# Relation between the Berry phase in quantum hermitian and non-hermitian
systems and the Hannay phase in the equivalent classical systems.

H. Fanchiotti  C.A. García Canal  IFLP(CONICET) and Departamento de Física,
Universidad Nacional de La Plata, C.C. 67 1900, La Plata, Argentina  M.
Mayosky  LEICI, Departamento de Electrotecnia Facultad de Ingeniería,
Universidad Nacional de La Plata, La Plata, Argentina and Comisión de
Investigaciones Científicas de la Provincia de Buenos Aires-CICpBA, Argentina
A. Veiga  LEICI, Departamento de Electrotecnia Facultad de Ingeniería,
Universidad Nacional de La Plata, La Plata, Argentina; CONICET  V. Vento
Departamento de Física Teórica-IFIC. Universidad de Valencia-CSIC, E-46100,
Burjassot (Valencia), Spain

###### Abstract

The well-known geometric phase present in the quantum adiabatic evolution
discovered by Berry many years ago has its analogue, the Hannay phase, in the
classical domain. We calculate the Berry phase with examples for quantum
hermitian and non-hermitian $PT$-symmetric Hamiltonians and compare with the
Hannay phase in their classical equivalents. We use the analogy to propose
resonant electric circuits which reproduce the theoretical solutions in
simulated laboratory experiments.

###### pacs:

03.65.-v, 03.65Vf, 45.05+x, 45.20-d

##  I Introduction

In a recent paper AJP  the measurement of the geometric Hannay angle Hannay
was studied in connection with the Foucault pendulum and an analog electric
circuit. This analysis induced us to find the explicit connection between the
Hannay phase and the well known quantum Berry phase Berry  on the basis of the
mathematical equivalence between classical and quantum dynamics presented in
ref. AP . In here we generalize the relation between the two phases for
several interesting quantum systems and describe the analog electric circuits.
For that purpose we start by recalling in section II the mathematical
formalism of decomplexification Arnold  and the construction of equivalent
classical equations of motion for effective hamiltonians AP . In section III
we calculate the Berry phase in a quantum adiabatic evolution process for
hermitian Hamiltonians by solving exactly the Schrödinger equation and compare
the result with previous work with classical Hamiltonians leading to the
Hannay phase. We find explicitly the well known relation between the Berry
phase and the classical Hannay angle Berry1 . We generalize the calculation
and the comparison to PT-symmetric quantum non-hermitian Hamiltonians. In
Section IV we establish the correspondence of the quantum computation with the
equivalent classical analogs which lead to equations which are similar to
those of resonant electric circuits coupled by gyrators. These results lead us
to propose and simulate laboratory experiments for measuring the equivalent
geometric phases in section V. We draw some conclusions of our study in the
final section.

##  II Classical analogs of quantum systems

We describe next the decomplexification formalism which defines a classical
analog of a quantum system in order to establish a connection between quantum
and classical dynamics, which result ultimately in an analog description of
the systems in terms of electric circuits. This analogy leads to feasible
laboratory experiments which helps one understand complex concepts like
geometric phases.

###  II.1 Decomplexification of quantum Hamiltonians

Let us recall the decomplexification procedure Arnold  of quantum
Hamiltonians. The quantum evolution of a system is governed by the Schrödinger
equation

$i\,\frac{\partial}{\partial t}\Psi=H\,\Psi.$ (1)

We use $\hbar=1$ unless a quantum effect needs to be emphasized. For a two
component system the Hamiltonian $H$ is a $2\times 2$ matrix with matrix
elements that in general are complex and eventually time-dependent. Moreover,
$H$ can be a hermitian or a non-hermitian operator treated as an effective
Hamiltonian. $\Psi$ in this case is a two component vector.

Let us write explicitly the Schrödinger equation in terms of real and
imaginary parts of both $\Psi$ and $H$,

$\Psi=\left[\begin{array}[]{c}\Psi_{1}\\\
\Psi_{2}\end{array}\right]=\left[\begin{array}[]{c}Re(\Psi_{1})\\\
Re(\Psi_{2})\end{array}\right]+i\left[\begin{array}[]{c}Im(\Psi_{1})\\\
Im(\Psi_{2})\end{array}\right]$

and

$H=\left[\begin{array}[]{cc}H_{11}&H_{12}\\\
H_{21}&H_{22}\end{array}\right]=\left[\begin{array}[]{cc}Re(H_{11})&Re(H_{12})\\\
Re(H_{21})&Re(H_{22})\end{array}\right]+i\left[\begin{array}[]{cc}Im(H_{11})&Im(H_{12})\\\
Im(H_{21})&Im(H_{22})\end{array}\right]$

In order to simplify the notation, we introduce

$Re(\Psi_{i})=x_{i}\,\,\,;\,\,\,Im(\Psi_{i})=y_{i}\,\,;\,i=1,2\,,$
$Re(H_{ij})=f_{ij}\,\,\,;\,\,\,Im(H_{ij})=g_{ij}\,\,;\,i,j=1,2\,.$

Separating real and imaginary parts we get

$\left[\begin{array}[]{c}\dot{x}_{1}\\\
\dot{x}_{2}\end{array}\right]=\left[\begin{array}[]{cc}g_{11}&g_{12}\\\
g_{21}&g_{22}\end{array}\right]\left[\begin{array}[]{c}x_{1}\\\
x_{2}\end{array}\right]+\left[\begin{array}[]{cc}f_{11}&f_{12}\\\
f_{21}&f_{22}\end{array}\right]\left[\begin{array}[]{c}y_{1}\\\
y_{2}\end{array}\right]$ (2)

and

$\left[\begin{array}[]{c}\dot{y}_{1}\\\
\dot{y}_{2}\end{array}\right]=\left[\begin{array}[]{cc}g_{11}&g_{12}\\\
g_{21})&g_{22}\end{array}\right]\left[\begin{array}[]{c}y_{1}\\\
y_{2}\end{array}\right]-\left[\begin{array}[]{cc}f_{11}&f_{12}\\\
f_{21}&f_{22}\end{array}\right]\left[\begin{array}[]{c}x_{1}\\\
x_{2}\end{array}\right].$ (3)

Using matrix notation we can write these equations as,

$\left[\begin{array}[]{c}\dot{x}_{1}\\\ \dot{y}_{1}\\\ \dot{x}_{2}\\\
\dot{y}_{2}\end{array}\right]=\left[\begin{array}[]{cccc}\;\;\,g_{11}&f_{11}&\;\;\,g_{12}&f_{12}\\\
-f_{11}&g_{11}&-f_{12}&g_{12}\\\ \;\;\,g_{21}&f_{21}&\;\;\,g_{22}&f_{22}\\\
-f_{21}&g_{21}&-f_{22}&g_{22}\end{array}\right]\left[\begin{array}[]{c}x_{1}\\\
y_{1}\\\ x_{2}\\\ y_{2}\end{array}\right].$ (4)

This result coincides with the general case presented in ref. Arnold  when
restricted to two dimensions.

Let us introduce further simplifying notation

$A=\left[\begin{array}[]{cc}g_{11}&g_{12}\\\
g_{21}&g_{22}\end{array}\right]\,\,\,\,\,\,;\,\,\,\,\,\,B=\left[\begin{array}[]{cc}f_{11}&f_{12}\\\
f_{21}&f_{22}\end{array}\right]$

and

$x=\left[\begin{array}[]{c}x_{1}\\\
x_{2}\end{array}\right]\,\,\,\,\,\,;\,\,\,\,\,\,y=\left[\begin{array}[]{c}y_{1}\\\
y_{2}\end{array}\right]$

Consequently, the first order differential equations Eqs.( 2) and (4) become

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle A\,x+B\,y$
$\displaystyle\dot{y}$ $\displaystyle=$ $\displaystyle A\,y-B\,x$

These equations can be easily transformed into two separated second order
differential equations for $x$ and $y$, namely

$\displaystyle\ddot{x}$ $\displaystyle=$
$\displaystyle(A+B\,A\,B^{-1})\,\dot{x}-(B\,A\,B^{-1}\,A\,+B\,B)\,x,$
$\displaystyle\ddot{y}$ $\displaystyle=$
$\displaystyle(A-B\,A\,B^{-1})\,\dot{y}+(B\,A\,B^{-1}\,A\,-B\,B)\,y.$ (5)

From the mathematical point of view, we have transformed the quantum evolution
equation into a pair of equations for coupled harmonic oscillators. Certainly,
the above procedure can be immediately extended to the case of a Hamiltonian
of finite arbitrary dimension. The connection between the Schrödinger equation
and harmonic oscillators has been obtained also by using different techniques
others .

###  II.2 Effective Hamiltonian

Let us apply the formalism to the standard expression for an effective
Hamiltonian of the form

$H=M-i\,\Gamma,$ (6)

where for two degrees of freedom $M$ and $\Gamma$ are $2\times 2$ hermitian
matrices. These matrices can be diagonalized and their corresponding
eigenvalues are real.

We are interested in the behavior of the probability densities in different
situations associated with the form of $M$ and of $\Gamma$. From the
Schrödinger equation and its adjoint, one immediately finds that the modulus
of the state function evolves in time according to

$\frac{d}{dt}\left(\Psi^{\dagger}\Psi\right)=-2\Psi^{\dagger}\Gamma\Psi.$

This expression shows that the probability density is controlled by $\Gamma$.,
i.e., the probability will stay constant or will decay in time according to
the structure of the $\Gamma$-matrix.

The characteristic polynomial of this matrix is given by

$P(\lambda)=\lambda^{2}-Tr(\Gamma)\lambda+\det(\Gamma)$

thus, the evolution of the probability density, driven by the eigenvalues of
$\Gamma$, depends on the trace and the determinant of the $\Gamma$ matrix.

Let us study first the case for which $Tr(\Gamma)>0$ and $\det(\Gamma)>0$
which imply that its eigenvalues are positive. Then, the most simple form of
the matrix is

$\Gamma=\left[\begin{array}[]{cc}s&0\\\ 0&s\end{array}\right],$ (7)

and with this shape of $\Gamma$ the probability density will decrease in time.

The other possibility is when $Tr(\Gamma)=0$ and $\det(\Gamma)<0$.
Consequently, the simple form of $\Gamma$ reads

$\Gamma=\left[\begin{array}[]{cc}-s&0\\\ 0&s\end{array}\right].$ (8)

This particular form implies that the evolution is given by

$\frac{d}{dt}\left(\Psi^{\dagger}\Psi\right)=-2\,s\left(\Psi_{1}^{\ast}\Psi_{1}-\Psi_{2}^{\ast}\Psi_{2}\right).$

Here, differently from the previous case, the term on the right has not a
definite sign, because we can also write a $\Gamma$-matrix with the same
properties with the positions of $s$ and of $-s$ exchanged,

$\Gamma=\left[\begin{array}[]{cc}s&0\\\ 0&-s\end{array}\right],$ (9)

with the result,

$\frac{d}{dt}\left(\Psi^{\dagger}\Psi\right)=2\,s\left(\Psi_{1}^{\ast}\Psi_{1}-\Psi_{2}^{\ast}\Psi_{2}\right).$

Consequently, both equalities can only be satisfied if

$\Psi_{1}^{\ast}\Psi_{1}-\Psi_{2}^{\ast}\Psi_{2}=0.$

Thus in this case the probability density is conserved, even if the
Hamiltonian has the imaginary part $\Gamma$, certainly of this very special
structure. This kind of Hamiltonian is an example of the well-known
$PT$-symmetric non-hermitian Hamiltonian PT  that we shall discuss in detail
below.

We continue with the analysis considering a hermitian two-dimensional
Hamiltonian with equal diagonal elements

$H=\left[\begin{array}[c]{cc}h&f-ig\\\ f+ig&h\end{array}\right]$ (10)

then, the matrices $A$ and $B$ above are simply

$A=\left[\begin{array}[c]{cc}0&-g\\\
g&0\end{array}\right]\,\,\,\,\,;\,\,\,\,\,B=\left[\begin{array}[c]{cc}h&f\\\
f&h\end{array}\right].$

These matrices define the equivalent coupled harmonic oscillator problem
Eq.(II.1).

Let us discuss some examples:

i) The case: $g=0$.

This case is related with the classical analog of two resonant electric
circuits coupled inductively Schindler . It has also a mechanical analog
consisting of two pendulums coupled by means of a spring. The analogy is
easily seen from the corresponding values of the $A$ and $B$ matrices,

$A=\left[\begin{array}[c]{cc}0&0\\\
0&0\end{array}\right]\,\,\,\,\,;\,\,\,\,\,B=\left[\begin{array}[c]{cc}h&f\\\
f&h\end{array}\right].$

ii) The case $f=0$.

This case has the classical analog of two resonant electric circuits coupled
by means of a gyrator. Remember that a gyrator is a non-reciprocal passive
element of two ports Bala  with a given conductance. In this case, again the
relation is fixed by the corresponding $A$ and $B$ matrices,

$A=\left[\begin{array}[c]{cc}0&-g\\\
g&0\end{array}\right]\,\,\,\,\,;\,\,\,\,\,B=\left[\begin{array}[c]{cc}h&0\\\
0&h\end{array}\right].$

In both cases, the differential equations are written in terms of the state
variables, voltage in each partial circuit. It is worth mentioning that this
circuit network is also related to a dimer, an oligomer consisting of two
monomers joined by bonds dimer . When the coupling is inductive, it appears
directly related to the voltage variable, while for the case of the gyrator,
the coupling is connected to the first derivative of voltages in the
differential equations. This last case was analyzed in AJP  in connection with
the Foucault pendulum.

##  III Berry and Hannay geometrical phases

We will come back to the relation between the quantum systems and the
classical analog of resonant electric circuits but next we proceed to study
the Berry phase, the geometric phase associated to the adiabatic quantum
evolution of our effective Hamiltonian, by solving the Schrödinger equation,
and will compare it with the Hannay phase, the classical geometric phase
analog Hannay .

###  III.1 Hermitian Hamiltonian

In conventional Hilbert spaces associated to quantum dynamical systems driven
by a hermitian Hamiltonian, a general two-dimensional quantum state can be
written as

$\left|\Psi(t)\right\rangle=a(t)\left|1\right\rangle+b(t)\left|2\right\rangle,$
(11)

where $\left|1\right\rangle$ and $\left|2\right\rangle$ stand for the
eigenstates of $H$ with eigenvalues $\lambda_{1}$ and $\lambda_{2}$
respectively. The initial state at $t=0$, when expressed on the Bloch sphere
of the Hilbert space reads

$\left|\Psi(0)\right\rangle=\cos\frac{\theta_{0}}{2}\left|1\right\rangle+\sin\frac{\theta_{0}}{2}\,e^{i\varphi_{0}}\left|2\right\rangle.$

We now replace in the time dependent Schrödinger equation

$i\frac{d}{dt}\left|\Psi(t)\right\rangle=H\,\left|\Psi(t)\right\rangle,$

the general solution (11) and get the following differential equations for
$a(t)$ and $b(t)$

$i\dot{a}(t)=\lambda_{1}a(t)\,\,\,\,\,;\,\,\,\,\,i\dot{b}(t)=\lambda_{2}b(t),$

whose solutions are

$a(t)=\alpha\,e^{-i\lambda_{1}\,t}\,\,\,\,\,;\,\,\,\,\,b(t)=\beta\,e^{-i\lambda_{2}\,t}.$

Thus the general solution results

$\left|\Psi(t)\right\rangle=\alpha\,e^{-i\lambda_{1}\,t}\left|1\right\rangle+\beta\,e^{-i\lambda_{2}\,t}\left|2\right\rangle$

where the parameters $\alpha$ and $\beta$ are fixed by the initial conditions
and become

$\alpha=\cos\frac{\theta_{0}}{2}\,\,\,\,\,;\,\,\,\,\,\beta=\sin\frac{\theta_{0}}{2}\,e^{i\varphi_{0}}.$

Finally, the general solution is

$\left|\Psi(t)\right\rangle=\cos\frac{\theta_{0}}{2}\,e^{-\lambda_{1}\,t}\left|1\right\rangle+\sin\frac{\theta_{0}}{2}\,e^{i\varphi_{0}}e^{-i\lambda_{2}\,t}\left|2\right\rangle,$

that we write as

$\left|\Psi(t)\right\rangle=e^{-\lambda_{1}i\,t}\left(\cos\frac{\theta_{0}}{2}\left|1\right\rangle+\sin\frac{\theta_{0}}{2}\,e^{i\varphi_{0}}e^{i\,t(\lambda_{1}-\lambda_{2})}\left|2\right\rangle\right).$
(12)

From this expression we can determine that the system under consideration will
return to the initial state after a time $T$ given by

$T\,(\lambda_{1}-\lambda_{2})=2\,\pi,$

when the state takes the form

$\left|\Psi(T)\right\rangle=e^{-i\lambda_{1}\,T}\left(\cos\frac{\theta_{0}}{2}\left|1\right\rangle+\sin\frac{\theta_{0}}{2}\,e^{i\varphi_{0}}\left|2\right\rangle\right)=e^{-i\lambda_{1}\,T}\left|\Psi(0)\right\rangle.$

This expression shows that after the evolution, the quantum system returns to
the initial state but having acquired a total extra phase

$\varphi_{total}=-\lambda_{1}\,T=-2\,\pi\frac{\lambda_{1}}{(\lambda_{1}-\lambda_{2})}.$
(13)

This total phase includes besides the standard dynamical phase, another one,
the Berry phase which is of geometrical origin. In order to detect this
geometrical phase we compute the dynamical phase associated with the adiabatic
motion

$\varphi_{dynamic}=-\int_{0}^{T}\left\langle\Psi(t)\right|H\left|\Psi(t)\right\rangle
dt,$

giving rise to

$\varphi_{dynamic}=-2\,\pi\,\frac{\lambda_{1}}{(\lambda_{1}-\lambda_{2})}+2\,\pi\,\sin^{2}\frac{\theta_{0}}{2}.$
(14)

The Berry geometric phase Berry , is obtained from Eqs. (13) and (14)

$\varphi_{geometric}=-2\,\pi\,\frac{\lambda_{1}}{(\lambda_{1}-\lambda_{2})}-\varphi_{dynamic}=-2\,\pi\,\sin^{2}\frac{\theta_{0}}{2}=\pi,(\cos\theta_{0}-1)$
(15)

where its geometric origin is evident since it is independent of the dynamics
determined by $H$. Moreover, this Berry phase coincides, except for a factor
$2$ and a change of sign, with the Hannay phase Hannay  of the classical
equivalent system, the Foucault pendulum, as it was determined in AJP ,

$\triangle\phi=2\,\pi-\eta=2\,\pi\,(1-\sin(\lambda))=2\,\pi\,(1-\cos(\theta))$
(16)

where the phase is written in terms of the colatitude $\theta$ as it is
usually presented.

Eq.(16) explicitly shows the connection between the classical Hannay angle and
the quantum Berry phase, namely

$\triangle\phi=-2\varphi_{geometric}.$ (17)

This is a particular case of the general result Berry1 ; BH

$\triangle\phi_{l}=\frac{\partial\varphi_{Hannay}}{\partial n_{l}}.$ (18)

The factor 2 in Eq.(17) comes from the quantum vacuum contribution to the
energy which in the harmonic oscillator is
$E_{n}\sim(n+\frac{1}{2})\hbar\omega$ Berry1 .

###  III.2 Non-hermitian $PT$-symmetric Hamiltonian

The basic property of the operators that represent observables in Quantum
Mechanics is hermiticity, because these operators have real eigenvalues that
can be considered measurable quantities related to the corresponding physical
magnitudes. Whenever open systems exhibit flows of energy, particles and
information, they are described by non-hermitian Hamiltonians, in general
associated with the decay of the norm of a quantum state. Among the non-
hermitian Hamiltonians, those that obey parity-time ($PT$) symmetry are of
particular interest because they can admit real eigenvalues while describing
physical open systems which present balanced loss into and gain from the
surrounding environment. Besides, as a parameter, let us call it $\gamma$,
that is associated with the degree of non-hermiticity of $H$, changes, a
spontaneous $PT$-symmetry breaking occurs (see the Appendix) and the real
properties of eigenvalues are lost, they become complex PT .

Let us study the two dimensional system characterized by

$H=M-i\,\Gamma=\left[\begin{array}[c]{cc}a&-ig\\\
ig&a\end{array}\right]-i\left[\begin{array}[c]{cc}-s&0\\\
0&s\end{array}\right]$ (19)

having eigenvalues

$a+\sqrt{\left(-s^{2}+g^{2}\right)}\,\,\,\,;\,\,\,\,a-\sqrt{\left(-s^{2}+g^{2}\right)}$

that can be written as

$a+g\sqrt{\left(-\gamma^{2}+1\right)}\,\,\,\,;\,\,\,\,a-g\sqrt{\left(-\gamma^{2}+1\right)}$

with

$\gamma=\frac{s}{g}$

Notice that in the limit $\gamma\rightarrow 0$ the hamiltonian becomes
hermitian. If $\gamma^{2}\leq 1$, both eigenvalues are real and the
$PT$-symmetry present in the Hamiltonian is also present in the solutions. The
symmetry is unbroken. The particular value $\gamma^{2}=1$, is known as an
exceptional point where the spontaneous breaking of the $PT$-symmetry appears
and the eigenvalues from then on become complex conjugate quantities.

In order to clarify the analysis of the geometric phase and due to the fact
that the $M$ and $\Gamma$ components of $H$ do not commute, it is necessary to
use the biorthogonal quantum formalism Brody .

Consider the symmetric situation. The dynamical equation is given by

$i\,\frac{\partial}{\partial
t}\left|\Psi\right\rangle=H\,\left|\Psi\right\rangle=(M-i\,\Gamma)\,\left|\Psi\right\rangle$

with the initial condition

$\left|\Psi\left(0\right)\right\rangle=\cos\left(\frac{\theta}{2}\right)\left|\phi_{1}\right\rangle+\sin\left(\frac{\theta}{2}\right)e^{i\varphi}\left|\phi_{2}\right\rangle$

where $\left|\phi_{1}\right\rangle$ and $\left|\phi_{2}\right\rangle$ are the
eigenstates of $H$. Recall that these states are not orthogonal but they
define a complete basis on the $2\times 2$ space.

The general solution of the dynamics is of the form

$\left|\Psi\left(t\right)\right\rangle=c_{1}e^{-i\kappa_{1}t}\left|\phi_{1}\right\rangle+c_{2}e^{-i\kappa_{2}t}\left|\phi_{2}\right\rangle$

where $\kappa_{1}$ and $\kappa_{2}$ stand for the eigenvalues of $H$
corresponding to the mentioned eigenfunctions. One can adjust the coefficients
$c_{1}$ and $c_{2}$ in order to reproduce the chosen initial conditions. Then

$c_{1}=\cos\left(\frac{\theta}{2}\right)\,\,\,\,;\,\,\,\,c_{2}=\sin\left(\frac{\theta}{2}\right)e^{i\varphi}$

in an entirely similar way as for the hermitian case.

Now we determine the time $T$ taken by the system after the evolution to go
back to the initial situation. To do this, we write the state as

$\left|\Psi(t)\right\rangle=e^{-\kappa_{1}i\,t}\left(\cos\frac{\theta}{2}\left|1\right\rangle+\sin\frac{\theta}{2}\,e^{i\varphi}e^{i\,t(\kappa_{1}-\kappa_{2})}\left|2\right\rangle\right)$

and conclude that

$T\,(\kappa_{1}-\kappa_{2})=2\pi$

and that after this time $T$ the state acquires and extra phase because

$\left|\Psi(T)\right\rangle=e^{-i\Phi}\left|\Psi\left(0\right)\right\rangle$

with

$\Phi=2\pi\frac{\kappa_{1}}{(\kappa_{1}-\kappa_{2})}$

The next step is to extract from $\Phi$ the geometric phase. In order to do so
we need to calculate the dynamical phase which is computed from an expression
whose mathematical structure is slightly different from the hermitian case due
to the non orthogonality of the used base vectors Brody , i.e.

$\Phi_{dynamic}=-\int_{0}^{T}\langle\tilde{\Psi}(t)|H|\Psi(t)\rangle/\langle\tilde{\Psi}(t)|\Psi(t)\rangle
dt$

where $\langle\tilde{\Psi}(t)|$ is the state but now expressed in the base of
$H^{{\dagger}}$, the hermitian conjugate of $H$,

$\langle\tilde{\Psi}(t)|=\tilde{c}_{1}e^{i\bar{\kappa}_{1}t}\langle\chi_{1}|+\tilde{c}_{2}e^{i\bar{\kappa}_{2}t}\langle\chi_{2}|$

and $\langle\chi_{1}|$ and $\langle\chi_{2}|$ are the members of the non-
orthogonal base of $H^{{\dagger}}$. Since
$\left\langle\chi_{n}|\phi_{m}\right\rangle=\delta_{nm}$
$\left\langle\chi_{n}|\phi_{n}\right\rangle$ and one can normalize the vector
product $\langle\chi_{n}|\phi_{n}\rangle=1$ Brody  , then

$\displaystyle\langle\tilde{\Psi}(t)|H|\Psi(t)\rangle$ $\displaystyle=$
$\displaystyle\tilde{c}_{1}c_{1}\kappa_{1}e^{i(\tilde{\kappa}_{1}-\kappa_{1})t}+\tilde{c}_{2}c_{2}\kappa_{2}e^{i(\tilde{\kappa}_{2}-\kappa_{2})t}$

Since we are in the region where the $PT$-symmetry is unbroken and
consequently the eigenvalues are real, one has

$\langle\tilde{\Psi}(t)|H\left|\Psi(t\right\rangle=\tilde{c}_{1}c_{1}\kappa_{1}+\tilde{c}_{2}c_{2}\kappa_{2}=\cos^{2}\left(\frac{\theta}{2}\right)\kappa_{1}+\sin\left(\frac{\theta}{2}\right).\kappa_{2}$

Since

$\langle\tilde{\Psi}(t)|\Psi(t)\rangle=1,$

one concludes that we have recovered the result of the the hermitian case.
Consequently the dynamical phase and the geometric phase are the same as in
the hermitian case. This conclusion is only valid in the $\gamma$-parameter
region where the $PT$-symmetry is valid.

##  IV Classical analogs with resonant electric circuits

We now revisit the geometric phase from the point of view of the classical
analog Eqs.(II.1) discussed previously. We recall initially the analysis of
the Hannay phase in the classical problem to proceed later on with the Berry
phase in quantum systems with a PT-symmetric non-hermitian Hamiltonian.

###  IV.1 Classical Foucault pendulum

The differential equations of the electric system equivalent to the Foucault
pendulum includes a coupling by means of a gyrator AJP ,

$\left[\begin{array}[c]{c}\ddot{x}_{1}\\\
\ddot{x}_{2}\end{array}\right]+\left[\begin{array}[c]{cc}0&-a\\\
a&0\end{array}\right]\left[\begin{array}[c]{c}\dot{x}_{1}\\\
\dot{x}_{2}\end{array}\right]+\left[\begin{array}[c]{cc}\omega^{2}&0\\\
0&\omega^{2}\end{array}\right]\left[\begin{array}[c]{c}x_{1}\\\
x_{2}\end{array}\right]=0$ (20)

with $a=\gamma=2\Omega\sin\lambda=\frac{G}{C}$ and $\omega^{2}=\frac{1}{LC}$.
The procedure is to diagonalize the coupling term to end with a pair of second
degree uncoupled equations.

Diagonalizing the coupling term

$\left[\begin{array}[c]{cc}0&-a\\\ a&0\end{array}\right],$

we obtain its eigenvalues $\pm ia$, and its eigenvectors that lead to the
equivalent uncoupled equations

$\left[\begin{array}[c]{c}\ddot{y}_{1}\\\
\ddot{y}_{2}\end{array}\right]+i\left[\begin{array}[c]{cc}a&0\\\
0&-a\end{array}\right]\left[\begin{array}[c]{c}\dot{y}_{1}\\\
\dot{y}_{2}\end{array}\right]+\left[\begin{array}[c]{cc}\omega^{2}&0\\\
0&\omega^{2}\end{array}\right]\left[\begin{array}[c]{c}y_{1}\\\
y_{2}\end{array}\right]=0.$ (21)

The new variables $y$ are related to the previous $x$ through

$\left[\begin{array}[c]{c}y_{1}\\\
y_{2}\end{array}\right]=\left[\begin{array}[c]{cc}1&i\\\
1&-i\end{array}\right]\left[\begin{array}[c]{c}x_{1}\\\
x_{2}\end{array}\right]=\allowbreak\left[\begin{array}[c]{c}x_{1}+ix_{2}\\\
x_{1}-ix_{2}\end{array}\right]$

The solution for the upper equation is $y_{1}=e^{i\Lambda t}$. Where the
values of $\Lambda$, solutions of $\Lambda^{2}+a\Lambda-\omega^{2}=0$ are

$\Lambda_{\pm}=\frac{1}{2}\,\left[-a\pm\sqrt{a^{2}+4\omega^{2}}\right]$

under the standard approximation $\omega\gg a$ imposed by the physics of the
considered systems, reduce to

$\Lambda_{\pm}\simeq-a/2\pm\sqrt{a^{2}+4\omega^{2}}/2=a/2\pm\omega$

Consequently, the general solution is

$y_{1}=Ae^{i\left(a/2+\omega\right)t}+Be^{i\left(a/2-\omega\right)t}=e^{iat/2}\,\left(A\,e^{-i\omega
t}+B\,e^{i\omega t}\right)$

It is clear that at $t=0$, $y_{1}\left(0\right)=A+B$, while at a time $T$ is

$y_{1}\left(T\right)=e^{iaT/2}\left(A\,e^{-i\omega T}+B\,e^{i\omega T}\right)$

if $T$ is chosen as $T=2\pi/\Omega$ (remember the Foucault pendulum) and due
to the fact that $\omega\gg\Omega$, one has

$\frac{\omega}{\Omega}=n\gg 1$

This shows that the solution comes back to the initial state but in the
evolution it has acquired an extra phase AJP

$y_{1}\left(T\right)=e^{iaT/2}y_{1}\left(0\right)=e^{i\,\Omega\,\sin\lambda\,T/2}y_{1}\left(0\right)$

###  IV.2 Classical analog of a simple quantum $PT$-symmetric model

In the case of $PT$-symmetric systems, the experimental study in terms of
electric circuits was pioneered in Ref. Schindler . We here consider the
quantum $PT$-symmetric model as a classic counterpart defined by two simple
resonant electric circuits but now coupled by a gyrator. In this case the
dynamical equations read

$\left[\begin{array}[c]{c}\ddot{x}_{1}\\\
\ddot{x}_{2}\end{array}\right]+\left[\begin{array}[c]{cc}-s&-g\\\
g&s\end{array}\right]\left[\begin{array}[c]{c}\dot{x}_{1}\\\
\dot{x}_{2}\end{array}\right]+\left[\begin{array}[c]{cc}\omega^{2}&0\\\
0&\omega^{2}\end{array}\right]\left[\begin{array}[c]{c}x_{1}\\\
x_{2}\end{array}\right]=0$ (22)

As before we diagonalize the coupling term

$\left[\begin{array}[c]{cc}-s&-g\\\ g&s\end{array}\right]$

to obtain the eigenvalues

$\Lambda_{+}=i\sqrt{\left(-s^{2}+g^{2}\right)}=i\tilde{a}\,\,\,\,;\,\,\,\,\Lambda_{-}=-i\sqrt{\left(-s^{2}+g^{2}\right)}=-i\tilde{a}$

where

$\tilde{a}=\sqrt{\left(g^{2}-s^{2}\right)}=g\sqrt{\left(1-\gamma^{2}\right)}$

where $\gamma=s/g$

In fact, we have reduced the equations to the previous hermitian case Eq.(21)
but now in terms of an effective $\tilde{a}$ that clearly goes into the
previous situation in the limit $\gamma=0$

The solution $y_{1}$ now will get the original form in terms of $\tilde{a}$
but with the important detail that in order to come back to the initial state
the required time is not $T$ but a modified one

$\tilde{T}=\frac{2\pi}{\Omega\sqrt{\left(1-\gamma^{2}\right)}}$

This modification implies that the acquired geometric phase is the same as in
the hermitian case, as it is easy to check from

$\tilde{a}\tilde{T}/2=\frac{1}{2}g\sqrt{\left(1-\gamma^{2}\right)}\frac{2\pi}{\Omega\sqrt{\left(1-\gamma^{2}\right)}}=\pi
g/\Omega$

It is important to note that $\tilde{T}$ increases as soon as one is
approaching, by modifying the parameter $s$, the exceptional point where the
$PT$-symmetry is spontaneously broken.

In summary, we have shown that the (classical) Hannay phase is related to the
Berry phase of the equivalent quantum system, in the case of $PT$ symmetry, as
it was in the standard hermitian situation.

Figure 1: Resonant circuits coupled by a Gyrator.

##  V Resonant Electric Circuits

The simulation of electric circuits with equations entirely identical to those
of the examples previously presented gives a further insight on the appearance
and behavior of the geometric phases as an holonomy effect.

Figure 2: Simulation results for two different values of the gyrator (that
correspond to different latitudes of the position of the Foucault pendulum).
Given that initial conditions of both oscillators are equal, both plots start
in the $45^{\circ}$ plane. The Hannay phase is the angle required to reach
$2\pi$ after a period $T$ (the complement of the latitude effect on the
pendulum)

We start with an electric network equivalent of the Foucault pendulum AJP
that is depicted in Fig. 1. It consist of two identical, ideal $LC$
oscillators (without losses) coupled by a gyrator. The simple equations for
voltages $V_{1}$ and $V_{2}$ are

$\left\\{\begin{array}[]{r}\ddot{V}_{1}+V_{1}\frac{1}{LC}-\dot{V}_{2}\frac{G_{g}}{C}=0\\\
\ddot{V}_{2}+V_{2}\frac{1}{LC}+\dot{V}_{1}\frac{G_{g}}{C}=0\\\
\end{array}\right.$ (23)

Defining

$x_{1}=V_{1}\,\,\,;\,\,\,x_{2}=V_{2}\,\,\,;\,\,\,\frac{1}{LC}=\omega_{0}^{2}\,\,\,;\,\,\,\frac{G_{g}}{C}=a$

the expressions take exactly the form of Eq. (20). In Fig. 2 the plot of
$V_{2}$ vs. $V_{1}$ is depicted. In this figure the Hannay phase is $2\,\pi$
minus the angle required to reach the initial position after an adiabatic
closed excursion of period $T$.

The electronic simile of a $PT$-symmetric system based upon a pair of coupled
$LC$ oscillators, one with loss (via $R$) and the other one with gain (via
-$R$) allows the detection of the transition between a real spectrum of
frequencies to a spontaneously broken $PT$-symmetry phase with complex
frequencies Schindler . This simple setup was performed with an inductive
coupling between the oscillators, but it can be performed also when the
coupling is through a gyrator, the non-reciprocal passive element of two ports
with a given conductance Bala . We notice that in this last case where a
gyroscopic effect is present, the classical analog of a quantum $PT$-symmetric
system reproduces all the known phenomena.

Figure 3: Non-hermitian $PT$-symmetric circuit using a Gyrator

The equivalent circuit to the $PT$-symmetric non-hermitic quantum system is
shown in Fig. 3. Resistor $R$ models the flow of energy outside the network,
while $-R$ of equal absolute value, accounts for energy income and the system
is $PT$-symmetric. The equations in this case are

$\left\\{\begin{array}[]{r}\ddot{V}_{1}+V_{1}\frac{1}{LC}-\dot{V}_{1}\frac{1}{RC}-\dot{V}_{2}\frac{G_{g}}{C}=0\\\
\ddot{V}_{2}+V_{2}\frac{1}{LC}+\dot{V}_{2}\frac{1}{RC}+\dot{V}_{1}\frac{G_{g}}{C}=0\\\
\end{array}\right.$ (24)

Again, defining

$x_{1}=V_{1}\,\,\,;\,\,\,x_{2}=V_{2}\,\,\,;\,\,\,\frac{1}{LC}=\omega_{0}^{2}\,\,\,;\,\,\,\frac{G_{g}}{C}=g\,\,\,;\,\,\,\frac{1}{RC}=s$

the equations take the form of Eq. (22). The parameter $s$ accounts for the
presence or not of the $PT$-symmetry. Fig. 4 shows the evolution of the real
and imaginary parts of the eigenvalues as $s$ changes, describing the path to
an exceptional point, where the spontaneous loss of $PT$-symmetry occurs and
the eigenvalues become complex conjugate. In this way our laboratory
simulation shows explicitly the occurrence of a phase transition in this non
hermitian system from a phase in which the $PT$-symmetry is realized to a
phase in which it is spontaneously broken.

Figure 4: Normalized imaginary an real parts vs. normalized degree of
$PT$-symmetry for the non-hermitic circuit

##  VI Conclusions

Both, the well-known Berry phase, the geometric phase present in the quantum
adiabatic evolution and its classical analogue, the Hannay phase where
computed and analyzed in the simple case of a finite dimensional dynamics. The
classical and quantum geometric phases for equivalent cases are related
through a very precise numerical factor that was previously predicted Berry1 .
The study was also done for the case of hermitian and non-hermitian but $PT$
symmetric quantum Hamiltonians and the corresponding classical mathematically
equivalent dynamics. It is worth mentioning that in the region where the
$PT$-symmetry is present in the solution, the geometric phase agrees with the
one obtained in the hermitian case.

Taking profit of the stricto sensu equivalence between classical and quantum
dynamics AP  was possible to build up resonant electric circuits, coupled by
means of gyrators, that reproduce exactly the theoretical solutions obtained.
In this way one is able to show explicitly a quantum mechanical phase
transition from a phase in which a symmetry is realized to a phase in which it
is spontaneously broken. Moreover, our construction allows one to discriminate
between couplings and to show that the gyrator coupling is favored not only by
its simpler analysis but also for pedagogical reasons, because, as it was
previously shown AJP ; M , in this case the systems can be put in one to one
correspondence with the Foucault pendulum.

## Acknowledgements

HF and CAGC were partially supported by ANPCyT, Argentina. VV was supported by
MCIN/AEI/10.13039/501100011033, European Regional Development Fund Grant No.
PID2019-105439 GB-C21 and by GVA PROMETEO/2021/083 .

## Appendix

As mentioned in the main text, the $PT$-symmetry is spontaneously broken in
terms of a parameter $\gamma$ when it goes through an exceptional point.

As it is well known an operator, as $PT$, represents a symmetry if it commutes
with the Hamiltonian of the system, namely

$\left[H,PT\right]=0$

In the case of $PT$, the Hamiltonian has real eigenvalues below the
exceptional point $\gamma^{\star}$ and complex conjugate ones above it.
Explicitly

$H\Psi_{1,2}=\lambda_{1,2}\Psi_{1,2}$

with $\lambda_{1}$ and $\lambda_{2}$ real for $\gamma<\gamma^{\star}$ and with
complex $\lambda_{2}=\lambda_{1}^{\ast}$ for $\gamma>\gamma^{\star}$

The presence of the symmetry implies that

$H\left(PT\right)\Psi_{1,2}=\left(PT\right)\left(\lambda_{1,2}\Psi_{1,2}\right)$

and in the region where the eigenvalues are real one has

$PT\left(\lambda_{1,2}\Psi_{1,2}\right)=\lambda_{1,2}PT\Psi_{1,2}$

meaning that $PT\Psi_{1,2}$ is also eigenfunction of $H$. Then $\Psi_{1,2}$,
the solution, has the same symmetry present in the Hamiltonian. In this region
the symmetry has a Wigner-Weyl realization.

Above the exceptional point the situation is different because the symmetry
operator also acts on the complex eigenvalues conjugating them.

$H\left(PT\right)\Psi_{1}=\left(PT\right)\left(\lambda_{1}\Psi_{1}\right)=\lambda_{1}^{\ast}\left(PT\right)\Psi_{1}=\lambda_{2}\left(PT\right)\Psi_{1}$

or

$\left(PT\right)\Psi_{1}=\Psi_{2}$

which explicitly shows that the $PT$-symmetry is not present in the solution
and for this reason the spontaneous symmetry breaking appears. The symmetry of
$H$ has a Nambu-Goldstone realization . Notice that we are dealing with a
discrete symmetry and therefore no Goldstone boson appears.

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