpha_{2}}+\frac{\gamma_{2}^{2}\alpha_{2}\beta_{1}}{\alpha_{1}}.$
(27)

In general, this transformed Hamiltonian is non-Hermitian and
non-$\mathcal{PT}$-symmetric. We choose the parameters $\gamma_{1}$ and
$\gamma_{2}$ of the $\hat{\eta}_{2}$ transformation in such a way that the
coefficients of the linear terms are zero; i.e., through the equations

$\displaystyle\gamma_{1}\lambda-\sqrt{\alpha_{1}\alpha_{2}}+\frac{2\gamma_{2}\alpha_{2}\beta_{1}}{\alpha_{1}}=0,\qquad\gamma_{2}\lambda-\sqrt{\alpha_{1}\alpha_{2}}+\frac{2\gamma_{1}\alpha_{1}\beta_{2}}{\alpha_{2}}=0,$
(28)

which gives

$\displaystyle\gamma_{1}=\frac{\alpha_{2}}{\sqrt{\alpha_{1}\alpha_{2}}}\frac{\alpha_{1}\lambda-2\alpha_{2}\beta_{1}}{\lambda^{2}-4\beta_{1}\beta_{2}},\qquad\gamma_{2}=\frac{\alpha_{1}}{\sqrt{\alpha_{1}\alpha_{2}}}\frac{\alpha_{2}\lambda-2\alpha_{1}\beta_{2}}{\lambda^{2}-4\beta_{1}\beta_{2}}.$
(29)

Substituting these values in the Hamiltonian $\hat{H}_{2}$, Eq. (6), it takes
the reduce form

$\hat{H}_{2}=\lambda\left(\hat{n}+\frac{1}{2}\right)+\frac{\alpha_{2}\beta_{1}}{\alpha_{1}}\hat{a}^{\dagger
2}+\frac{\alpha_{1}\beta_{2}}{\alpha_{2}}\hat{a}^{2}+\delta,$ (30)

where

$\delta=\frac{\alpha^{2}_{1}\beta_{2}+\alpha^{2}_{2}\beta_{1}-\lambda\alpha_{1}\alpha_{2}}{\lambda^{2}-4\beta_{1}\beta_{2}}.$
(31)

This Hamiltonian is non-Hermitian but it is $\mathcal{PT}$-symmetric, and
possess the same structure as the Hamiltonian of the Swanson oscillator.
In order to solve the Schrodinger equation associated with the Hamiltonian
$\hat{H}_{2}$, we make the third and final transformation
$\hat{\eta}_{3}=\exp\left(\zeta\hat{a}^{2}\right)$. If we select

$\zeta=\frac{-\lambda\alpha_{1}+\sqrt{\alpha_{1}^{2}\lambda^{2}-4\alpha_{1}^{2}\beta_{1}\beta_{2}+4\alpha_{2}^{2}\beta_{1}^{2}}}{4\alpha_{2}\beta_{1}},$
(32)

the transformed Hamiltonian is

$\displaystyle\hat{H}_{3}=\hat{\eta}_{3}\hat{H}_{2}\hat{\eta}_{3}^{-1}=\tilde{\lambda}\left(\hat{n}+\frac{1}{2}\right)+\frac{\alpha_{2}\beta_{1}}{\alpha_{1}}\left(\hat{a}^{\dagger
2}+\hat{a}^{2}\right)+\delta,$ (33)

where

$\tilde{\lambda}=\lambda\sqrt{1-\frac{4\beta_{1}}{\alpha_{1}^{2}\lambda^{2}}\left(\alpha_{1}^{2}\beta_{2}-\alpha_{2}^{2}\beta_{1}\right)}.$
(34)

The Hamiltonian $\hat{H}_{3}$, Eq. (33), is Hermitian if and only if
$\frac{4\beta_{1}}{\alpha_{1}^{2}\lambda^{2}}\left(\alpha_{1}^{2}\beta_{2}-\alpha_{2}^{2}\beta_{1}\right)<1$;
in this case, the transformation
$\hat{\eta}_{3}=\exp\left(\zeta\hat{a}^{2}\right)$ is non-unitary as $\zeta$
can take two real values from the different signs of its square-root
expression. This can be appreciated more clearly in Fig. 6, where in the
regime
$\frac{4\beta_{1}}{\alpha_{1}^{2}\lambda^{2}}\left(\alpha_{1}^{2}\beta_{2}-\alpha_{2}^{2}\beta_{1}\right)<1$,
the model has real values (solid blue line) corresponding to the unbroken
${\mathcal{P}T}$ symmetry. On the other hand, the situation is rather
different if
$\frac{4\beta_{1}}{\alpha_{1}^{2}\lambda^{2}}\left(\alpha_{1}^{2}\beta_{2}-\alpha_{2}^{2}\beta_{1}\right)>1$;
the Hamiltonian $\hat{H}_{3}$, Eq. (33), is not hermitian anymore, since
$\tilde{\lambda}$ appears in conjugate pairs of purely imaginary values, and
the values of $\zeta$ become complex conjugate of each other; above abrupt
transition indicates a broken ${\mathcal{PT}}$-symmetry of the system. In the
case
$\frac{4\beta_{1}}{\alpha_{1}^{2}\lambda^{2}}\left(\alpha_{1}^{2}\beta_{2}-\alpha_{2}^{2}\beta_{1}\right)=1$,
the eigenvalues coalesce and we are at the exceptional point; there we have
only one eigenvalue. Thus, we restrict ourselves to the range of parameters
where $\zeta$ is real and positive (positive sign of the square root), where
the Hamiltonian Eq. (33) is Hermitian and we can guarantee real eigenvalues.
This is also consistent with the conjecture of Bender[12, 13, 14] and in exact
accordance with the typical Swanson’s Hamiltonian which admits a purely real
positive spectrum.

Figure 6: Real and imaginary values of $\tilde{\lambda}/\lambda$ as a function
of
$\frac{4\beta_{1}}{\alpha_{1}^{2}\lambda^{2}}\left(\alpha_{1}^{2}\beta_{2}-\alpha_{2}^{2}\beta_{1}\right)$.
The function $\tilde{\lambda}/\lambda$ exhibits three different regimes
depending on
$\frac{4\beta_{1}}{\alpha_{1}^{2}\lambda^{2}}\left(\alpha_{1}^{2}\beta_{2}-\alpha_{2}^{2}\beta_{1}\right)$
is less, greater or equal to one.

The exact solution of the Schrödinger equation corresponding to the
Hamiltonian (33) is

$\ket{\psi_{3}\left(t\right)}=\exp\left(-i\delta
t\right)\exp\left(-2i\frac{\alpha_{2}\beta_{1}}{\alpha_{1}}\hat{\mathscr{H}}t\right)\ket{\psi_{3}\left(0\right)},$
(35)

where

$\hat{\mathscr{H}}=\hat{K}^{+}+\frac{\alpha_{1}\tilde{\lambda}}{\alpha_{2}\beta_{1}}\hat{K}^{0}+\hat{K}^{-},$
(36)

being $\hat{K}^{+}=\frac{1}{2}\hat{a}^{\dagger 2}$,
$\hat{K}^{0}=\frac{1}{2}\left(\hat{n}+\frac{1}{2}\right)$ and
$\hat{K}^{-}=\frac{1}{2}\hat{a}^{2}$. The commutation relations of these
operators are $\left[\hat{K}^{+},\hat{K}^{-}\right]=-2\hat{K}^{0}$,
$\left[\hat{K}^{0},\hat{K}^{\pm}\right]=\pm\hat{K}^{\pm}$, and they are the
generators of the su(1,1) algebra [53, 54, 55, 56, 57, 58, 59]. Nonetheless,
one may easily convert the Hamiltonian $\hat{H}_{3}$ into the diagonal
Hamiltonian,
$\hat{\tilde{H}}_{3}=\hat{S}^{-1}\hat{H}_{3}\hat{S}=2\sqrt{\lambda^{2}-4\beta_{1}\beta_{2}}\hat{K}^{0}+\delta$
by applying the squeeze-like transformation

$\hat{S}=\exp\left[-\frac{1}{2}\operatorname{arctanh}\left(\frac{2\alpha_{2}\beta_{1}}{\alpha_{1}\tilde{\lambda}}\right)\left(\hat{K}^{+}-\hat{K}^{-}\right)\right].$
(37)

Since the eigenstates of $\hat{K}^{0}$ are $\ket{n}$ with eigenvalues
$\frac{1}{2}\left(n+\frac{1}{2}\right)$, then, the Hamiltonian $\tilde{H}_{3}$
corresponds to the harmonic oscillator with frequency
$\sqrt{\lambda^{2}-4\beta_{1}\beta_{2}}$. Consequently, we at once give to the
energy spectrum of our original Hamiltonian (25) from the eigenequation of the
Hamiltonian $\hat{\tilde{H}}_{3}$, i.e,
$\hat{\tilde{H}}_{3}\ket{n}=E_{n}\ket{n}$, where

$E_{n}=\sqrt{\lambda^{2}-4\beta_{1}\beta_{2}}\left(n+\frac{1}{2}\right)+\frac{\alpha^{2}_{1}\beta_{2}+\alpha^{2}_{2}\beta_{1}-\lambda\alpha_{1}\alpha_{2}}{\lambda^{2}-4\beta_{1}\beta_{2}},$
(38)

being real as long as $\lambda^{2}-4\beta_{1}\beta_{2}\gg 0$. Meanwhile the
eigenstates of $\hat{H}_{GSW}$ are given by
$\ket{\tilde{n}}=\hat{\eta}_{1}^{-1}\hat{\eta}_{2}^{-1}\hat{\eta}_{3}^{-1}\hat{S}\ket{n}$,
where $\hat{\eta}_{1}^{-1},\hat{\eta}_{2}^{-1},\hat{\eta}_{3}^{-1}$and
$\hat{S}$ act like intertwining operators, in the sense that they transform an
eigenstate of $\hat{\tilde{H}}_{3}$ to eigenstate of $\hat{H}_{GSW}$ with the
same spectrum through these operators. In other words, the eigenfunctions of
$\hat{H}_{GSW}$ can be derived from the eigenfunctions of the harmonic
oscillator.
Let us finish this section by mentioning that $\hat{H}_{GSW}$ can be recast in
terms of position and momentum operators by the well-known relationships,
$\hat{a}=\sqrt{\frac{m\omega}{2}}\left(\hat{x}+i\frac{\hat{p}}{m\omega}\right)$
and
$\hat{a}^{\dagger}=\sqrt{\frac{m_{0}\omega_{0}}{2}}\left(\hat{x}-i\frac{\hat{p}}{m_{0}\omega_{0}}\right)$,
being $\omega$ and $m$ the constant frequency and the mass; this leads us to
the Hamiltonian form

$\hat{H}_{GSW}=\nu_{1}\hat{p}^{2}+\nu_{2}\hat{x}^{2}+i\nu_{3}\left(\hat{x}\hat{p}+\hat{p}\hat{x}\right)+i\nu_{4}\hat{p}+\nu_{5}\hat{x},$
(39)

with energy spectrum

$E_{n}=\sqrt{\nu_{1}\nu_{2}+\nu^{2}_{3}}\left(2n+1\right)+\frac{\nu^{2}_{4}\nu_{2}-\nu^{2}_{5}\nu_{1}-2\nu_{3}\nu_{4}\nu_{5}}{4\left(\nu_{1}\nu_{2}+\nu^{2}_{3}\right)},$
(40)

being

$\displaystyle\nu_{1}=$
$\displaystyle\frac{1}{2m\omega}\left(\lambda-\beta_{1}-\beta_{2}\right),\quad\nu_{2}=\frac{m\omega}{2}\left(\lambda+\beta_{1}+\beta_{2}\right),\quad\nu_{3}=\frac{1}{2}\left(\beta_{2}-\beta_{1}\right),$
$\displaystyle\nu_{4}=$
$\displaystyle\frac{1}{\sqrt{2m\omega}}\left(\alpha_{2}-\alpha_{1}\right)\qquad\;\nu_{5}=\sqrt{\frac{m\omega}{2}}\left(\alpha_{1}+\alpha_{2}\right).$
(41)

Using the above Ansatz , one can get the corresponding solution for the
Schrödinger equation governed by the Hamiltonian $\hat{H}_{GSW}$ in the time
dependent case as well as in the time-independent scenario, as already
reported in [60].

###  6.1 Simple photonic lattice analog

Lastly, a simple analog of the previous model for the particular situation
$\beta_{1}=\beta_{2}$ can be realized by using a zigzag waveguide with
nonuniform nearest-neighbor hopping that depend on the square root of the site
number. The light evolution in the non-Hermitian lattice satisfies the
differential equation set

$\displaystyle i\frac{d\Psi_{n}(Z)}{dZ}$
$\displaystyle+\lambda\left(n+\frac{1}{2}\right)\Psi_{n}(Z)+\alpha_{1}\sqrt{n}\;\Psi_{n-1}(Z)+\alpha_{2}\sqrt{n+1}\;\Psi_{n+1}(Z)$
$\displaystyle+\beta\sqrt{n(n-1)}\;\Psi_{n-2}(Z)+\beta\sqrt{(n+1)(n+2)}\;\Psi_{n+2}(Z)=0,\quad
n=0,1,2,...,$ (42)

where $\Psi_{n}(Z)$ is the complex field amplitude in the site $n$ at the
dimensionless propagation distance $Z$, and we adopt the convention that
$\Psi_{n}(Z)=0$ for $n<0$; the term $\lambda n$ is the refractive index and it
varies gradually with the site number. The parameters $\alpha_{1}$ and
$\alpha_{2}$ denote the left and right nearest-neighbor hopping and $\beta$
represents the next-nearest interaction. In order to describe the evolution of
nonclassical light in the above waveguide system, it is convenient to use a
simplified notation in which each single-mode waveguides are arranged in a
vector given by
$\ket{\psi\left(Z\right)}=\sum_{n=0}^{\infty}\Psi_{n}(Z)\ket{n}$, where
$\ket{n}$ plays an analogous role to Fock states. In this form, Eq.(6.1) can
be rewritten into a rather simple and suggestive Schrödinger-like equation
form,

$\displaystyle i\frac{d\ket{\psi\left(Z\right)}}{dZ}=$
$\displaystyle-\left[\lambda\left(\hat{n}+\frac{1}{2}\right)+\alpha_{1}\hat{a}^{\dagger}+\alpha_{2}\hat{a}+\beta\left(\hat{a}^{\dagger
2}+\hat{a}^{2}\right)\right]\ket{\psi\left(Z\right)},$ (43)

whose Hamiltonian is the same as (25), when $\beta_{1}=\beta_{2}$. In fact,
one can easily check that substituting this vector proposal into Eq. (43)
yields the waveguide system given by Eq. (6.1).
A flagship example of how the lattice can be engineered is given in [61];
there, the zigzag shape consists of two interleaved waveguides (see Fig. 7)
where one layer forms a scalene triangle with two adjacent waveguides in the
down layer; such scalene configuration gives rise to unequal cross coupling
nearest-neighbor, $\alpha_{1}$ and $\alpha_{2}$, whereas the next neighbor
hopping, $\beta$, is due to the coupling of waveguides in the same layer.

Figure 7: Sketch of zigzag waveguide array with nearest,
$\left(\alpha_{1},\alpha_{2}\right)$, and next-nearest,$\left(\beta\right)$,
neighbor hopping.

Further, the choice of a different geometrical setting, such as a one-
dimensional linear chain of waveguides arrays, leads to the next-nearest
interaction effects starting to become completely insignificant, $\beta=0$; as
a result, the problem is reduced to the Non-Hermitian Glauber–Fock lattice
[62, 63, 64] with a transverse ramp of refractive index [65] when $\lambda\neq
0$. Finally, for the case $\alpha_{1}=\alpha_{2}=\alpha$ and
$\beta_{1}=\beta_{2}=\beta$, the Eq (6.1) turns out to be the zigzag lattice
reported in [55].

##  7 Conclusion

We have tried to show, that by using non-unitary transformations, non-
Hermitian Hamiltonians may be produced. We have used in most of our results a
non-unitary ”squeezed operator”. However, such operator may be greatly
extended; for instance, consider the time dependent harmonic oscillator
Hamiltonian (we set the frequency equal to one for simplicity)

$H(t)=\hat{a}^{\dagger}\hat{a}+\frac{\lambda}{\sqrt{2}}(\hat{a}e^{it}+\hat{a}^{\dagger}e^{-it}),$
(44)

with the driving amplitude given by $\frac{\lambda}{\sqrt{2}}$. The simple
unitary transformation, $e^{-i\hat{a}^{\dagger}\hat{a}}$ may be used to take
such Hamiltonian to the time independent one

$H=\frac{\lambda}{\sqrt{2}}\hat{x}.$ (45)

This Hamiltonian has position as eigenstates eigenfunctions that,
unfortunately, are not normalized and therefore they are not proper
wavefunctions. We may produce the nonunitary transformation [66]
$T=e^{-{\frac{\hat{p}^{2}}{2}}}$ such that we obtain the non-Hermitian
Hamiltonian

$H_{T}=THT^{\dagger}=\frac{\lambda}{\sqrt{2}}(\hat{x}+i\hat{p})=\lambda\hat{a},$
(46)

whose eigenfunctions are coherent states, i.e., properly normalized
wavefunctions. Both Hamiltonians, $H$ and $H_{T}$ share the same eigenvalues
because they are related by a transformation, but the first has unnormalized
eigenfunctions while the last has normalized ones. The transformation, $T$,
has as argument of the exponential the operator $\hat{p}^{2}$ that form an
algebraic group, namely, $SU(2)$, with $\hat{x}^{2}$ and
$\hat{x}\hat{p}+\hat{p}\hat{x}$. This last operator we have used in Section 4
to produce a Hermitian Hamiltonian from the paradigmatic example of
$\mathcal{PT}$ symmetric infinite dimensional Hamiltonians. The fact that the
operator involved in such transformation belongs to the $SU(2)$ group, just as
the one used in this Section, to transform the position operator into the
annihilation operator makes it clear that there are no concerns over the
domain in which solutions are valid, but only the usual concerns related to
the use of non-Hermitian Hamiltonians that obviously produce not proper
wavefunctions as they do not conserve probability.
Many more nonunitary transformations may be used to produce non-Hermitian
Hamiltonians from Hermitian ones, for instance, $R_{x}=\exp(g[\hat{x}])$ or
$R_{p}=\exp(f[\hat{p}])$ where $g$ and $f$ are in general a complex functions.
Moreover, we have applied such non-unitary transformations even to the case
where an effective potential may be considered a function of position and
momentum. In particular, note that the Swanson Hamiltonian (25) is more
general than the Hamiltonian (22), in the sense that it has terms of the form
$\hat{a}^{2}$, and therefore it implies a potential that not only depends on
the position operator, $\hat{x}$, but also on the momentum operator,
$\hat{p}$. We have associated the complex time produced by such non-unitary
transformations to Wick rotations.

##  8 Acknowledgments

B.M. Villegas-Martínez wishes to express his gratitude to CONACyT as well as
to the National Institute of Astrophysics, Optics and Electronics (INAOE) for
financial support.

##  9 Author Contributions

All authors have contributed equally to this work. All authors have read and
agreed to the published version of the manuscript.

##  10 Conflicts of Interest

The authors declare no conflict of interest.

##  11 Data Availability Statement

No Data associated in the manuscript

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As outlined in Section IV, DIDs and VCs have demonstrated their versatility
and applicability across a wide array of domains for distinct purposes.
Nevertheless, upon a comprehensive review of the existing literature, it
becomes evident that there exists a notable disparity in the adoption of these
technologies across various application domains. In some domains, we find a
robust body of literature and practical implementations, reflecting a
consistent and well-established presence of DIDs and VCs. Conversely, in
certain other fields, the utilization of DIDs and VCs remains comparatively
limited, despite the potential these technologies hold to effectively address
prevailing privacy and security challenges. To bridge this gap and fully
unlock their potential, future research should also focus on under-considered
domains. In the following, we discuss some of them.

####  VII-D1 E-voting

We claim that DIDs and VCs are very promising in the development of novel
e-voting systems that securely store all the preferences without leaking any
information about the individuals [155]. In this context, DIDs are used to
uniquely identify voters, while VCs are issued by trusted entities, such as
government agencies or identity providers, confirming a voter’s eligibility
without exposing their details. DIDs and VCs, however, allow for the creation
of pseudonymous, unlinkable digital identities. Voters can cast their ballots
without revealing their real-world identity, and these votes can be securely
recorded on the blockchain, ensuring privacy. Furthermore, the use of DIDs and
VCs opens up opportunities for remote and mobile voting. Indeed, voters can
potentially participate from anywhere, reducing geographical barriers and
increasing overall voter turnout. These technologies provide a strong
foundation for end-to-end verifiable e-voting systems, ensuring that votes are
accurately counted and users can verify to have been included in the count
[156].

However, it is important to note that the adoption of DIDs and VCs in e-voting
systems also brings challenges. Standardization, trust frameworks, and robust
authentication mechanisms must be established to create a resilient and
trustworthy e-voting ecosystem.

####  VII-D2 Digital Twin

In the context of DTs, there is only a work that proposes the use of such
technologies despite their capacity to establish trustworthiness among
entities, which is a crucial concern in implementing digital replicas of
objects and environments [157]. DIDs and VCs provide a standardized and secure
way to verify the authenticity of data sources, ensuring that the data used to
create and update the digital twin is accurate and reliable. Trustworthiness
also encompasses the assurance that the data collected from the physical
entity and mirrored in the digital twin remains unaltered and secure
throughout its lifecycle. DIDs provide immutable linking of data sources to
DTs, reducing the risk of unauthorized tampering or data breaches. On the
other hand, VCs offer a mechanism to ensure that the entities providing data
to the digital twin can be authenticated, enhancing the overall security of
the system. Moreover, DTs often interact with a multitude of systems and
entities, making interoperability and cross-domain trust essential. DIDs and
VCs offer a standardized way to establish trust and data exchange across
domains.

####  VII-D3 Smart Grid

Furthermore, there are also suitable to enhance next-generation services such
as Grid-to-Go [158], which consider a scenario where a landlord of an Airbnb
has to pay the energy bill for their tenants although they may consume an
excessive amount of energy (e.g., charging electric vehicles). To avoid such a
concern, smart meters can be programmed to charge the tenant by routing
consumption measurements from the landlord’s utility to the tenant’s utility.
The use of DIDs can be utilized to provide among participants in a
decentralized way, avoiding placing trust in specific entities.

###  VII-E Security and Privacy

Although DIDs and VCs offer many security benefits especially in bolstering
privacy and providing users greater control over their data, some security
issues need to be further explored.

####  VII-E1 DID Management

Granting third parties access to manage DID documents introduces potential
vulnerabilities that must be approached with caution. One such risk is the
possibility of the DID controller impersonating the DID owner by generating a
new key pair and modifying the public key in the DID document. Authorization
and auditing mechanisms are needed to detect and prevent unauthorized
modifications. The public key associated with a DID is usually registered to a
blockchain enabling access to other nodes of the network. The compromising of
the corresponding private key is then followed by key rotation that generates
and registers a new key pair. To replace the current key pair, the DID owner
publishes the novel public key signed with the previous one [159]. Since the
moment at which the private key cannot be known in advance, to avoid malicious
attacks based on the compromised private key, entities can only update the key
pair regularly to reduce the vulnerability window. This is a primary concern
for a DID-based system since compromising the private key will lead to
complete control of the attacker over that identity. Furthermore, the
continuous generation of novel key pairs also exhibits usability issues. To
tackle such challenges, DIDs should support flexible key rotation [160]. The
security credentials of the DID can be enriched with a one-time password (OTP)
based on the hash chain that serves as nonce and second-factor authentication
for key rotation. In this way, the attacker must obtain the OTP and the
private key to steal the victim’s digital identity.

####  VII-E2 VC Revocation

Despite the growing interest in VCs, the current landscape lacks a substantial
body of research focused on developing novel and efficient approaches for
their revocation. At the time of this paper, there is only one specification
proposed, which does not hold the status of a W3C standard, nor is it on the
W3C standards track. This specification, known as ”Revocation List 2020”
[161], revolves around a bitstring-based mechanism for managing revocations.
In this arrangement, each bit within the bitstring corresponds to an index
stored in the VCs. When the bit is set to 1 the VC is revoked, otherwise it is
still valid. It is worth noting that the degree of privacy achievable through
this approach is directly linked to the size of the revocation list. A large
number of VCs issued makes it more difficult to correlate a VC to the owner.
Another favorable facet of this strategy is its potential for compression.

Given that the majority of credentials remain valid, lengthy stretches of bits
often retain identical values. This trait lends itself to compression
techniques like ZLIB [162], allowing for the reduction of the bitstring’s
size. For instance, a 16KB bitstring housing only a handful of revoked VCs
could potentially be compressed down to a mere few hundred bytes. To visualize
this, refer to Fig. 15 for an illustrative representation of Revocation List
2020 in action.

Figure 15: Revocation List 2020.

####  VII-E3 Accountability

Ensuring privacy while simultaneously adhering to existing regulations, such
as Know-Your-Customer (KYC) [163] and Anti-Money-Laundering (AML) [164], poses
a significant challenge for DIDs. DIDs are designed with privacy preservation
as a core objective, but reconciling this feature with the need to screen
users presents several complexities. Modern systems leveraging DIDs should
meet these regulations and be able to screen users of the system.
Specifically, they should be able to identify and validate credentials
associated with users. This becomes crucial for screening individuals against
various lists, such as sanctions lists, to determine if any preventative
actions, like blacklisting, are warranted.

####  VII-E4 Privacy

Research in this domain should explore novel methods and technologies that can
seamlessly integrate privacy-preserving features within DIDs while
concurrently enabling compliance with regulatory measures. This involves
developing cryptographic techniques or privacy-enhancing protocols that
empower DIDs to provide verifiable information without compromising the
sensitive details of users.

##  VIII Conclusions

Digital identification has always been a persistent concern since the first
inception of the Internet. Over the years, a multitude of solutions have been
proposed, and the contemporary landscape reflects a shift towards
decentralization, prioritizing end-user control over their data and aligning
with evolving privacy regulations. However, the concept of digital
identification is not limited to individuals, being potentially extendible to
anyone and anything, including cloud, edge, and IoT resources.

In this article, we present a comprehensive survey on DIDs and VCs, two
building blocks to authenticate and authorize entities in the modern digital
landscape, enabling secure and trustworthy communication. This survey aims to
comprehensively analyze DIDs and VCs in terms of implementations, application
domains, and regulations. Through the investigation of the available
implementations available, we provide developers with valuable insights into
the features of existing libraries, facilitating informed choices across
different projects. While the analysis of the application DIDs and VCs
demonstrates that their utility goes far beyond SSI systems. Furthermore, our
survey also explores global initiatives and projects spearheaded by
governments and political institutions, offering a tangible glimpse into the
trajectory DIDs and VCs are taking on a global scale. Finally, as these
technologies are relatively novel, we outline emerging challenges and
highlight avenues for future research, serving as a valuable starting point
for ongoing and future research endeavors in this field.

## Acknowledgments

This work was partially supported by the project SERICS (PE00000 014) under
the MUR National Recovery and Resilience Plan program funded by the European
Union - NextGenerationEU, the U.S. National Science Foundation (Award: NSF-
EAGER-2219920), and Microsoft. The views expressed by the authors in this
paper are their own, not those of the funding entities.

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|  Carlo Mazzocca received his M.Sc. and B.Sc. degrees in Computer Engineering
in 2018 and 2020, respectively, both from the University of Naples Federico
II, Italy. He is currently a Ph.D. student in Computer Science and Engineering
at the University of Bologna, Bologna, Italy. His research interests mainly
include digital identity, security mechanisms based on distributed ledger
technologies, and authentication and authorization solutions for the cloud-to-
thing continuum.
---|---
|  Abbas Acar received his MSc and Ph.D. degrees in the Department of
Electrical and Computer Engineering at Florida International University in
2019 and 2020, respectively. Before that, he received his B.S. degree in
Electrical and Electronics Engineering from Middle East Technical University
in 2015. His research interests include continuous authentication, IoT
security/privacy, and homomorphic encryption. More information can be obtained
from https: //web.eng.fiu.edu/aacar/
---|---
|  Selcuk Uluagac is an Eminent Scholar Chaired Professor and the director of
the Cyber-Physical Systems Security Lab in the School of Computing and
Information Sciences at Florida International University (FIU), Miami,
Florida, USA. Before FIU, he was a Senior Research Engineer at Georgia
Institute of Technology and at Symantec. He holds an M.S. and Ph.D. from
Georgia Tech and an M.S. from Carnegie Mellon University. He is an expert on
security and privacy topics with hundreds of scientific/creative works in
practical and applied aspects of these areas. He received the US NSF CAREER
Award (2015), the US Air Force Office of Sponsored Research’s Summer Faculty
Fellowship (2015), and the University of Padova’s (Italy) Summer Faculty
Fellowship (2016). His research in cybersecurity has been funded by numerous
government agencies and industry. He has served on the program committees of
top-tier security conferences such as ACM CCS, IEEE Security & Privacy
(“Oakland”), NDSS, and Usenix Security, inter alia. He was the General Chair
of the ACM Conference on Security and Privacy in Wireless and Mobile Networks
(ACM WiSec) in 2019. Currently, he serves on the editorial boards of IEEE
Transactions on Information Forensics and Security as Deputy Editor-inChief,
and IEEE Transactions on Mobile Computing and Elsevier Computer Networks
Journal as associate editor. More information can be obtained from
http://nweb.eng.fiu.edu/selcuk/.
---|---
|  Rebecca Montanari full professor at the University of Bologna since 2020
carries out her research in the area of information security and of the
design/development of middleware solutions for the provision of services in
mobile and IoT systems. Her research is currently focused on blockchain
technologies to support various supply chains, including agrifood,
manufacturing and fashion and on security systems for Industry 4.0.
---|---
|  Paolo Bellavista received the Ph.D. degree in computer science engineering
from the University of Bologna, Italy, in 2001. He is currently a Full
Professor with the University of Bologna. His research interests include
middleware for mobile computing, QoS management in the cloud continuum,
infrastructures for big data processing in industrial environments, and
performance optimization in wide-scale and latency-sensitive deployment
environments. He serves on the Editorial Boards of IEEE COMMUNICATIONS SURVEYS
AND TUTORIALS, IEEE TRANSACTIONS ON NETWORK AND SERVICE MANAGEMENT, IEEE
TRANSACTIONS ON SERVICES COMPUTING, ACM CSUR, ACM TIOT, and PMC (Elsevier). He
is the Scientific Coordinator of the H2020 IoTwins Project (https:
www.iotwins.eu).
---|---
|  Mauro Conti received the Ph.D. degree from the Sapienza University of Rome,
Italy, in 2009. He is a Full Professor at the University of Padua, Italy. He
is also affiliated with TU Delft and the University of Washington, Seattle.
His research in the area of Security and Privacy is also funded by companies,
including Cisco, Intel, and Huawei. He published more than 450 papers in
topmost international peer-reviewed journals and conferences. He is the
Editor-in-Chief for IEEE Transactions on Information Forensics and Security
and has been an Associate Editor for several journals, including IEEE
Communications Surveys and Tutorials, IEEE Transactions on Dependable and
Secure Computing, and IEEE Transactions on Network and Service Management. He
is a Fellow of YAE and a Senior Member of ACM.
---|---