The Geometric Phase in Classical Systems and in the Equivalent Quantum Hermitian and Non-Hermitian PT-Symmetric Systems

The decomplexification procedure allows one to show mathematically (stricto sensu) the equivalence (isomorphism) between the quantum dynamics of a system with a finite number of basis states and a classical dynamics system. This unique way of connecting different dynamics was used in the past to analyze the relationship between the well-known geometric phase present in the quantum evolution discovered by Berry and its generalizations, with their analogs, the Hannay phases, in the classical domain. In here, this analysis is carried out for several quantum hermitian and non-hermitian PT-symmetric Hamiltonians and compared with the Hannay phase analysis in their classical isomorphic equivalent systems. As the equivalence ends in the classical domain with oscillator dynamics, we exploit the analogy to propose resonant electric circuits coupled with a gyrator, to reproduce the geometric phase coming from the theoretical solutions, in simulated laboratory experiments.


Introduction
We are mainly interested in the connection between classical and quantum systems, and for this reason, we start noticing that the Schrödinger dynamics is defined in a complex space, while the classical dynamics of interest implies real evolution equations.Consequently, it is necessary to take profit from a decomplexification procedure that allows one to work in a common space [1].Using this procedure, it was proved [2] the mathematical equivalence between quantum and classical dynamics defining in a unique way an isomorphism that establishes a correspondence between the description of both systems.
In a recent paper [3], the measurement of the geometric Hannay angle [4] was studied in connection with the Foucault pendulum taking profit of an analog electric circuit with a gyrator that provided a very way of having an analogous of the pendulum in the laboratory.This analysis and connection between electric oscillators and geometric phases induced us to understand and exemplify in simple cases the explicit connection between the classical geometric phase and the well-known quantum phase [5] or its generalizations [6] based on the previously mentioned mathematical equivalence between classical and quantum dynamics presented in refs.[2,7].
In here, we discuss the relation between the quantum and classical geometric phases for several interesting quantum systems described in terms of particular analog electric circuits with a gyrator.We understand that our analysis generalizes and simplifies the results of [8] and [9][10][11].To do so, we start by recalling in Section 2 the mathematical formalism of decomplexification [1] and the construction of equivalent classical equations of motion for effective Hamiltonians [2].In Section 3, we calculate the geometric phase in a quantum 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.In these examples, we find explicitly the verification 143 Page 2 of 11 of the relation between the quantum geometric phase and the classical angle [12].
The non-Hermitian PT-symmetric Hamiltonians [14] have been of continuous interest for more than 20 years because in some range of parameters, they possess real eigenvalues.One possible question, partially answered in Gu et al. [15], is how the well-known geometric phase (Berry [5] or Aharanov and Anandan phase [6]) of this extended quantum mechanics compares with the one for standard hermitian Hamiltonians.It is also of interest to find the relation between these geometric phases and the classical one (Hannay angle [4]) of the mathematically equivalent classical systems correctly obtained [2] via the decomplexification procedure [1].Consequently, we generalize the calculation and the comparison of geometric phases to PT-symmetric quantum non-hermitian Hamiltonians.
In Section 4, we establish the correspondence of the quantum analysis with the equivalent classical analogs which lead to equations that are entirely similar to those of resonant electric circuits coupled by gyrators.These results lead us to propose and simulate simple laboratory experiments for measuring the equivalent geometric phases in Section 5. Similar attempts were carried out in the past with dimer and trimer systems [16].We draw some conclusions of our study in the final section.

Classical Analogs of Quantum Systems
We start by describing the decomplexification formalism which defines a classical analog of a quantum system to establish the connection between quantum and classical dynamics, which results ultimately in an analog description of the systems in terms of electric circuits.This analogy leads to feasible laboratory experiments which help understand complex concepts like geometric phases.

Decomplexification of Quantum Hamiltonians
Let us recall the decomplexification procedure [1] of quantum Hamiltonians.The quantum evolution of a system is governed by the Schrödinger equation We use ℏ = 1 unless a quantum effect needs to be empha- sized.For a two-component system, the Hamiltonian H is a 2 × 2 matrix with matrix elements that are in general com- plex and eventually time-dependent.Moreover, H can be a Hermitian or a non-Hermitian operator treated as an effective Hamiltonian.In this case, Ψ is a two-component vector.
( (4) 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 [17].

Effective Hamiltonian
Let us apply the formalism to the standard expression for an effective Hamiltonian of the form where for two degrees of freedom M and Γ are 2 × 2 Hermi- tian 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 Γ .From the Schrödinger equation and its adjoint, one immediately finds that the modulus of the state function evolves in time according to This expression shows that the probability density is controlled by Γ , i.e., the probability will stay constant or will decay in time according to the structure of the Γ-matrix.
The characteristic polynomial of this matrix is given by thus, the evolution of the probability density, driven by the eigenvalues of Γ , depends on the trace and the determinant of the Γ matrix.
Let us study the cases of physical interest.First, the case for which Tr(Γ) > 0 and det(Γ) > 0 which implies that its eigen- values are positive.Then, the most simple form of the matrix is s > 0 , and with this shape of Γ , the probability density will decrease in time.
A second case is when Tr(Γ) = 0 and det(Γ) < 0 .A simple form of Γ with these properties is This particular form implies that the evolution is given by ( 5) Here, differently from the previous case, the term on the right has not a definite sign.However, a Γ-matrix with the same properties can be written by exchanging the positions of s and of −s, with the result Consequently, both equalities can only be satisfied if Thus, in this case, the probability density is conserved, even if the Hamiltonian has an imaginary part Γ , certainly of this very special structure.This kind of Hamiltonian is an example of the well-known PT-symmetric non-Hermitian Hamiltonian [13] that we shall discuss in detail below.

Let us continue our analysis by considering a Hermitian two-dimensional Hamiltonian with equal diagonal elements
In this case, the matrices A and B above become These matrices define the equivalent coupled harmonic oscillator problem Eq. ( 5).
Let us discuss some examples: (i) The case: g = 0 .This case is related to the classi- cal analog of two resonant electric circuits coupled inductively [8].It has also a mechanical analog consisting of two pendulums coupled by a spring.The analogy is easily seen from the corresponding values of the A and B matrices, (ii) The case f = 0 .This case has the classical analog of two resonant electric circuits coupled by a gyrator.Remember that a gyrator is a non-reciprocal passive element of two ports [22] with a given conductance.
In this case, again, the relation is fixed by the corresponding A and B matrices, In both cases, the differential equations are written in terms of the state variables, a 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 [23].When the coupling is inductive, it appears directly related to the voltage variable, while in 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 [3] in connection with the Foucault pendulum.

Quantum and Classical 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 geometric phase associated with the quantum evolution of our effective Hamiltonian by using the approach of Aharonov and Anandan [6] solving the Schrödinger equation and will compare it with the Hannay angle, the classical geometric phase analog [4].

Hermitian Hamiltonian
In conventional Hilbert spaces associated with quantum dynamical systems driven by a Hermitian Hamiltonian, a general two-dimensional quantum state can be written as where �1⟩ and �2⟩ stand for the eigenstates of H with eigen- values 1 and 2 respectively.The initial state at t = 0 , when expressed on the Bloch sphere of the Hilbert space reads We now replace in the time-dependent Schrödinger equation the general solution Eq. ( 10) and get the following differential equations for a(t) and b(t) whose solutions are Thus, the general solution results where the parameters and are fixed by the initial conditions and become Finally, the general solution is that we write as From this expression, we can determine that the system under consideration will return to the initial state after a time T given by when the state takes the form This expression shows that after the evolution, the quantum system returns to the initial state but having acquired a total extra phase This total phase includes besides the standard dynamical phase, another one, the Berry-like phase which is of geometrical origin.To detect this geometrical phase, we compute the dynamical phase associated with the evolution giving rise to The quantum geometric phase [5,6] is obtained from Eqs. ( 12) and ( 13) .
where its geometric origin is evident since it is independent of the dynamics determined by H.Moreover, this phase coincides, except for a factor 2 and a change of sign, with the Hannay angle [4] of the classical equivalent system, the Foucault pendulum, as it was determined in [3], where the phase is written in terms of the colatitude as it is usually presented.Equation ( 15) explicitly shows the connection between the classical Hannay angle and the quantum phase, namely This is a particular case of the general result [12,24] The factor 2 in Eq. ( 16) comes from the quantum vacuum contribution to the energy which in the harmonic oscillator is

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 [18][19][20].Besides, as a parameter, let us call it , 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 [13].In the PT-symmetric phase, the PT-symmetric Hamiltonian is just an alternative representation of the same system described by a Hermitian Hamiltonian [21].This explains why in the PT-symmetric phase the eigenvalues are real.
Let us study the two-dimensional system characterized by having eigenvalues that can be written as with Notice that in the limit → 0 , the Hamiltonian becomes Hermitian.If 2 ≤ 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 2 = 1 is known as an exceptional point where the sponta- neous breaking of the PT-symmetry appears and the eigenvalues from then on become complex conjugate quantities.
To clarify the analysis of the geometric phase and since the M and Γ components of H do not commute, it is nec- essary to use the biorthogonal quantum formalism [25].
Consider the symmetric situation.The dynamical equation is given by with the initial condition where � � 1 ⟩ and � � 2 ⟩ are the eigenstates of H. Recall that these states are not orthogonal, but they define a complete basis on the 2 × 2 space.
The general solution of the dynamics is of the form where 1 and 2 stand for the eigenvalues of H corresponding to the mentioned eigenfunctions.One can adjust the coefficients c 1 and c 2 to reproduce the chosen initial conditions.Then, in an entirely similar way to 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 and conclude that and that after this time T, the state acquires an extra phase because with The next step is to extract from Φ the geometric phase.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 nonorthogonality of the base vectors used [25], i.e., where ⟨ Ψ(t)� is the state but now expressed in the base of H † , the Hermitian conjugate of H, and ⟨ 1 � and ⟨ 2 � are the members of the non-orthogonal base of H † .Since ⟨ n � m ⟩ = nm ⟨ n � n ⟩ and one can nor- malize the vector product ⟨ n � n ⟩ = 1 [25], then Since we are in the region where the PT-symmetry is unbroken and consequently the eigenvalues are real, one has Because one concludes that we have recovered the result of 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 -parameter the PT-symmetry is valid.

Classical Analogs with Resonant Electric Circuits
We now revisit the geometric phase from the point of view of the classical analog Eq. ( 5) discussed previously.We recall initially the analysis of the Hannay phase in the classical problem to proceed later on with the geometric phase in quantum systems with a PT-symmetric non-Hermitian Hamiltonian.

Classical Foucault Pendulum
The differential equations of the electric system equivalent to the Foucault pendulum include a coupling by a gyrator [3], with a = = 2Ω sin = G C and 2 = 1 LC .The procedure is to diagonalize the coupling term to end with a pair of seconddegree uncoupled equations.
Diagonalizing the coupling term we obtain its eigenvalues ±ia , and its eigenvectors that lead to the equivalent uncoupled equations The new variables y are related to the previous x through The solution for the upper equation is Where the values of Λ , solutions of Λ 2 + aΛ − 2 = 0 are under the standard approximation  ≫ a imposed by the physics of the considered systems (pendulum frequency very much larger than the Earth rotation frequency), reduce to Consequently, the general solution is It is clear that at t = 0 , y 1 (0) = A + B , while at a time T is if T is chosen as T = 2 ∕Ω (remember the Foucault pendu- lum) and since  ≫ Ω , one has This shows that the solution comes back to the initial state, but in the evolution, it has acquired an extra phase [3] 4.2 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 Schindler et al. [8].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 As before, we diagonalize the coupling term to obtain the eigenvalues where where = s∕g In fact, we have reduced the equations to the previous hermitian case Eq. ( 20) but now in terms of an effective ã that clearly goes into the previous situation in the limit = 0 The solution y 1 now will get the original form in terms of ã but with the important detail that to come back to the initial state, the required time is not T but a modified one This modification implies that the acquired geometric phase is the same as in the Hermitian case, as it is easy to check from It is important to note that T increases as soon as one is approaching, by modifying the parameter , the exceptional point where the PT-symmetry is spontaneously broken.y 1 (T) = e iaT∕2 y 1 (0) = e i Ω sin T∕2 y 1 (0) In summary, we have shown that the (classical) Hannay phase is related to the geometric phase of the equivalent quantum system, in the case of PT symmetry, as it was in the standard Hermitian situation, due to the one-to-one relationship between the quantum dynamic equation and the classical one.

Resonant Electric Circuits
The simulation of electric circuits with equations entirely identical to those of the examples previously presented gives further insight on the appearance and behavior of the geometric phases as a holonomy effect.
We start with an electric network equivalent of the Foucault pendulum [3] that is depicted in Fig. 1.It consists of two identical, ideal LC oscillators (without losses) coupled by a gyrator.The simple equations for voltages V 1 and V 2 are Defining the expressions take exactly the form of Eq. ( 19).In Fig. 2, the plot of V 2 vs. V 1 is depicted.In this figure, the Hannay phase is 2 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 [8].This simple setup was performed with 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 [22].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.
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 Again, defining the equations take the form of Eq. ( 21).The parameter s accounts for the presence or not of the PT-symmetry.Figure 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.

Conclusions
The well-known geometric phase discovered by Berry and its generalizations, in particular the Aharanov Anandan phase, present in the quantum evolution and its classical analog, the Hannay angle, were computed, analyzed, and compared in the case of finite-dimensional dynamics.This comparison makes sense when one takes into account the mathematical equivalence between quantum and classical dynamics based on the decomplexification procedure.We explicitly found that the classical and quantum geometric phases for equivalent cases are related through a very precise numerical factor as was previously predicted [12].The study was also done for the case of non-Hermitian but PT symmetric quantum Hamiltonians and the corresponding classical mathematically equivalent dynamics.It is worth mentioning that, as expected, in the region where the PT-symmetry is present in the solution, the geometric phase coincides with the one obtained in the Hermitian case.Taking advantage of the stricto sensu equivalence between classical and quantum dynamics [2] that guarantees the correctness of the procedure, it was possible to build up resonant electric circuits, coupled by gyrators, that reproduce exactly both the equations of movement and the theoretical solutions obtained.In this way, one can 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 new proposal of simulation 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 [3,26], in this case, the systems can be put in one-to-one correspondence with the Foucault pendulum.In summary, we have presented a realization of the equivalence between quantum hermitian and PT-symmetric and classical dynamics that also allows one to confirm that the formal connection includes the behavior of the corresponding geometric phases.Moreover, a novel electric circuit was proposed that is able to simulate the entire process and allows one to do a quantitative analysis.

Appendix
As mentioned in the main text, the PT-symmetry is spontaneously broken in terms of a parameter 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 In the case of PT, the Hamiltonian has real eigenvalues below the exceptional point  ⋆ and complex conjugate ones above it.Explicitly,  with 1 and 2 real for  <  ⋆ and with complex 2 = * 1 for  >  ⋆  The presence of the symmetry implies that and in the region where the eigenvalues are real one has meaning that PTΨ 1,2 is also eigenfunction of H.Then, Ψ 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.or which explicitly shows that the PT-symmetry is not present in the solution and changes one solution into the other one, 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.
Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.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.

Fig. 2 Fig. 3
Fig.2Simulation 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,

Fig. 4
Fig.4 Normalized imaginary and real parts vs. normalized degree of PT-symmetry for the non-Hermitic circuit