# Real energies and Berry phases in all PT-regimes in time-dependent non-
Hermitian theories

Andreas Fring∙, Takanobu Taira∘ and Rebecca Tenney∙
$\bullet$ Department of Mathematics, City, University of London, Northampton
Square,
London EC1V 0HB, UK
$\circ$ Research Fellow of Japan Society for Promotion of Science, Institute
of Industrial
Science, The University of Tokyo 5-1-5 Kashiwanoha, Kashiwa 277-8574, Japan
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract:

We demonstrate that the existence of a Hermitian time-dependent intertwining
operator that maps the non-Hermitian time-dependent energy operator to its
Hermitian conjugate and its right to its left eigenstates guarantees the
reality of the instantaneous energies. This property holds throughout all
three $\cal{PT}$-regimes, in the time-independent scenario referred to as the
$\cal{PT}$-symmetric regime, the exceptional point and the spontaneously
broken $\cal{PT}$-regime. We also propose a modified adiabatic approximation
consisting of an expansion of the wavefunctions in terms the instantaneous
eigenstates of the energy operator, instead of the usually used eigenfunctions
of the Hamiltonian. We show that this proposal always leads to real Berry
phases. We illustrate the working of our general proposals with two explicit
examples for a time-dependent non-Hermitian spin model.

††conference: Real energies and Berry phases in all PT-regimes in non-
Hermitian theories

##  1 Introduction

Time-independent non-Hermitian $\cal{PT}$-symmetric/quasi-Hermitian quantum
systems are characterised by three different regimes in their parameter space
in which they exhibit fundamentally distinct behaviour. In their
$\cal{PT}$-symmetric regime the eigenspectra of the Hamiltonian are all real,
in the spontaneously broken $\cal{PT}$-regime at least two of the eigenvalues
occur in complex conjugate pairs and at the transition point between the two
regimes, the exceptional point, the eigenvalues and eigenstates coalesce. Thus
this point is different from standard degeneracy where only the eigenvalues
coalesce. It is well-known that the explanation for this behaviour can be
attributed to the existence of an antilinear symmetry operator [1] of which
$\cal{PT}$ [2], a simultaneous reflection in space and time is an example. In
the symmetric regime the $\cal{PT}$-operator commutes with the Hamiltonian and
their eigenstates are identical up to a phase, whereas in the broken regime
the latter property no longer holds.

For systems described by explicitly time-dependent Hamiltonians it was
recently observed [3, 4, 5, 6, 7, 8, 9] that they are no longer separated by
exceptional points, even though the qualitative behaviour in these regimes in
parameter space might still differ. Especially remarkable is the observation
that the instantaneous energy eigenvalues become real in the spontaneously
broken regime. Other physical quantities, such as the von Neumann entropy,
inherit this behaviour and lead to new physical effects when crossing from one
regime to the other [4, 6]. Up to now these features were only observed in
case-by-case studies of specific models, but a generic explanation for such a
behaviour was still missing. The main purpose of this manuscript is to provide
such an explanation.

Central to the proper treatment of explicitly time-dependent non-Hermitian
Hamiltonian systems is a clear distinction between the energy operator and the
Hamiltonian. Unlike as in the Hermitian case they are no longer identical with
the latter not even being an observable quantity [10, 11, 12, 13, 14, 15, 16,
7, 17, 18, 19], see also [20] and additional references therein. Here we build
further on this difference and propose a new type of adiabatic approximation
that will always lead to real Berry phases for non-Hermitian
$\cal{PT}$-symmetric/quasi-Hermitian quantum systems.

Our manuscript is organised as follows: In section 2 we provide the general
argument that the existence of a time-dependent intertwining operator with
certain properties guarantees the reality of the instantaneous energy
eigenstates. In section 3 we propose an alternative adiabatic expansion in
terms of the eigenvectors of the energy operator that will always lead to real
geometric phases when the inner product is appropriately taken with a new
metric operator. In section 4 we present a worked out example for a spin model
in terms of Pauli matrices with complex time-dependent coefficients for two
different types of metric operators. Our conclusions are stated in section 5.

##  2 Real instantaneous energy eigenvalues in time-dependent non-Hermitian
systems

We start by noticing that the time-dependent quasi-Hermiticity relation for a
non-Hermitian Hamiltonian $H(t)$ can be re-written formally as the standard
quasi-Hermiticity relation for the instantaneous energy operator
$\tilde{H}(t)$

$i\hbar\partial_{t}\rho=H^{\dagger}\rho-\rho
H,\qquad\Leftrightarrow\qquad\tilde{H}^{\dagger}\rho=\rho\tilde{H},\quad\text{with}\,\,\,\tilde{H}:=H+i\hbar\eta^{-1}\partial_{t}\eta.$
(1)

Here $\rho(t)=\eta^{\dagger}(t)\eta(t)$ is a time-dependent positive definite
metric and $\eta$ denotes the time-dependent Dyson map, see recent review [20]
and references therein for more details on these quantities. When given the
Hamiltonian $H(t)$ as a starting point, one needs to stress that the left
relation in (1) has to be solved first for $\rho(t)$ before the expression for
the energy operator $\tilde{H}$ can actually be written down. Thus, unless
very strong assumptions on the Dyson map are made, the operator $\tilde{H}$ is
a meaningless starting point. There might also be solutions to the right
version in (1) that are not positive definite, which we denote as
$\tilde{{\cal P}}(t)$ in loose analogy to the time-independent version where
the corresponding operator can be identified as the parity operator. However,
we do not require this operator to be an involution, thus in general we have
$\tilde{{\cal P}}^{2}(t)\neq\mathbb{I}$. Notice further that due to the fact
that $\tilde{{\cal P}}(t)$ is negative definite, $\tilde{{\cal P}}(t)<0$, it
does not factorise into Dyson maps $\tilde{{\cal
P}}(t)\neq\eta^{\dagger}\eta$, so that the left relation in (1) does not hold
when $\rho$ is replaced with $\tilde{{\cal P}}(t)$.

We can now define two types of time-dependent ${\cal C}(t)$-operators, which
has been introduced as the operator that relates negative definite to positive
definite solutions of the quasi-Hermiticity relation in the time-independent
[21] as well as in the time-dependent scenario [22]. The first option for the
latter case was introduced in [22], where ${\cal C}(t)$ was expanded in terms
of the right and left eigenstates, $|\hat{\psi}(t)\rangle$ and
$|\hat{\phi}(t)\rangle$, of the non-Hermitian Hamiltonian $H(t)$

$H(t)|\hat{\psi}(t)\rangle=E|\hat{\psi}(t)\rangle,\qquad
H^{\dagger}|\hat{\phi}(t)\rangle=E|\hat{\phi}(t)\rangle,\qquad\langle\hat{\phi}(t)|\hat{\psi}(t)\rangle=\mathbb{I},$
(2)

as

${\cal
C}(t):=\sum_{n}\hat{s}_{n}|\hat{\psi}_{n}(t)\rangle\langle\hat{\phi}_{n}(t)|=\sum_{n}\hat{s}_{n}|\hat{\phi}_{n}(t)\rangle\langle\hat{\psi}_{n}(t)|={\cal
P}\hat{\rho}(t)=\hat{\rho}^{-1}(t){\cal P}.$ (3)

Here $\hat{s}_{n}=\pm 1$ are the signatures of this expansion and ${\cal P}$
is the time-independent parity operator with ${\cal P}^{2}=\mathbb{I}$. The
operator ${\cal{C}}$ is involutory but, unlike as in the time-independent
case, does not commute with the time-dependent Hamiltonian

${\cal{C}}^{2}(t)=\mathbb{I},\qquad\text{and}\qquad
i\hbar\partial_{t}{\cal{C}}(t)=[H(t),{\cal{C}}(t)].$ (4)

Besides relating a time-independent negative definite solution ${\cal P}$ to
the pseudo-Hermiticity relation to a positive time-dependent definite metric
operator $\hat{\rho}(t)$, due to the first relation in (7) this ${\cal
C}$-operator can formally also be identified with a Lewis-Riesenfeld invariant
as shown in [22] when we identify $\hat{\rho}(t)$ with $\rho(t)$.

Alternatively, we can define a different operator in terms of the right and
left eigenstates, $|\tilde{\psi}(t)\rangle$ and $|\tilde{\phi}(t)\rangle$, of
the energy operator $\tilde{H}$ defined in (1)

$\tilde{H}|\tilde{\psi}(t)\rangle=\tilde{E}|\tilde{\psi}(t)\rangle,\qquad\tilde{H}^{\dagger}|\tilde{\phi}(t)\rangle=\tilde{E}|\tilde{\phi}(t)\rangle,\qquad\langle\tilde{\phi}(t)|\tilde{\psi}(t)\rangle=\mathbb{I},$
(5)

that relates the non-positive definite time-dependent to the positive definite
time-dependent solutions as

$\tilde{{\cal
C}}(t):=\sum_{n}\tilde{s}_{n}|\tilde{\psi}_{n}(t)\rangle\langle\tilde{\phi}_{n}(t)|=\sum_{n}\tilde{s}_{n}|\tilde{\phi}_{n}(t)\rangle\langle\tilde{\psi}_{n}(t)|=\tilde{{\cal
P}}^{-1}(t)\rho(t)=\rho^{-1}(t)\tilde{{\cal P}}(t).$ (6)

Here $\tilde{s}_{n}=\pm 1$ are the signatures of this expansion. The two
expansions (3) and (6) are obviously different. In the time-independent
version this ambiguity does not exist as the two operators $H$ and
$\tilde{H}$, and therefore its eigenstates coincide. It is easily verified
that $\tilde{{\cal C}}(t)$, as defined in (6), is an involution operator that
commutes with the energy operator,

$\tilde{{\cal C}}^{2}(t)=\mathbb{I},\qquad\text{and}\qquad[\tilde{{\cal
C}}(t),\tilde{H}(t)]=0.$ (7)

We now make the following observation: When the time-dependent non-Hermitian
energy operator $\tilde{H}(t)$ is quasi-Hermitian with regard to the action of
the time-dependent Hermitian intertwining operator $\tilde{{\cal P}}$ that
relates its right eigenvectors to its left eigenvectors, then the
instantaneous energies $\tilde{E}(t)$ are real. Thus, if the properties

$i)\,\,\tilde{{\cal P}}\tilde{H}=\tilde{H}^{\dagger}\tilde{{\cal P}},\qquad
ii)\,\,\tilde{{\cal P}}|\tilde{\psi}\rangle=\alpha|\tilde{\phi}\rangle,\qquad
iii)\,\,\tilde{{\cal P}}=\tilde{{\cal
P}}^{\dagger},\,\,\,\,\alpha\in\mathbb{R},$ (8)

hold, then $\tilde{E}(t)\in\mathbb{R}$ at any time $t$.

We easily prove this statement using these three properties together with the
standard properties of left and right eigenvectors (5). We have

$\tilde{E}\\!\\!\overset{\text{(\ref{evequn})}}{=}\\!\\!\langle\tilde{\phi}|\tilde{H}|\tilde{\psi}\rangle\overset{ii,iii}{=}\frac{1}{\alpha}\langle\tilde{\psi}|\tilde{{\cal
P}}\tilde{H}|\tilde{\psi}\rangle\overset{i}{=}\frac{1}{\alpha}\langle\tilde{\psi}|\tilde{H}^{\dagger}\tilde{{\cal
P}}|\tilde{\psi}\rangle\overset{\text{(\ref{evequn})}}{=}\frac{E^{*}}{\alpha}\langle\tilde{\psi}|\tilde{{\cal
P}}|\tilde{\psi}\rangle\overset{ii,iii}{=}\tilde{E}^{*}\langle\tilde{\phi}|\tilde{\psi}\rangle\overset{\text{(\ref{evequn})}}{=}\tilde{E}^{*}.$
(9)

We notice that we no longer require the use of the time-reversal operator and
moreover that property $ii)$ in (8) is slightly different from the usual
requirement used in the time-independent case where ${\cal{PT}}$ maps
right/left eigenvector to right/left eigenvectors up to a phase, see e.g.
section 2.4.1 in [20]. Thus, in that case $\cal{PT}$ is an involution
operator, which is no longer the case for the $\tilde{{\cal P}}$-operator,
i.e. $\tilde{{\cal P}}^{2}\neq\mathbb{I}$. This is also the reason why we do
not refer to $\tilde{{\cal P}}$ as a time-dependent parity operator. As we
stated, the time-dependent quasi-Hermiticity relation holds in general for the
energy operator $\tilde{H}$ whereas the time-independent ${\cal PT}$-symmetry
holds for the Hamiltonian $H$, which explains why the spontaneously broken
${\cal PT}$-regime can be mended in the time-dependent scenario. This feature
was previously observed for an number of systems [3, 4, 5, 6, 7, 8, 9], but
left unexplained up to now.

##  3 Real Berry phases in non-Hermitian systems

We briefly recall how the geometrical (Berry) phase [23] originates in
standard Hermitian systems in order to set the scene for a non-Hermitian
generalisation. One considers a quantum mechanical system described by an
explicitly time-dependent Hermitian Hamiltonian $h[q(t)]$, where the time-
dependence is acquired through a set of parameters
$q(t)=(q_{1}(t),q_{2}(t),\ldots,q_{n}(t))$ with period $T$, i.e. $q(0)=q(T)$.
The time-dependent Schrödinger equation (TDSE) then governs the wave function
$\left|\chi(t)\right\rangle$ as

$h[q(t)]\left|\chi(t)\right\rangle=i\hbar\partial_{t}\left|\chi(t)\right\rangle.$
(10)

Crucially, the system is assumed to evolve adiabatically so that at each
moment in time one can expand the system in terms of instantaneous orthonormal
energy eigenstates as

$h[q(t)]|\tilde{\chi}_{n}(t)\rangle=\tilde{E}_{n}[q(t)]|\tilde{\chi}_{n}(t)\rangle.$
(11)

The states $\left|\chi(t)\right\rangle$ are then assumed to be expanded as

$\left|\chi(t)\right\rangle=\sum_{n}c_{n}(0)e^{i\gamma_{n}(t)}e^{i\alpha_{n}(t)}|\tilde{\chi}_{n}(t)\rangle,\quad\alpha_{n}(t)=-\frac{1}{\hbar}\int_{0}^{t}\tilde{E}_{n}[q(t)],\quad
c_{n}(0)=const\in\mathbb{C},$ (12)

where one distinguishes between the time-dependent dynamical phases
$\alpha_{n}(t)$ and the geometrical phases $\gamma_{n}(t)$ at level $n$.
Substituting the expansion of $\left|\chi(t)\right\rangle$ from (12) into the
TDSE (10), together with the orthonormality relation of the instantaneous
energy eigenstates
$\langle\tilde{\chi}_{n}(t)|\tilde{\chi}_{m}(t)\rangle=\delta_{n,m}$, one
derives that the geometrical phase has to satisfy

$\dot{\gamma}_{n}(t)=i\langle\tilde{\chi}_{n}[q(t)]|\partial_{t}\tilde{\chi}_{n}[q(t)]\rangle.$
(13)

The phase becomes a physical observable when the ray
$|\tilde{\chi}_{n}[q(t)]\rangle$ returns to its initial state, i.e. when
$|\tilde{\chi}_{n}[q(0)]\rangle=|\tilde{\chi}_{n}[q(T)]\rangle$, one may have
picked up a non-vanishing phase difference

$\gamma_{n}=i\int_{0}^{T}\left\langle\tilde{\chi}_{n}[q(t)]\right|\left.\partial_{t}\tilde{\chi}_{n}[q(t)]\right\rangle
dt=i\oint_{C}\left\langle\tilde{\chi}_{n}[q(t)]\right|\left.\nabla_{q}\tilde{\chi}_{n}[q(t)]\right\rangle
dq,$ (14)

where $C$ is a closed path traced out in parameter space. It is the latter
expression which makes it clear why $\gamma_{n}$ is referred to as a
geometrical phase. In a Hermitian system $\gamma_{n}$ is known to be always
real.

Let us now explain how the above extends to a real geometric phase for pseudo-
Hermitian systems. For this purpose we consider now a non-Hermitian
Hamiltonian $H[q(t)]$ depending on the same time-dependent parameter set
$q(t)$ as the Hermitian Hamiltonian $h[q(t)]$ satisfying the TDSE

$H[q(t)]\left|\psi(t)\right\rangle=i\hbar\partial_{t}\left|\psi(t)\right\rangle,$
(15)

and the time-dependent Dyson equation

$h[q(t)]=\eta(t)H[q(t)]\eta^{-1}(t)+i\hbar\partial_{t}\eta(t)\eta^{-1}(t),\qquad\text{with}\,\,\,|\chi(t)\rangle=\eta(t)|\psi(t)\rangle.$
(16)

Thus, we are not considering here open non-Hermitian systems as for instance
in [24, 25, 26]. Just as in the Hermitian case we assume that the system
evolves adiabatically at each moment in time. However, unlike as for the
Hermitian case we have now formally two options to implement the adiabatic
assumption, we may either expand $|\psi(t)\rangle$ in terms of the
instantaneous eigenstates of the Hamiltonian $H$ or the energy operator
$\tilde{H}$. This is the same ambiguity already encountered previously for the
expansion of the $\cal{C}$ and $\tilde{\cal{C}}$-operators. In [24, 25, 13]
the former option was chosen leading to complex Berry phases, here we chose
the latter obtaining always real phases. In [27] the authors employ the
eigenstates of the Lewis-Riesenfeld invariant, obtaining real Berry phases. In
[28] the authors showed that one may also obtain real phases by imposing the
constraint $H=-i\hbar\eta^{-1}\dot{\eta}$, which is, however, an unnecessary
limitation for our definition of the Berry phase. Of course one may also
consider non-adiabatic phases and avoid the above expansions [30, 29, 19].
Here we use

$\tilde{H}[q(t)]|\tilde{\psi}_{n}(t)\rangle=\tilde{E}_{n}[q(t)]|\tilde{\psi}_{n}(t)\rangle,\qquad\text{with}\,\,\,\tilde{H}[q(t)]=H[q(t)]+i\hbar\eta^{-1}\partial_{t}\eta,$
(17)

where $\tilde{H}[q(t)]$ is the energy operator already encountered in (1),
rather than the Hamiltonian $H[q(t)]$. The second relation in (16) together
with the orthonormality of the states $|\tilde{\phi}_{n}(t)\rangle$ implies
that the states $|\tilde{\psi}_{n}(t)\rangle$ are orthonormal with regard to
the new time-dependent metric $\rho(t)=\eta^{\dagger}(t)\eta(t)$, that is
$\langle\tilde{\psi}_{n}(t)|\rho(t)\tilde{\psi}_{m}(t)\rangle=\delta_{n,m}$.
The solutions of the TDSE $|\psi(t)\rangle$ then expanded as

$\left|\psi(t)\right\rangle=\sum_{n}c_{n}(0)e^{i\gamma_{n}(t)}e^{i\alpha_{n}(t)}|\tilde{\psi}_{n}(t)\rangle,$
(18)

with both phases $\alpha_{n}$ and $\gamma_{n}$ being identical to those in the
Hermitian case.

Substituting the expansion (18) into the TDSE (15), together with the
orthonormality relation for $|\tilde{\phi}_{n}(t)\rangle$, we derive that the
geometrical phase has to satisfies

$\dot{\gamma}_{n}(t)=i\left\langle\tilde{\psi}_{n}[q(t)]\right|\rho(t)\left(\partial_{t}+\eta^{-1}\partial_{t}\eta\right)\left|\tilde{\psi}_{n}[q(t)]\right\rangle.$
(19)

Relation (19) also follows directly when using
$|\tilde{\chi}(t)\rangle=\eta(t)|\tilde{\psi}(t)\rangle$ in (13). Thus we have
two alternative ways to compute the geometrical phase $\gamma_{i}(t)$, i.e.
either in terms of quantities related to the non-Hermitian system (19) or in
terms of the quantities of the equivalent Hermitian system (14). As both
expressions can be converted into each other and the phase (14) can be shown
to be always real, this implies that the phase computed for the non-Hermitian
system in (19) must also be real. Regarding the physical interpretation
associated to the two expansions one should note that the expansion in terms
of the eigenvalues of the energy operator provides a clear picture identical
to the interpretation in the Hermitian case. However, it is unclear what the
commonly used expansion in terms of the eigenstates of the Hamiltonian should
be as the Hamiltonian is not even an observable operator and has no chance of
becoming one as it is not quasi-Hermitian with regard to a time-dependent
metric.

##  4 An example: an explicitly time-dependent two level system

As a sample system we consider the explicitly time-dependent non-Hermitian
Hamiltonian of the most general $(2\times 2)$-matrix form describing a spin
system

$H(t)=-\frac{1}{2}\left[\omega\mathbb{I}+\alpha(t)\sigma_{x}+\mu(t)\sigma_{y}+\tau(t)\sigma_{z}\right],\qquad\quad\text{with}\,\,\,\alpha,\mu,\tau\in\mathbb{C},$
(20)

with $\sigma_{x},\sigma_{y},\sigma_{z}$ denoting standard Pauli matrices.
Treating this Hamiltonian initially as time-independent, i.e. we take the
parameter $\alpha,\mu,\tau$ for a fixed time, we may identify the parity
operator as the non-positive definite solution of the pseudo-Hermiticity
equation ${\cal P}H=H^{\dagger}{\cal P}$ with ${\cal P}^{2}=\mathbb{I}$.
Separating real and imaginary parts as $x=x_{r}+ix_{i}$ for
$x=\alpha,\mu,\tau$, $x_{r},x_{i}\in\mathbb{R}$ and imposing the constraints

$\alpha_{r}\alpha_{i}=-\mu_{r}\mu_{i},\qquad\text{and}\qquad\tau_{r}=0,$ (21)

we find the solution

${\cal
P}=\left(\begin{array}[]{cc}0&\frac{\alpha_{r}-i\mu_{r}}{\sqrt{\alpha_{r}^{2}+\mu_{r}^{2}}}\\\
\frac{\alpha_{r}+i\mu_{r}}{\sqrt{\alpha_{r}^{2}+\mu_{r}^{2}}}&0\\\
\end{array}\right).$ (22)

The eigenvalues of ${\cal P}$ are $\pm 1$ so that it is indeed negative
definite. The energy eigenvalues of $H$

$E_{\pm}=\frac{1}{2}\left[-\omega\pm\frac{1}{\alpha_{r}}\sqrt{\Delta}\right]\qquad\Delta:=(\alpha_{r}^{2}+\mu_{r}^{2})(\alpha_{r}^{2}-\mu_{i}^{2})-\alpha_{r}^{2}\tau_{i}^{2},$
(23)

may become real, complex conjugate or coalesce depending on the value of the
discriminant, exhibiting the three possible ${\cal PT}$-regimes in parameter
space. These are the $\cal{PT}$-symmetric regime when $\Delta>0$, the
exceptional point when $\Delta=0$ and the spontaneously broken
$\cal{PT}$-regime when $\Delta<0$.

###  4.1 Hermitian Dyson map, $[\tilde{{\cal P}}{\cal T},H]=0$

Let us next turn to the fully time-dependent case and solve the time-dependent
Dyson equation (16) for $\eta(t)$ and $h(t)$. Substituting the Hermitian
Ansatz

$\eta(t)=\eta_{0}(t)\mathbb{I}+\eta_{z}(t)\sigma_{z},\qquad\eta_{0},\eta_{z}\in\mathbb{R}$
(24)

into the (16) and demanding the left hand side to be Hermitian, leads to the
two coupled first order differential equations with an additional constraints
(21) and

$\dot{\eta}_{0}=\frac{1}{2}\eta_{z}\tau_{i},\qquad\dot{\eta}_{z}=\frac{1}{2}\eta_{0}\tau_{i},\qquad\mu_{i}=-\frac{2\alpha_{r}\eta_{0}\eta_{z}}{\eta_{0}^{2}+\eta_{z}^{2}}.$
(25)

The first two equations can be combined into the second order equation

$\ddot{\eta}_{0}-\frac{\dot{\tau}_{i}}{\tau_{i}}\dot{\eta}_{0}-\frac{1}{4}\tau_{i}^{2}\eta_{0}=0,$
(26)

which is solved by

$\eta_{0}(t)=c_{1}\sinh\left[\frac{\delta(t)}{2}\right]+c_{2}\cosh\left[\frac{\delta(t)}{2}\right],\qquad\text{with}\quad\delta(t):=\int^{t}\tau_{i}(s)ds,$
(27)

and real integration constants $c_{1}$ and $c_{2}$. It then follows directly
from the constraints (25) that
$\eta_{z}(t)=c_{1}\cosh\left[\delta(t)/2\right]+c_{2}\sinh\left[\delta(t)/2\right]$
and therefore the time-dependent Dyson map becomes

$\eta(t)=\left(\begin{array}[]{cc}(c_{1}+c_{2})\exp\left[\frac{\delta(t)}{2}\right]&0\\\
0&(c_{1}-c_{2})\exp\left[-\frac{\delta(t)}{2}\right]\\\ \end{array}\right).$
(28)

Since $\eta(t)$ is Hermitian the corresponding metric operator is immediately
obtained as

$\rho(t)=\eta^{2}(t)=\left(\begin{array}[]{cc}(c_{1}+c_{2})^{2}e^{\delta(t)}&0\\\
0&(c_{1}-c_{2})^{2}e^{-\delta(t)}\\\ \end{array}\right),$ (29)

which is obviously positive definite with
$\det\rho=(c_{1}^{2}-c_{2}^{2})^{2}$.

The Hermitian Hamiltonian results from (16) to

$h(t)=\left(\begin{array}[]{cc}-\frac{\omega}{2}&\frac{\left(c_{1}^{2}-c_{2}^{2}\right)\left[\alpha_{r}(t)-i\mu_{r}(t)\right]}{4c_{1}c_{2}\sinh[\delta(t)]+2\left(c_{1}^{2}+c_{2}^{2}\right)\cosh[\delta(t)]}\\\
\frac{\left(c_{1}^{2}-c_{2}^{2}\right)\left[\alpha_{r}(t)+i\mu_{r}(t)\right]}{4c_{1}c_{2}\sinh[\delta(t)]+2\left(c_{1}^{2}+c_{2}^{2}\right)\cosh[\delta(t)]}&-\frac{\omega}{2}\\\
\end{array}\right).$ (30)

The corresponding instantaneous energy eigenvalues together with their
normalised eigenstates are

$\tilde{E}_{\pm}(t)=\pm\frac{\sqrt{2}\left(c_{1}^{2}-c_{2}^{2}\right)\sqrt{\alpha_{r}^{2}(t)+\mu_{r}^{2}(t)}}{4c_{1}c_{2}\sinh[\delta(t)]+2\left(c_{1}^{2}+c_{2}^{2}\right)\cosh[\delta(t)]}-\frac{\omega}{2},\quad|\tilde{\chi}_{\pm}(t)\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\frac{\sqrt{\alpha_{r}^{2}(t)+\mu_{r}^{2}(t)}}{\alpha_{r}(t)+i\mu_{r}(t)}\\\
1\\\ \end{array}\right).$ (31)

With these expressions we can now directly compute the time-dependent
$\tilde{{\cal P}}$, $\tilde{{\cal C}}$ operators and the Berry phase (19).
From the definition (6), together with
$|\tilde{\phi}_{\pm}(t)\rangle=\eta^{-1}(t)|\tilde{\chi}_{\pm}(t)\rangle$ and
the signature $s_{\pm}=\pm 1$ we compute

$\tilde{{\cal
C}}(t)=\left(\begin{array}[]{cc}0&\frac{\left(c_{1}-c_{2}\right)e^{-\delta(t)}\sqrt{\alpha_{r}^{2}(t)+\mu_{r}^{2}(t)}}{\left(c_{1}+c_{2}\right)\left[\alpha_{r}(t)+i\mu_{r}(t)\right]}\\\
\frac{\left(c_{1}+c_{2}\right)\left[\alpha_{r}(t)+i\mu_{r}(t)\right]e^{\delta(t)}}{\left(c_{1}-c_{2}\right)\sqrt{\alpha_{r}^{2}(t)+\mu_{r}^{2}(t)}}&0\\\
\end{array}\right),$ (32)

which is an involution operator that commutes with the non-Hermitian energy
operator

$\tilde{H}(t)=\left(\begin{array}[]{cc}-\frac{\omega}{2}&-\frac{\left(c_{1}-c_{2}\right)^{2}\left[\alpha_{r}(t)-i\mu_{r}(t)\right]}{\left(c_{1}+c_{2}\right)^{2}e^{2\delta(t)}+\left(c_{1}-c_{2}\right){}^{2}}\\\
-\frac{\left(c_{1}+c_{2}\right){}^{2}e^{\delta(t)}\left[\alpha_{r}(t)+i\mu_{r}(t)\right]}{4c_{2}c_{1}\sinh\left[\delta(t)\right]+2(c_{1}^{2}+c_{2}^{2})\cosh\left[\delta(t)\right]}&-\frac{\omega}{2}\\\
\end{array}\right)$ (33)

as stated in (7). Next we calculate the time-dependent operator $\tilde{{\cal
P}}(t)$ from relation (6) to

$\tilde{{\cal P}}(t)=\rho(t)\tilde{{\cal
C}}(t)=\frac{c_{1}^{2}-c_{2}^{2}}{\sqrt{\alpha_{r}^{2}(t)+\mu_{r}^{2}(t)}}\left(\begin{array}[]{cc}0&\alpha_{r}(t)-i\mu_{r}(t)\\\
\alpha_{r}(t)+i\mu_{r}(t)&0\\\ \end{array}\right).$ (34)

We notice that $\tilde{{\cal P}}^{2}(t)\neq\mathbb{I}$, although this can be
achieved with a particular choice for the constants, e.g. $c_{1}=1,c_{2}=0$.
We also verify that $\tilde{{\cal P}}(t)$ does indeed satisfy the second
version of the time-dependent quasi-Hermiticity relation in (1). However, with
eigenvalues $(c_{1}^{2}-c_{2}^{2})$ and $-(c_{1}^{2}-c_{2}^{2})$, it is not
positive and therefore not a metric. We verify that the non-Hermitian time-
dependent energy operator is i) quasi-Hermitian with regard to the action of
the intertwining operator $\tilde{{\cal P}}(t)$, ii) the $\tilde{{\cal
P}}$-operator converts right eigenstates of $\tilde{H}$ into their left
eigenstates as $\tilde{{\cal
P}}(t)|\tilde{\psi}_{\pm}\rangle=\pm|\tilde{\phi}_{\pm}\rangle$, with the bi-
orthonormality preserved when scaling the states as
$|\tilde{\psi}\rangle\rightarrow 1/\alpha|\tilde{\psi}\rangle$,
$|\tilde{\phi}\rangle\rightarrow\alpha|\tilde{\phi}\rangle$ , and iii) that
$\tilde{{\cal P}}(t)$ is Hermitian. This means all three relations in (8) hold
and hence the instantaneous energy eigenvalues of the non-Hermitian energy
operator are guaranteed to be real.

In addition we notice that each term in the energy operator is $\tilde{{\cal
P}}(t)$-quasi Hermitian, so that relations $i)$ and $iii)$ also hold for the
Hamiltonian $H(t)$. However, the second relation in (8) does not hold for
$H(t)$ as

${\cal P}(t)|\hat{\psi}_{\pm}\rangle\neq\alpha|\hat{\phi}_{\pm}\rangle,$ (35)

so that we can not apply the above argument to the Hamiltonian. Thus in this
sense the broken ${\cal P}{\cal T}$-regime in the time-independent scenario
has been mended in the time-dependent case, a possibility first observed in
[3].

Having computed the eigenstates of $\tilde{H}$ and the Dyson map $\eta(t)$ we
may now also directly calculate the Berry phase by means of (19). We find

$\gamma_{\pm}=\frac{1}{2}\int_{0}^{T}\frac{\alpha_{r}\dot{\mu_{r}}-\mu_{r}\dot{\alpha}_{r}}{\alpha_{r}^{2}+\mu_{r}^{2}}dt=\left.\frac{1}{2}\arctan\left(\frac{\mu_{r}}{\alpha_{r}}\right)\right|_{0}^{T}.$
(36)

Evidently $\gamma_{\pm}$ is always real.

###  4.2 Non-Hermitian Dyson map, $[\tilde{{\cal P}}{\cal T},H]\neq 0$

In the previous section we have constructed a relatively simple diagonal
solution for a Hermitian Dyson map $\eta(t)$. We also found that
$[\tilde{{\cal P}}{\cal T},H]=0$ and for a particular choice of the constants
$\tilde{{\cal P}}(t)$-operator became identical to the parity operator in the
time-independent case. However, it is well-known that Dyson maps are not
unique in general and one may even construct infinite series in the time-
dependent case [18]. We will now exploit this ambiguity and see if one can
construct $\tilde{{\cal P}}(t)$-operator that are more distinct from the
parity operator by making a more generic non-Hermitian Ansatz and solve (16)
once more. We assume now

$\eta(t)=\eta_{0}(t)\mathbb{I}+\eta_{z}(t)\sigma_{z}+i\eta_{y}(t)\sigma_{y},\qquad\eta_{0},\eta_{y},\eta_{z}\in\mathbb{R},$
(37)

and proceed as in the previous example to solve the time-dependent Dyson
equation. In this case we find the solution

$\eta(t)=\left(\begin{array}[]{cc}\frac{c_{1}\mu_{i}}{\alpha_{r}}-2c_{1}&\frac{c_{1}\mu_{i}}{\alpha_{r}}\\\
-\frac{c_{1}\mu_{i}}{\alpha_{r}}&-\frac{c_{1}\mu_{i}}{\alpha_{r}}-2c_{1}\\\
\end{array}\right)=-2c_{1}\mathbb{I}+\frac{c_{1}\mu_{i}}{\alpha_{r}}\sigma_{z}+i\frac{c_{1}\mu_{i}}{\alpha_{r}}\sigma_{y},$
(38)

with $c_{1}$ being an integration constant and the additional constraints

$\mu_{r}=-\tau_{i}-2A,\qquad\alpha_{i}=2\frac{\mu_{i}}{\alpha_{r}}A,\qquad\tau_{r}=\mu_{i},\qquad
A:=\frac{\tau_{i}\alpha_{r}^{2}}{\mu_{i}^{2}}-\frac{\dot{\alpha}_{r}}{\mu_{i}}+\frac{\alpha_{r}\dot{\mu}_{i}}{\mu_{i}^{2}},$
(39)

together with the time-dependent Hermitian Hamiltonian

$h=\left(\begin{array}[]{cc}-\frac{\omega}{2}&-\frac{\alpha_{r}}{2}-iA\\\
-\frac{\alpha_{r}}{2}+iA&-\frac{\omega}{2}\\\ \end{array}\right).$ (40)

We have omitted here the calculation, but the critical reader may convince
themselves by simply substituting the solution back into the time-dependent
Dyson equation. Notice also that the solution (38) does not reduce to the
previous one, as the elimination of the non-Hermitian term by
$\mu_{i}\rightarrow 0$ will simply lead to the trivial solution proportional
to $\mathbb{I}$.

The time-dependent instantaneous energy eigenvalues together with their
normalised eigenstates as introduced in (11) are found to

$\tilde{E}_{\pm}=\frac{1}{2}\left(-\omega\pm\sqrt{4A^{2}+\alpha_{r}^{2}}\right),\qquad|\tilde{\chi}_{\pm}\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\frac{\mp
i\sqrt{4A^{2}+\alpha_{r}^{2}}}{2A+i\alpha_{r}}\\\ 1\\\ \end{array}\right).$
(41)

Evidently $\tilde{E}_{\pm}\in\mathbb{R}$, such that the broken regime from the
time-independent regime has been mended once more. As in the previous section
we use the relation
$|\tilde{\psi}_{\pm}(t)\rangle=\eta^{-1}(t)|\tilde{\chi}_{\pm}(t)\rangle$
between the eigenstates of $h(t)$ and $\tilde{H}(t)$ to compute the states in
the expansion (3). For the signature $s_{\pm}=\pm 1$ we obtain

$\tilde{{\cal
C}}=\frac{-1}{\sqrt{4A^{2}+\alpha_{r}^{2}}}\left(\begin{array}[]{cc}\mu_{i}+i\frac{\mu_{i}^{2}}{\alpha_{r}^{2}}A&\alpha_{r}+\mu_{i}+i\left(\frac{\mu_{i}^{2}}{\alpha_{r}^{2}}+\frac{2\mu_{i}}{\alpha_{r}}+2\right)A\\\
\alpha_{r}-\mu_{i}-i\left(\frac{\mu_{i}^{2}}{\alpha_{r}^{2}}-\frac{2\mu_{i}}{\alpha_{r}}+2\right)A&-\mu_{i}-i\frac{\mu_{i}^{2}}{\alpha_{r}^{2}}A\\\
\end{array}\right).$ (42)

We verify that this operator squares to $\mathbb{I}$ and does indeed commute
with the energy operator

$\tilde{H}=\left(\begin{array}[]{cc}-\frac{\mu_{i}}{2}-\frac{\omega}{2}-i\frac{A\mu_{i}^{2}}{2\alpha_{r}^{2}}&-\frac{\mu_{i}}{2}-\frac{\alpha_{r}}{2}-iA\left(\frac{\mu_{i}^{2}}{2\alpha_{r}^{2}}+\frac{\mu_{i}}{\alpha_{r}}+1\right)\\\
\frac{\mu_{i}}{2}-\frac{\alpha_{r}}{2}+iA\left(\frac{\mu_{i}^{2}}{2\alpha_{r}^{2}}-\frac{\mu_{i}}{\alpha_{r}}+1\right)&\frac{\mu_{i}}{2}-\frac{\omega}{2}+i\frac{A\mu_{i}^{2}}{2\alpha_{r}^{2}}\\\
\end{array}\right),$ (43)

as required in (7).

The corresponding metric is directly computed to

$\rho=\eta^{\dagger}\eta=\left(\begin{array}[]{cc}\frac{2c_{1}^{2}\left(\mu_{i}^{2}-2\mu_{i}\alpha_{r}+2\alpha_{r}^{2}\right)}{\alpha_{r}^{2}}&\frac{2c_{1}^{2}\mu_{i}^{2}}{\alpha_{r}^{2}}\\\
\frac{2c_{1}^{2}\mu_{i}^{2}}{\alpha_{r}^{2}}&\frac{2c_{1}^{2}\left(\mu_{i}^{2}+2\mu_{i}\alpha_{r}+2\alpha_{r}^{2}\right)}{\alpha_{r}^{2}}\\\
\end{array}\right).$ (44)

The eigenvalues of $\rho$ are

$\lambda_{\pm}^{\rho}=\frac{2c_{1}^{2}\left[\mu_{i}^{2}\alpha_{r}^{4}+2\alpha_{r}^{6}\pm\sqrt{\mu_{i}^{2}\alpha_{r}^{8}\left(\mu_{i}^{2}+4\alpha_{r}^{2}\right)}\right]}{\alpha_{r}^{6}}\geq
0,$ (45)

so that the metric is indeed positive.

We can now compute the time-dependent parity operator according to (6) as

$\tilde{{\cal P}}(t)=\rho(t)\tilde{{\cal
C}}(t)=\frac{2c_{1}^{2}}{\alpha_{r}\sqrt{4A^{2}+\alpha_{r}^{2}}}\left(\begin{array}[]{cc}\mu_{i}\left(\mu_{i}-2\alpha_{r}\right)&\mu_{i}^{2}-2\alpha_{r}^{2}-4iA\alpha_{r}\\\
\mu_{i}^{2}-2\alpha_{r}^{2}+4iA\alpha_{r}&\mu_{i}\left(\mu_{i}+2\alpha_{r}\right)\\\
\end{array}\right).$ (46)

The two eigenvalues of $\tilde{{\cal P}}$ are

$\lambda_{\pm}^{{\cal
P}}=\frac{2c_{1}^{2}}{\alpha_{r}\sqrt{4A^{2}+\alpha_{r}^{2}}}\left(\mu_{i}^{2}\pm\sqrt{16A^{2}\alpha_{r}^{2}+4\alpha_{r}^{4}+\mu_{i}^{4}}\right),$
(47)

so that $\tilde{{\cal P}}$ is negative definite with $\det\tilde{{\cal
P}}=-16c_{1}^{4}$. Moreover, we verify that the first relation in (8) is
satisfied, that $\tilde{{\cal
P}}|\tilde{\psi}_{\pm}\rangle=\pm|\tilde{\phi}_{\pm}\rangle$ and that
$\tilde{{\cal P}}=\tilde{{\cal P}}^{\dagger}$. These properties suffice to
guarantee the reality of the instantaneous energy eigenvalues
$\tilde{E}_{\pm}$.

The Berry phase is computed in a similar way as in the previous section from
(19) to

$\gamma_{\pm}=-\frac{1}{2}\int_{0}^{T}\frac{2\alpha_{r}\dot{A}-2A\dot{\alpha}_{r}}{4A^{2}+\alpha_{r}^{2}}dt=\left.-\frac{1}{2}\arctan\left(\frac{2A}{\alpha_{r}}\right)\right|_{0}^{T},$
(48)

which is always real.

##  5 Conclusions

We have identified a new operator $\tilde{{\cal P}}$ as the non-involutory
non-positive definite solution of the quasi-Hermiticity relation (1) involving
the energy operator. The existence of this operator will ensure the reality of
the instantaneous energies when it satisfies the three properties stated in
(8). As this operator differs in general from the operator ${\cal P}{\cal
T}$-operator of the time-independent case, the characteristic regimes this
operator classifies are naturally different from those in the time-dependent
case.

Furthermore, we showed that when changing the commonly used adiabatic
approximation from expanding the solutions to the TDSE in terms of the
eigenstates of the energy operator rather than the Hamiltonian, the Berry
phase will be identical to the one computed for the Hermitian counterpart and
therefore real.

Acknowledgments: RT is supported by EPSRC grant EP/W522351/1. TT is supported
by JSPS KAKENHI Grant Number JP22J01230.

## References

  * [1] E. Wigner,  Normal form of antiunitary operators,  J. Math. Phys. 1, 409–413 (1960).
  * [2] C. M. Bender and S. Boettcher,  Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry,  Phys. Rev. Lett. 80, 5243–5246 (1998).
  * [3] A. Fring and T. Frith,  Mending the broken PT-regime via an explicit time-dependent Dyson map,  Phys. Lett. A , 2318 (2017).
  * [4] A. Fring and T. Frith,  Eternal life of entropy in non-Hermitian quantum systems,  Phys. Rev. A 100(1), 010102 (2019).
  * [5] A. Fring and R. Tenney,  Time-independent approximations for time-dependent optical potentials,  Eur. Phys. J. Plus 135(2), 163 (2020).
  * [6] T. Frith,  Exotic entanglement for non-Hermitian Jaynes–Cummings Hamiltonians,  J. of Phys. A: Math. and Theor. 53(48), 485303 (2020).
  * [7] A. Fring and R. Tenney,  Perturbative approach for strong and weakly coupled time-dependent for non-Hermitian quantum systems,  Phys. Scr. 96(4), 045211 (2021).
  * [8] A. Alsalam Abu Moise, G. Cox, and M. Merkli,  Dynamics of entropy in bipartite quasi-Hermitian systems and their Hermitian counterparts,  arXiv e-prints , arXiv–2206 (2022).
  * [9] M. Huang, R.-K. Lee, Q.-H. Wang, G.-Q. Zhang, and J. Wu,  Solvable dilation model of time-dependent PT-symmetric systems,  Phys. Rev. A 105(6), 062205 (2022).
  * [10] C. Figueira de Morisson Faria and A. Fring,  Time evolution of non-Hermitian Hamiltonian systems,  J. Phys. A 39, 9269–9289 (2006).
  * [11] A. Mostafazadeh,  Time-dependent pseudo-Hermitian Hamiltonians defining a unitary quantum system and uniqueness of the metric operator,  Phys. Lett. B 650(2), 208–212 (2007).
  * [12] M. Znojil,  Time-dependent version of crypto-Hermitian quantum theory,  Phys. Rev. D 78(8), 085003 (2008).
  * [13] J. Gong and Q.-H. Wang,  Time-dependent PT-symmetric quantum mechanics,  J. Phys. A: Math. and Theor. 46(48), 485302 (2013).
  * [14] A. Fring and M. H. Y. Moussa,  Unitary quantum evolution for time-dependent quasi-Hermitian systems with nonobservable Hamiltonians,  Phys. Rev. A 93(4), 042114 (2016).
  * [15] A. Mostafazadeh,  Time-dependent pseudo-Hermitian hamiltonians and a hidden geometric aspect of quantum mechanics,  Entropy 22(4), 471 (2020).
  * [16] A. Fring and R. Tenney,  Spectrally equivalent time-dependent double wells and unstable anharmonic oscillators,  Phys. Lett. A , 126530 (2020).
  * [17] A. Fring and R. Tenney,  Exactly solvable time-dependent non-Hermitian quantum systems from point transformations,  Phys. Lett. B 410, 127548 (2021).
  * [18] A. Fring and R. Tenney,  Infinite series of time-dependent Dyson maps,  J. Phys. A: Math. and Theor. 54(48), 485201 (2021).
  * [19] A. Mostafazadeh,  Energy observable for a quantum system with a dynamical Hilbert space and a global geometric extension of quantum theory,  Phys. Rev. D 98(4), 046022 (2018).
  * [20] A. Fring,  An introduction to PT-symmetric quantum mechanics–time-dependent systems,  preprint arXiv:2201.05140, to appear J. Phys.: Conf. Ser. (2022).
  * [21] C. M. Bender, D. C. Brody, and H. F. Jones,  Complex Extension of Quantum Mechanics,  Phys. Rev. Lett. 89, 270401(4) (2002).
  * [22] A. Fring, T. Taira, and R. Tenney,  Time-dependent C-operators as Lewis-Riesenfeld invariants in non-Hermitian theories,  Phys. Lett. A 452, 128458 (2022).
  * [23] M. V. Berry,  Quantal phase factors accompanying adiabatic changes,  Proc. Royal Soc. A 392(1802), 45–57 (1984).
  * [24] J. C. Garrison and E. Wright,  Complex geometrical phases for dissipative systems,  Phys. Lett. A 128(3-4), 177–181 (1988).
  * [25] S.-D. Liang and G.-Y. Huang,  Topological invariance and global Berry phase in non-Hermitian systems,  Phys. Rev. A 87(1), 012118 (2013).
  * [26] E. J. Pap, D. Boer, and H. Waalkens,  A unified view on geometric phases and exceptional points in adiabatic quantum mechanics,  SIGMA 18, 003 (2022).
  * [27] N. Amaouche, M. Sekhri, R. Zerimeche, M. Maamache, and J.-Q. Liang,  Non-Hermitian Hamiltonian beyond PT-symmetry for time-dependant SU (1, 1) and SU (2) systems–exact solution and geometric phase in pseudo-invariant theory,  arXiv preprint arXiv:2207.02477 (2022).
  * [28] D.-J. Zhang, Q.-H. Wang, and J. Gong,  Time-dependent PT-symmetric quantum mechanics in generic non-Hermitian systems,  Phys. Rev. A 100(6), 062121 (2019).
  * [29] S. Cheniti, W. Koussa, A. Medjber, and M. Maamache,  Adiabatic theorem and generalized geometrical phase in the case of pseudo-Hermitian systems,  J. Physics A: Math. and Theor. 53(40), 405302 (2020).
  * [30] M. Maamache,  Periodic pseudo-Hermitian Hamiltonian: nonadiabatic geometric phase,  Phys. Rev. A 92(3), 032106 (2015).