# Non-Hermitian Floquet-Free Analytically Solvable Time Dependant Systems

Hamed Ghaemi-Dizicheh, Hamidreza Ramezani<EMAIL_ADDRESS>Department of Physics and Astronomy, University of Texas Rio Grande Valley,
Edinburg, Texas 78539, USA

###### Abstract

The non-Hermitian models, which are symmetric under parity ($P$) and time-
reversal ($T$) operators, are the cornerstone for the fabrication of new
ultra-sensitive optoelectronic devices. However, providing the gain in such
systems usually demands precise contorol of nonlinear processes, limiting
their application. In this paper, to bypass this obstacle, we introduce a
class of time-dependent non-Hermitian Hamiltonians (not necessarily Floquet)
that can describe a two-level system with temporally modulated on-site
potential and couplings. We show that implementing an appropriate non-Unitary
gauge transformation converts the original system to an effective one with a
balanced gain and loss. This will allow us to derive the evolution of states
analytically. Our proposed class of Hamiltonians can be employed in different
platforms such as electronic circuits, acoustics, and photonics to design
structures with hidden $PT$-symmetry potentially without imaginary onsite
amplification and absorption mechanism to obtain an exceptional point.

##  I Introduction

The non-Hermitian models in physics have become a vast research area in recent
years, including condensed matter physics [1], photonics [2, 3], biophysics
[4], and acoustic [5]. Among them, the non-Hermitian generalization of
topological tight-binding systems introduces new phenomena absent in Hermitian
ones, such as non-Hermitian skin effect [6], distinct transport effects [7],
relocation of topological edge states [8], and noise-resilient [9], to name a
few.
A well-known approach to making a tight-binding model non-Hermitian is
introducing complex onsite potential, playing the role of gain (loss) in
photonic models [10]. One can also stagger inter (intra) cell hopping
amplitudes and develop non-reciprocity through the system [11, 6, 12].
Among non-Hermitian hamiltonians with gain and loss, the ones with balanced
amplification and absorption are known as parity time-reversal ($PT$)
symmetric hamiltonians and attracted more intention. $PT$-symmetric systems
have been studied in both continuous and discrete models [13, 14] and
extensively investigated theoretically and experimentally in photonic lattices
[15, 16, 17, 18], microring resonator [19, 20, 21], and electrical circuit
[22, 23, 24].
In $PT$-symmetric non-Hermitian systems, specific singular points are known as
exceptional points (EPs) where at least two or more eigenvalues and
corresponding eigenvectors of the hamiltonians coalesce. In recent years, the
EPs have led to many interesting phenomena observed in different physical
systems [25, 26, 27, 28, 29, 30].
Indeed, EPs separate the broken $PT$-symmetric regions, in which the
eigenvalues of the non-Hermitian Hamiltonian become complex, from the unbroken
$PT$-symmetric phase with real-valued eigenvalues. Many nontrivial features
such as unidirectional invisibility [31], invisible sensor [32], single-mode
lasing [33], sensitive readout [34] and robust wireless power transfer [35]
can be investigated by creating and developing $PT$-symmetric structure.
However, from a practical point of view, constructing delicate balanced gain
and loss is a rigorous requirement that hinders the possibility of
implementing $PT$-symmetry and studying its interesting features in
specifically quantum systems.
In addition, fundamental obstacles such as gain-induced noise and
instabilities arise motivation to look for a different route of achieving
$PT$-symmetry without gain. The gain-free models with a hidden $PT$-symmetry
capture various research interests [36, 37, 38, 39, 40, 19, 41].
The key approaches to design a system benefit hidden $PT$-symmetry, are based
on nonlinear phenomena [42] or by making use of a temporally modulated
coupling. In the latter method, a unitary gauge transformation leads to an
effective system whose Hamiltonian is invariant under parity and time
reversal. A well-known platform to explore such a gauge transformation is an
electronic system. For example, a transient $PT$-symmetry can be triggered by
the switching on and off of electronic devices [43].
A larger class of Hamiltonian with hidden $PT$-symmetry can be developed and
studied by releasing the unitarity condition on the transformation to reach
the effective Hamiltonian. This approach shapes the main idea of this paper.
In this study, we introduce a general class of time-dependent two-level non-
Hermitian Hamiltonians with a hidden $PT$-symmetry which can be revealed by
implementing a non-unitary gauge transformation. The main aspect of this
approach is its generality. Indeed, our proposed Hamiltonian is not
necessarily floquet which makes it applicable to the vast range of time-
dependent two-level systems. In addition, we formulated our two-level
Hamiltonian with an arbitrary onsite potential. In the case of onsite
potential with zero imaginary part, the system can be transformed into the
effective one with balanced gain and loss. Therefore our proposed method can
bypass the usual difficulties in realizing amplification and absorption
mechanism. The two-level Hamiltonian investigated in this letter can model
acoustics, electronics, and photonics systems whose couplings are time or
coordinate dependant.
Besides, we show that the analysis of state evolution described by this class
of Hamiltonian can be analytically obtained regardless of adiabatic condition.
The exact solutions are helpful to predict and manipulate the system’s
behavior precisely when it encircles EPs [44].
We organize the paper as follows. In section II, we introduce the Hamiltonian,
non-unitary time dependant transformation and manifest the $PT$-symmetry in
the model. In section III, we apply our method to an electronic system to
illustrate different $PT$-symmetric phases. The system’s time evolution is
studied analytically in the appendix.

##  II Time-dependant model

Figure 1: A two-level model which is described by the Hamiltonian 3 transforms
to a model with balanced gain and loss by a time-dependent transformation
$A(t)$. This system can describe two optical resonators coupled to each other
nonreciprocally in which their coupling is modulated in time. In this model,
the resonator 1 or 2 provides dynamical gain (loss) for ${\rm
Im}(\omega_{1,2})<0$ ( ${\rm Im}(\omega_{1,2})>0$).

Let us consider a class of two-level systems described by the following time-
dependent Hamiltonian

$\displaystyle H(t)=\left(\begin{array}[]{cc}\omega_{1}(t)&\dfrac{\nu
f_{1}(t)}{f_{2}(t)}\\\
\dfrac{\nu^{\prime}f_{2}(t)}{f_{1}(t)}&\omega_{2}(t)\end{array}\right),$ (3)

where $\nu,\nu^{\prime}$ are real-valued constants representing time-
independent coupling, $\omega_{1}(t)$ and $\omega_{2}(t)$ are complex-valued
time-dependent onsite potentials, and we introduce nonzero finite-range timely
modulated couplings $f_{1,2}(t)$ which are generally linearly independent and
in an electronic system they represents time-dependent capacitor and
inductance [43]. In the following, we will show how this particular type of
coupling, combined with an appropriate non-unitary gauge transformation,
results in an effective Hamiltonian with balanced gain and loss. The time-
dependent Schrödinger equation reads

$i\partial_{t}\Psi(t)=H(t)\Psi(t),$ (4)

where $\Psi(t)=(\psi_{1}(t),\psi_{2}(t))^{T}$ is the state vector such that
for a given initial value $\Psi(t_{0})$, the dynamical evolution of modes
$\psi_{1,2}$ is given via linear differential equation (4).
We introduce time-dependent transformation $\Psi(t)=A(t)\chi(t)$, which
converts the Hamiltonian (3) into the effective Hamiltonian

$H\rightarrow H_{\text{eff}}(t):=A^{-1}HA-i\partial_{t}\ln A,$ (5)

where $A$ is an invertible ($\det A\neq 0$) matrix. For a closed two-level
system with a reciprocal coupling between their sites, the Hamiltonian is
given by a Hermitian matrix (i.e., $H=H^{\dagger}$). For such a system, the
transformation $A$ needs to be given by a unitary matrix to conserve the norm
of state (i.e., $\|\Psi(t)\|=\|\chi(t)\|$), in this case, the two terms on the
right side of (5) are also Hermitian. However, in our study, we do not
restrict our approach to the similarity condition and consider a general time-
dependent transformation. Specifically, by choosing a suitable form of time-
dependent transformation $A(t)$, one can reduce the Hamiltonian $H(t)$ to a
traceless matrix (i.e., Tr($H_{\text{eff}})=0$) which describes the effective
system with balanced gain and loss. This can be achieved by

$\displaystyle
A(t)=\exp\left(-i\int^{t}\Omega_{-}(t^{\prime})dt^{\prime}\right)\left(\begin{array}[]{cc}\sqrt{\frac{f_{1}(t)}{f_{2}(t)}}&0\\\
0&\sqrt{\frac{f_{2}(t)}{f_{1}(t)}}\end{array}\right),$ (8)

where, $\Omega_{\pm}(t):=(\omega_{1}(t)\pm\omega_{2}(t))/2$. By substituting
$A(t)$ into (5), the off-diagonal terms become time-independent, and the
effective Hamiltonian turns to

$\displaystyle
H_{\text{eff}}(t)=-i\Gamma\sigma_{z}+\nu\sigma_{+}+\nu^{\prime}\sigma_{-},$
(9)

where

$\Gamma(t):=i\Omega_{-}+\frac{1}{2}\partial_{t}\ln\frac{f_{1}}{f_{2}},$ (10)

is an complex-valued effective onsite potential and
$\sigma_{\pm}:(\sigma_{x}\pm i\sigma_{y})/2$ are given in terms of Pauli
matrices $\sigma_{x,y,z}$ (see Fig. 1 for the schematic of the
transformation). For the effective Hamiltonian (9), Schrödinger equation then
reads as

$i\partial_{t}\chi(t)=H_{\text{eff}}(t)\chi(t),$ (11)

where the transformed vector state is now introduced with
$\chi(t)=(\zeta_{-}(t),\zeta_{+}(t))^{T}$.
By choosing a specific form of time transformation $A$ given by (8), the
Schrödinger equation (11) can be recasted into the following second-order
differential equation for $\zeta_{\mp}$

$\dfrac{d^{2}\zeta_{\mp}}{d\tau^{2}}\pm\left[\partial_{\tau}\Gamma(\tau)\mp
i\Gamma(\tau)^{2}\pm 1\right]\zeta_{\mp}=0.$ (12)

In driving the above differential equations, we scaled variables such as

$\left(\dfrac{\Gamma(t)}{\sqrt{\nu^{\prime}\nu}},\sqrt{\nu^{\prime}\nu}t\right)\rightarrow(\Gamma(\tau),\tau).$
(13)

If we suppose $\omega_{2}(t)=0$, then, the vector state $\Psi(t)$ is given by

$(\psi_{1},\psi_{2})=e^{\int^{\tau}\Gamma(\tau^{\prime})d\tau^{\prime}}(f_{1}f_{2}^{-1}\zeta_{-},\zeta_{+}).$
(14)

For the nonzero $\omega_{2}(t)\neq 0$, the evolution of $\psi_{1,2}$ are given
by the above equation by replacing
$\Gamma(\tau)\rightarrow\omega_{2}(\tau)-\Gamma(\tau)$.
In the appendix, by considering three different forms for the effective onsite
potential $\Gamma(t)$, we show that the evolution of the system governed by
(11) can be obtained analytically. From there, implementation of an inverse
time transformation leads to the dynamics of the original system given by (3).
To conclude the section, we are looking through the eigenbasis and eigenvalues
of the original Hamiltonian (3) and its effective counterpart (9). For the
effective system, the time-dependent eigenvalues are given by

$\bar{\lambda}_{\mp}=\mp\sqrt{\nu\nu^{\prime}-\Gamma^{2}(t)}.$ (15)

In the case of static $PT$-symmetric where $\Gamma$ is time independent and
$\nu=\nu^{\prime}$, the system is in $PT$ unbroken (broken) phase for
$|\nu|>\Gamma$ ($|\nu|<\Gamma$), where $|\nu|=\Gamma$ is an exceptional point.
In the case of the periodic Hamiltonian, then the $PT$ (un)broken phase can be
found by looking at the exceptional points of the Floquet Hamiltonian.
For the general time-dependant $\Gamma(t)$, as a consequence of non-Hermicity,
the eigenbasis of the system is not orthogonal, but one can define
biorthogonal eigenbasis by using its right and left eigenvectors which are
defined via

$\displaystyle
H_{\text{eff}}\chi_{\mp}^{r}=\bar{\lambda}_{\mp}\chi_{\mp}^{r},$
$\displaystyle
H_{\text{eff}}^{T}\chi_{\mp}^{l}=\bar{\lambda}_{\mp}\chi_{\mp}^{l},$ (16)

where $T$ stands for the transpose of the matrix. However, for the reciprocal
system where $\nu=\nu^{\prime}$ and the invariant effective hamiltonian under
transpose (i.e., $H_{\text{eff}}^{T}=H_{\text{eff}}$), one can find the
similar right and left eigenbasis and construct a biorthogonal pair just by
picking up one of them. In this case, then the eigenbasis is given as [45]

$\displaystyle\chi_{-}^{r}=\chi_{-}^{l}=(-\sin\frac{\bar{g}}{2},\cos\frac{\bar{g}}{2})^{T},$
$\displaystyle\chi_{+}^{r}=\chi_{+}^{l}=(\cos\frac{\bar{g}}{2},\sin\frac{\bar{g}}{2})^{T},$
(17)

with $\bar{g}$ such that $\cot\bar{g}=i\Gamma/\nu$. On the other side, for the
original Hamiltonian, the eigenvalues are

$\lambda_{\mp}=\Omega_{+}\mp\sqrt{\Omega_{-}^{2}+\nu\nu^{\prime}},$ (18)

and the right eigenbasis is given by

$\displaystyle\Psi_{-}^{r}=(-f_{1}(t)\sin\frac{g}{2},f_{2}(t)\cos\frac{g}{2})^{T},$
$\displaystyle\Psi_{+}^{r}=(f_{1}(t)\cos\frac{g}{2},f_{2}(t)\sin\frac{g}{2})^{T},$
(19)

with $g$ given by $\cot g=\Omega_{-}/\nu$. The relation between $g$ and
$\bar{g}$ can be found such as

$\cot(\bar{g})=\cot(g)-\frac{iW(f_{1}f_{2})}{2\nu f_{1}f_{2}},$ (20)

where $W(f_{1}f_{2})$ is the Wronskian of two differentiable functions $f_{1}$
and $f_{2}$. From Eq. (20), one can find $g$ in terms of $\bar{g}$ from

$\displaystyle
g=i\coth^{-1}(\frac{1}{2\nu}\partial_{t}\ln\frac{f_{1}}{f_{2}}+i\cot\bar{g})+\pi
k,$ $\displaystyle k\in\mathbb{Z}.$ (21)

For the original system, if $\Omega_{-}=\pm i\nu$, it reaches exceptional
points (EPs). By following the same calculations for the effective system
given by the Hamiltonian $H_{\text{eff}}$, the EPs exist if

$\Omega_{-}=\mp i\nu+\frac{i}{2}\partial_{t}\ln(f_{1}f_{2}).$ (22)

To study the dynamic of EPs under time-dependent transformation, we
furthermore define the following complex parameter

$B(t):=\Omega_{-}(t)/\nu=\cot g.$ (23)

In regarding the original Hamiltonian (3), the complex parameter $B$ takes a
complex-valued number in the parameter space (${\rm Re}(B)$-${\rm Im}(B)$),
which are

$B^{P}=\pm i.$ (24)

Under the transformation $A(t)$, the loci of EPs for the effective system in
its parameter space (${\rm Re}(\bar{B})$-${\rm Im}(\bar{B})$) are given by
$\bar{B}^{P}=\pm i$ where

$\bar{B}(t):=\dfrac{i\Gamma(t)}{\nu}=\cot\bar{g}.$ (25)

Tracking the loci of EPs for the effective system in the $B$-parameter space
shows that they are no longer constant and given in time as

$\bar{B}^{P}=B^{P}-ib(t),$ (26)

where
$b(t):=\frac{\partial_{t}\ln\frac{f_{1}}{f_{2}}}{2\nu}=b_{r}(t)+ib_{i}(t)$. To
derive (26), we put $\cot g=\cot\bar{g}=i$ in Eq. (20). It shows that, in the
$B$-parameter space, the real (imaginary) coordinate at $t=t_{1}$ of EPs shift
by the factor of $-b_{i}(t_{1})$($b_{r}(t_{1})$).
In the following section, by making use of an electronic platform, we
illustrate the hidden $PT$-symmetric in the system whose dynamics are given by
the Hamiltonian (3).

##  III Application: $PT$ symmetry in an electronic system

We implement our approach to design a $PT$-symmetric model using an electronic
platform consisting of an RLC circuit. By following the Kirchhoff’s laws, one
can find the coupled equation for the voltage in capacitor $V(t)$ and the
current across the inductor $I(t)$:

$\displaystyle\partial_{t}V(t)=-\frac{I(t)}{C},$ (27)
$\displaystyle\partial_{t}I(t)=\frac{V(t)}{L}+\frac{R}{L}I(t).$ (28)

By taking $\Psi(t)=(V(t),I(t))^{T}$, the dynamic of $RLC$ circuit can be given
with $H(t)$

$\displaystyle H(t)=i\left(\begin{array}[]{cc}0&-1/C\\\
1/L&R/L\end{array}\right).$ (31)

For a time-dependent resistance $R(t)$, capacitance $C(t)$ and inductance
$L(t)$, the above Hamiltonian-like matrix can be written in the form of the
Hamiltonian (3) if we set

$\displaystyle L(t)=L_{0}f(t),$ $\displaystyle C(t)=C_{0}f^{-1}(t),$
$\displaystyle R(t)=g(t).$ (32)

Figure 2: Time evolution of the circuit’s energy for LC circuit (a,b) and RLC
circuit (c,d). In $\gamma<\omega_{0}$ ($\gamma<2\omega_{0}$) regime, the
system is in the unbroken $PT$-symmetric phase, and the energy oscillates.
This oscillation is depicted in plots (a) and (c), where, in the latter, the
resistance causes the damping oscillation. In the broken phase, the energy
amplifies for both electronic systems.

Then in terms of our denotations, we have

$\displaystyle f_{1}(t):=f(t),~{}~{}~{}f_{2}(t):=1,$ (33)
$\displaystyle\omega_{1}(t):=0,~{}~{}~{}~{}~{}~{}\omega_{2}(t)=ig(t)/L_{0}f(t),$
(34) $\displaystyle\nu:=-i/C_{0},~{}~{}~{}~{}\nu^{\prime}:=i/L_{0}.$ (35)

If we modulate the resistance such that $g(t)=L_{0}\partial_{t}f$, then the
effective Hamiltonian (9) reduces to

$H_{\text{eff}}=i\partial_{t}\ln f\sigma_{z}+\omega_{0}\sigma_{y},$ (36)

where $\omega_{0}=1/\sqrt{L_{0}C_{0}}$. To get the above relation, we modified
the time transformation (8) such that
$A(t)\rightarrow\text{diag}(C_{0}^{-1/2},L_{0}^{-1/2})A(t)$ [46]. We first
study the time-independent, effective Hamiltonian corresponding to case (a) in
the appendix. This can be achieved by setting $f(t)=e^{-\gamma t}$. Then, the
electronic $RLC$ circuit stimulate a static $PT$-symmetric system whose
Hamiltonian is given by
$H_{\text{eff}}^{PT}=i\gamma\sigma_{z}+\omega_{0}\sigma_{y}$. The eigenvalues
of this Hamiltonian are given by
$\lambda_{\pm}=\pm\sqrt{\omega_{0}^{2}-\gamma^{2}}$. From these eigenvalues,
we can find the $PT$-symmetry breaking threshold where the system is referred
to as being in a $PT$-broken phase for $\gamma>\omega_{0}$. Fig. 2
demonstrates the time evolution of the circuit’s energy $U_{L}+U_{C}$ for both
LC ($R(t)=0$) and RLC circuit where

$\displaystyle U_{L}:=\frac{1}{2}L(t)I^{2}(t),$ $\displaystyle
U_{C}:=\frac{1}{2}C(t)V^{2}(t).$ (37)

In the regime $\gamma<\omega_{0}$ ($\gamma<2\omega_{0}$), which defines
unbroken $PT$-symmetric phase, one can see from the diagram that the energy of
the RLC (LC) system oscillates while in the broken phase, the energy of both
systems amplifies. Note that in the RLC system, the oscillation is damping by
the factor of $e^{-R/2L}$ because of the resistance.
In the second step, we consider a periodic time-dependent
$f(t)=\epsilon_{1}\cos(\Omega_{0}t)+\epsilon_{2}$ with the period
$T=2\pi/\Omega_{0}$. In this case, the effective Hamiltonian becomes periodic,
and therefore its dynamics are determined by the one-period, non-unitary time
evolution operator obtained from the following time-ordered product [47],

$U(T)=\mathbb{T}e^{-i\int_{0}^{T}H_{\text{eff}}(\tau^{\prime})d\tau^{\prime}}\equiv
e^{-iTH_{F}},$ (38)

where the time-independent operator $H_{F}$ is the so-called Floquet
Hamiltonian in which their eigenvalues and eigenstates are called quasi-
energies, and quasi-states [48]. The corresponding Floquet Hamiltonian $H_{F}$
can determine the $PT$ broken and unbroken phases of the system. To show this,
we consider the eigenvalues of Floquet Hamiltonian $\varepsilon_{1,2}$ and the
time evolution $\lambda_{1,2}$ that related to each other through
$\lambda_{1,2}=e^{-iT\varepsilon_{1,2}}$. The quasi-energies
$\varepsilon_{1,2}$ take real value in the phase where the system is invariant
under parity and time-reversal transformation. In this phase, the eigenvalues
of the evolution operator $U_{F}$ satisfies $|\lambda_{1}|$=$|\lambda_{2}|=1$
(see Fig. 3). The energy of the time-dependent RLC circuit oscillates in this
phase while it is damping when the Floquet Hamiltonian enters the broken $PT$
phase.
We note that for time-dependent coupling in the form of the step function, the
$PT$ (un)broken phases have been studied in [49] for the LC circuit. However,
our approach can encompass a general class of time-dependent systems with
hidden $PT$-symmetric phase.

Figure 3: The magnitude of the eigenvalues $\lambda_{k}$ belong to the time
evolution operator $U_{F}$ versus $\Omega_{0}$. Orange and blue regions
represent the $PT$ unbroken and broken phases of the system, respectively.

##  IV Conclusion

In summary, in this paper, we propose a class of time-dependent Hamiltonian
describing a two-level system with the hidden $PT$-symmetry. The hidden
symmetric can be revealed by implementing a specific non-Unitary gauge
transformation which transforms the original Hamiltonian with temporally
modulated couplings and onsite potentials to an effective one with balanced
gain and loss. The dynamics of the effective system can be analytically
extracted. Time-dependent systems are not usually analytically solvable. Thus,
using our method we can propose a class of systems that one can analytically
drive the condition for the existence of exceptional points and thus use it
for determining other applications of exceptional points including sensing.
The approach of this paper can provide a protocol to design and develop a
different type of system with a $PT$-symmetric characteristic without
encountering the obstacles of injecting gain. We demonstrated how our method
can be applied to electronic devices. However, photonics models may have a
potential application. The main experimental challenge in developing a
photonics application is designing a system with time-varying couplings and
onsite potentials. Microwave Smoothly deformed metallic waveguides in the
microwave and helical waveguides in optics have the potential to be used in
the design of driven photonics systems. In such a system, the deformed
waveguide is extended along the $x$-axis, allowing us to achieve the spatially
modulated couplings [50, 51], where $x$ now represents time in our approach.

###### Acknowledgements.

H. R. acknowledge the support by the Army Research Office Grant No.
W911NF-20-1-0276 and NSF Grant No. PHY-2012172 and OMA-2231387.

##  Appendix A Appendix: Analytic Solution

In this appendix, we study the analytic solutions of (9) by considering
different forms for $\Gamma(t)$.
Case (a). We choose the effective potential such that the original Hamiltonian
(3) transforms to the time-independent effective Hamiltonian. It corresponds
to set $\Gamma(\tau)=\gamma$, where $\gamma\in\mathbb{C}$ is a real-valued
constant. Since, in general, the time-dependent transformation $A(t)$ in (8)
is a non-similarity transformation, then the eigenvalues of the system do not
remain unchanged. Regarding this, we can design a system with linearly, time-
modulated Hamiltonian such as $H(t)$, which effectively acts as a time-
independent system whose eigenenergies are constant. One practical way to
apply this protocol is to make use of an electronic platform. Regarding this
choice of effective on-site potential, from differential equation (12), one
can find the following solution for $\zeta_{\mp}$:

$\zeta_{\mp}=c_{1}^{(\mp)}e^{i\bar{\gamma}\tau}+c_{2}^{(\mp)}e^{-i\bar{\gamma}\tau},$
(39)

where $\bar{\gamma}:=\sqrt{1-\gamma^{2}}$, and $c_{j}^{(\mp)}$ are constants
given by the initial values of $\zeta_{-}(0):=a$ and $\zeta_{+}(0):=b$ such as

$\displaystyle c_{1}^{(-)}=\dfrac{\bar{\gamma}(1+i)a+i\nu
b}{2\bar{\gamma}},~{}~{}c_{2}^{(-)}=\dfrac{\bar{\gamma}(1-i)a-i\nu
b}{2\bar{\gamma}},$ $\displaystyle
c_{1}^{(+)}=\dfrac{i\nu^{\prime}a+\bar{\gamma}(1-i)b}{2\bar{\gamma}},~{}~{}c_{2}^{(+)}=\dfrac{-i\nu^{\prime}a+\bar{\gamma}(1+i)b}{2\bar{\gamma}}.$

If $\omega_{2}(t)=0$, from (14), the vector state $\Psi(\tau)$ is given by

$\Psi(\tau)=e^{-\gamma\tau}(f_{1}f_{2}^{-1}\zeta_{-},\zeta_{+})^{T}.$ (41)

Now let $\gamma:=(\gamma_{1}+\gamma_{2})/2\sqrt{\nu\nu^{\prime}}$, then in
light of the definition of $\Omega(\tau)$ and $\Gamma(\tau)$, one can find the
relation between the one-site potential $\omega_{j}(t)$ and coupling
$f_{j}(t)$ such that:

$\displaystyle\omega_{1}(t)-\dfrac{if^{\prime}_{1}(t)}{f_{1}(t)}=-i\gamma_{1},$
$\displaystyle\omega_{2}(t)-\dfrac{if^{\prime}_{2}(t)}{f_{2}(t)}=-i\gamma_{2}.$
(42)

These lead to the following equation

$\dfrac{f_{1}(t)}{f_{2}(t)}=f_{0}u(t)e^{i\int[(\omega_{1}(t)-\omega_{2}(t))]dt},$
(43)

where $f_{0}:=\dfrac{f_{1}(0)}{f_{2}(0)}$ and
$u(t):=e^{2(\gamma_{1}-\gamma_{2})t}$.

Figure 4: The quasienergies and the mode density for the toy model given with
the parameter given in relations (44) and (45) in the unbroken (a) and broken
(b) $PT$ phase.

This shows that, for a given $f_{j}(t)$, we can find a corresponding on-site
potential $\omega_{j}(t)$ in which the system transforms to the time-
independent model. Besides, for the given time-dependant $\omega_{j}(t)$, Eq.
(43) gives the corresponding time-dependent couplings in which we can
transform the system to a time-independent one. One can find this protocol for
designing a canonical $PT$-symmetric system.
To show this, we consider a toy model that can describe a couple two
waveguides whose dynamic is given by Hamiltonian (3) with the following
functions

$\displaystyle f_{1}(t)=\sin(\omega
t)+\epsilon_{1},~{}~{}~{}~{}~{}~{}f_{2}(t)=\cos(\omega t)+\epsilon_{2},$ (44)
$\displaystyle\omega_{1}=\dfrac{i\omega\cos(\omega t)}{\sin(\omega
t)+\epsilon_{1}}+i\gamma,~{}~{}\omega_{2}=-\dfrac{i\omega\sin(\omega
t)}{\cos(\omega t)+\epsilon_{2}}-i\gamma,$ (45)

where $\gamma$, $\epsilon_{1}$, $\epsilon_{2}$ and $\omega$ are real-valued
constant and, we also suppose $\nu=\nu^{\prime}$. By making use of
transformation $A(t)$, the system transforms to the effective one with static
$PT$-symmetric Hamiltonian located in unbroken phase for $\nu>\gamma$ and
broken phase for $\nu<\gamma$. These phases can be observed by looking at the
quasienergies of the $H(t)$ and mode profile in unbroken and broken phases. In
fig. 4, we demonstrated the phases of the system.
Case (b). In the second scenario, we choose $\Gamma(\tau)$ such that the term
inside bracket in Eq. (12) vanishes for each effective vector state
$\zeta_{\mp}$ corresponding to stationary solution. It leads to the following
choice

$\Gamma^{(\mp)}=\mp\tanh(\tau).$ (46)

One can show that for effective on-site potential $\Gamma^{(-)}$, the vector
states $\zeta_{\pm}$ are given by

$\displaystyle\zeta_{-}=c_{1}^{(-)}\tau+c_{2}^{(-)},$
$\displaystyle\zeta_{+}=c_{1}^{(+)}\tanh\tau+c_{2}^{(+)}(\tau\tanh\tau-1),$
(47)

where in this case, the constants of the system are

$\displaystyle c_{1}^{(-)}=-i\nu b,~{}~{}~{}~{}~{}~{}~{}c_{2}^{(-)}=-a,$
$\displaystyle c_{1}^{(+)}=-i\nu^{\prime}a+b,~{}c_{2}^{(+)}=-b.$ (48)

For $\Gamma^{(+)}$, the vector states are same as (47) where the constants
given by (48) with $a\leftrightarrow b$ and $\nu\leftrightarrow\nu^{\prime}$.
If $\omega_{2}(\tau)=0$, then for the original vector state we find

$\Psi(\tau)_{\mp}=(\cosh\tau)^{\pm
1}(f_{1}f_{2}^{-1}\zeta_{-},\zeta_{+})^{T}.$ (49)

Case (c). The following relation gives the third model we study here for
effective on-site potential

$\Gamma(\tau)=\alpha e^{i\gamma\tau}-\beta.$ (50)

By introducing $\zeta_{\mp}:=e^{-\eta_{\mp}/2}W_{\mp}(\eta_{\mp})$, the
equation (12) transforms to

$\eta_{\mp}^{2}\dfrac{d^{2}W_{\mp}}{d\eta_{\mp}^{2}}+\eta_{\mp}(1-\eta_{\mp})\dfrac{dW_{\mp}}{d\eta_{\mp}}\pm\left[\dfrac{\alpha_{1}^{-1}+\alpha_{2}}{\gamma}\mp\frac{\alpha_{2}^{2}}{\gamma^{2}}\right]W_{\mp}=0,$
(51)

where

$\displaystyle\eta_{\mp}:=\mp\dfrac{2i\alpha}{\gamma}e^{i\gamma\tau},$
$\displaystyle\alpha_{1}:=(-i\beta+\sqrt{1-\beta^{2}})^{-1},$ (52)
$\displaystyle\alpha_{2}:=\sqrt{1-\beta^{2}}.$ (53)

From here, one can obtain the general solution for $W_{\mp}$ such that

$\displaystyle
W_{-}=\eta_{-}^{\alpha_{2}/\gamma}[c_{1}^{(-)}U(\dfrac{\alpha_{1}}{\gamma},1+\dfrac{2\alpha_{2}}{\gamma},\eta_{-})+$
$\displaystyle~{}~{}~{}~{}~{}~{}c_{2}^{(-)}L(\dfrac{\alpha_{1}^{-1}+2\alpha_{2}}{\gamma},\dfrac{2\alpha_{2}}{\gamma},\eta_{-})],$
(54) $\displaystyle
W_{+}=\eta_{+}^{\alpha_{2}/\gamma}[c_{1}^{(+)}U(\dfrac{\alpha_{1}^{-1}}{\gamma},1+\dfrac{2\alpha_{2}}{\gamma},\eta_{+})+$
$\displaystyle~{}~{}~{}~{}~{}~{}c_{2}^{(+)}L(-\dfrac{\alpha_{1}^{-1}}{\gamma},\dfrac{2\alpha_{2}}{\gamma},\eta_{+})],$
(55)

where $U$ represents confluent hypergeometric functions of the second kind,
and $L$ stands for Laguerre polynomials which satisfy the following properties
[52]

$\displaystyle\frac{d}{dz}U(m_{1},m_{2},z)=-m_{1}U(m_{1}+1,m_{2}+2,z),$ (56)
$\displaystyle\frac{d}{dz}L(m_{3},m_{2},z)=-m_{3}L(m_{3}-1,m_{2}+1,z),$ (57)
$\displaystyle
L(m_{3},m_{2},z)=\frac{(m_{2}+1)_{m_{3}}}{m_{1}!}F_{1}(-m_{3},m_{2}+1,z).$
(58)

In the third relation, $(m_{2}+1)_{m_{1}}$ is Pochhammer symbol, and $F_{1}$
represents confluent hypergeometric functions of the first kind. The
coefficients $c_{1,2}^{({\mp})}$ depend on initial conditions. By following
the method given in [44], we can obtain constants by using the Transfer Matrix
method

$\left(\begin{array}[]{cc}c_{1}^{(-)}\\\
c_{2}^{(-)}\end{array}\right)=z_{0}^{-m}e^{z_{0}}\left[\mathbf{M}_{0}-z_{1}\mathbf{M}_{1}\right]^{-1}\left(\begin{array}[]{cc}a\\\
b\end{array}\right),$ (59)

where the transfer matrix $\mathbf{M}_{j}$ is obtained as

$\displaystyle\mathbf{M}_{0}=\left(\begin{array}[]{cc}\mathbf{U}^{(0)}&\mathbf{L}^{(0)}\\\
z_{3}\mathbf{U}^{(0)}&z_{3}\mathbf{L}^{(0)}\end{array}\right),$
$\displaystyle\mathbf{M}_{2}=\left(\begin{array}[]{cc}0&0\\\
\mathbf{U}^{(1)}&\mathbf{L}^{(1)}\end{array}\right).$ (64)

In the above relations, we introduce

$\displaystyle z_{0}:=\dfrac{-2i\alpha}{\gamma},$ $\displaystyle
z_{1}:=2\alpha,$ (66) $\displaystyle
z_{3}:=(\dfrac{z_{1}m-z_{1}z_{0}}{2z_{0}}-i\beta),$ $\displaystyle
m:=\dfrac{\alpha_{2}}{\gamma},$ (67)

and in the entries of transfer matrices, $\mathbf{U}$ and $\mathbf{L}$ stands
for

$\displaystyle\mathbf{U}^{(j)}:=U(m_{1}+j,m_{2}+j,z=0),$ (68)
$\displaystyle\mathbf{L}^{(j)}:=L(m_{3}-j,m_{2}+j,z=0).$ (69)

For $\alpha=-\rho$ and $\beta=-1$, the exact solution is reduced to the one
given in [44].

## References

  * Rui  _et al._ [2019] W. B. Rui, M. M. Hirschmann, and A. P. Schnyder, $\mathcal{PT}$-symmetric non-hermitian dirac semimetals, Phys. Rev. B 100, 245116 (2019).
  * Longhi [2010] S. Longhi, Spectral singularities and bragg scattering in complex crystals, Phys. Rev. A 81, 022102 (2010).
  * Ding  _et al._ [2015] K. Ding, Z. Q. Zhang, and C. T. Chan, Coalescence of exceptional points and phase diagrams for one-dimensional $\mathcal{P}\mathcal{T}$-symmetric photonic crystals, Phys. Rev. B 92, 235310 (2015).
  * Chou  _et al._ [2011] T. Chou, K. Mallick, and R. Zia, Non-equilibrium statistical mechanics: from a paradigmatic model to biological transport, Reports on progress in physics 74, 116601 (2011).
  * Zhu  _et al._ [2014] X. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, $\mathcal{P}\mathcal{T}$-symmetric acoustics, Phys. Rev. X 4, 031042 (2014).
  * Yao and Wang [2018] S. Yao and Z. Wang, Edge states and topological invariants of non-hermitian systems, Phys. Rev. Lett. 121, 086803 (2018).
  * Ghaemi-Dizicheh and Schomerus [2021] H. Ghaemi-Dizicheh and H. Schomerus, Compatibility of transport effects in non-hermitian nonreciprocal systems, Phys. Rev. A 104, 023515 (2021).
  * Ramezani [2022] H. Ramezani, Anomalous relocation of topological states, arXiv preprint arXiv:2207.12193 (2022).
  * Tuxbury  _et al._ [2022] W. Tuxbury, R. Kononchuk, and T. Kottos, Non-resonant exceptional points as enablers of noise-resilient sensors, Communications Physics 5, 210 (2022).
  * Schomerus [2013] H. Schomerus, Topologically protected midgap states in complex photonic lattices, Opt. Lett. 38, 1912 (2013).
  * Lee [2016] T. E. Lee, Anomalous edge state in a non-hermitian lattice, Phys. Rev. Lett. 116, 133903 (2016).
  * Lieu [2018] S. Lieu, Topological phases in the non-hermitian su-schrieffer-heeger model, Phys. Rev. B 97, 045106 (2018).
  * Feng  _et al._ [2017] L. Feng, R. El-Ganainy, and L. Ge, Non-hermitian photonics based on parity–time symmetry, Nature Photonics 11, 752 (2017).
  * El-Ganainy  _et al._ [2018] R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, Non-hermitian physics and pt symmetry, Nature Physics 14, 11 (2018).
  * El-Ganainy  _et al._ [2007] R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, Theory of coupled optical pt-symmetric structures, Opt. Lett. 32, 2632 (2007).
  * Makris  _et al._ [2008] K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Beam dynamics in $\mathcal{P}\mathcal{T}$ symmetric optical lattices, Phys. Rev. Lett. 100, 103904 (2008).
  * Rüter  _et al._ [2010] C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Observation of parity–time symmetry in optics, Nature physics 6, 192 (2010).
  * Regensburger  _et al._ [2012] A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Parity–time synthetic photonic lattices, Nature 488, 167 (2012).
  * Peng  _et al._ [2014a] B. Peng, Ş. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. Bender, F. Nori, and L. Yang, Loss-induced suppression and revival of lasing, Science 346, 328 (2014a).
  * Hodaei  _et al._ [2014] H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, Parity-time–symmetric microring lasers, Science 346, 975 (2014).
  * Peng  _et al._ [2014b] B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, Parity–time-symmetric whispering-gallery microcavities, Nature Physics 10, 394 (2014b).
  * Schindler  _et al._ [2011] J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, Experimental study of active lrc circuits with $\mathcal{PT}$ symmetries, Phys. Rev. A 84, 040101 (2011).
  * Chitsazi  _et al._ [2017] M. Chitsazi, H. Li, F. M. Ellis, and T. Kottos, Experimental realization of floquet $\mathcal{P}\mathcal{T}$-symmetric systems, Phys. Rev. Lett. 119, 093901 (2017).
  * León-Montiel  _et al._ [2018] R. d. J. León-Montiel, M. A. Quiroz-Juárez, J. L. Domínguez-Juárez, R. Quintero-Torres, J. L. Aragón, A. K. Harter, and Y. N. Joglekar, Observation of slowly decaying eigenmodes without exceptional points in floquet dissipative synthetic circuits, Communications Physics 1, 1 (2018).
  * Berry [2004] M. V. Berry, Physics of nonhermitian degeneracies, Czechoslovak journal of physics 54, 1039 (2004).
  * Bender [2007] C. M. Bender, Making sense of non-hermitian hamiltonians, Reports on Progress in Physics 70, 947 (2007).
  * Rotter [2009] I. Rotter, A non-hermitian hamilton operator and the physics of open quantum systems, Journal of Physics A: Mathematical and Theoretical 42, 153001 (2009).
  * Moiseyev [2011] N. Moiseyev,  _Non-Hermitian quantum mechanics_ (Cambridge University Press, 2011).
  * Heiss [2012] W. Heiss, The physics of exceptional points, Journal of Physics A: Mathematical and Theoretical 45, 444016 (2012).
  * Cao and Wiersig [2015] H. Cao and J. Wiersig, Dielectric microcavities: Model systems for wave chaos and non-hermitian physics, Rev. Mod. Phys. 87, 61 (2015).
  * Lin  _et al._ [2011] Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Unidirectional invisibility induced by $\mathcal{P}\mathcal{T}$-symmetric periodic structures, Phys. Rev. Lett. 106, 213901 (2011).
  * Fleury  _et al._ [2015] R. Fleury, D. Sounas, and A. Alu, An invisible acoustic sensor based on parity-time symmetry, Nature communications 6, 1 (2015).
  * Feng  _et al._ [2014] L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, Single-mode laser by parity-time symmetry breaking, Science 346, 972 (2014).
  * Dong  _et al._ [2019] Z. Dong, Z. Li, F. Yang, C.-W. Qiu, and J. S. Ho, Sensitive readout of implantable microsensors using a wireless system locked to an exceptional point, Nature Electronics 2, 335 (2019).
  * Assawaworrarit and Fan [2020] S. Assawaworrarit and S. Fan, Robust and efficient wireless power transfer using a switch-mode implementation of a nonlinear parity–time symmetric circuit, Nature Electronics 3, 273 (2020).
  * Li  _et al._ [2020a] H. Li, A. Mekawy, A. Krasnok, and A. Alù, Virtual parity-time symmetry, Phys. Rev. Lett. 124, 193901 (2020a).
  * Li  _et al._ [2020b] H. Li, H. Moussa, D. Sounas, and A. Alù, Parity-time symmetry based on time modulation, Phys. Rev. Applied 14, 031002 (2020b).
  * Guo  _et al._ [2009] 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).
  * Ornigotti and Szameit [2014] M. Ornigotti and A. Szameit, Quasi-symmetry in passive photonic lattices, Journal of Optics 16, 065501 (2014).
  * Feng  _et al._ [2013] L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies, Nature materials 12, 108 (2013).
  * Liu  _et al._ [2019] H. Liu, D. Sun, C. Zhang, M. Groesbeck, R. Mclaughlin, and Z. V. Vardeny, Observation of exceptional points in magnonic parity-time symmetry devices, Science advances 5, eaax9144 (2019).
  * Jiang  _et al._ [2019] Y. Jiang, Y. Mei, Y. Zuo, Y. Zhai, J. Li, J. Wen, and S. Du, Anti-parity-time symmetric optical four-wave mixing in cold atoms, Phys. Rev. Lett. 123, 193604 (2019).
  * Yang  _et al._ [2022] X. Yang, J. Li, Y. Ding, M. Xu, X.-F. Zhu, and J. Zhu, Observation of transient parity-time symmetry in electronic systems, Phys. Rev. Lett. 128, 065701 (2022).
  * Hassan  _et al._ [2017] A. U. Hassan, B. Zhen, M. Soljačić, M. Khajavikhan, and D. N. Christodoulides, Dynamically encircling exceptional points: Exact evolution and polarization state conversion, Phys. Rev. Lett. 118, 093002 (2017).
  * Milburn  _et al._ [2015] T. J. Milburn, J. Doppler, C. A. Holmes, S. Portolan, S. Rotter, and P. Rabl, General description of quasiadiabatic dynamical phenomena near exceptional points, Phys. Rev. A 92, 052124 (2015).
  * [46] One can consider a rigorous version of kirchhoff’s laws where the time derivative of capacitance ($C(t)$) is taken into account. then there is an extra onsite potential (i.e., $\omega_{1}$ is no longer zero). however, in our approach to transforming the original hamiltonian into an effective one with balanced gain and loss, we consider an arbitrary onsite potentials. as a result, hidden pt symmetry can still be revealed in such a system with different broken and unbroken phases, but the location of exceptional point shifts.
  * Shirley [1965] J. H. Shirley, Solution of the schrödinger equation with a hamiltonian periodic in time, Phys. Rev. 138, B979 (1965).
  * Barone  _et al._ [1977] S. Barone, M. Narcowich, and F. Narcowich, Floquet theory and applications, Physical Review A 15, 1109 (1977).
  * Quiroz-Juárez  _et al._ [2021] M. A. Quiroz-Juárez, Z. A. Cochran, J. L. Aragón, Y. N. Joglekar, and R. d. J. León-Montiel, Parity-time symmetry via time-dependent non-unitary gauge fields, arXiv preprint arXiv:2109.03846 (2021).
  * Rechtsman  _et al._ [2013] M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Photonic floquet topological insulators, Nature 496, 196 (2013).
  * Doppler  _et al._ [2016] J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, Dynamically encircling an exceptional point for asymmetric mode switching, Nature 537, 76 (2016).
  * Zwillinger and Jeffrey [2007] D. Zwillinger and A. Jeffrey,  _Table of integrals, series, and products_ (Elsevier, 2007).