Dynamics of finite dimensional non-hermitian systems with indefinite metric

We discuss the time evolution of physical finite dimensional systems which are modelled by non-hermitian Hamiltonians. We address both general non-hermitian Hamiltonians and pseudo-hermitian ones. We apply the theory of Krein Spaces to construct metric operators and well-defined inner products. As an application, we study the stationary behaviour of dissipative One Axis Twisting Hamiltonians. We discuss the effect of decoherence under different coupling schemes.


I. INTRODUCTION
In the last years, the dynamics of non-hermitian hamiltonians has renewed interest, particularly in dealing with open physical systems [1][2][3]. Among this type of hamiltonians, pseudo-hermitian operators play a central role. The formal beginning of this subject was due to Bender and Boettcher [4] in 1998. The Authors of [4] have proposed the study of the celebrated Hamiltonian H 1 = p 2 + x 2 (ix), which has real spectrum and is not self-adjoint. The more relevant characteristic of this hamiltonian, a particular one of the parametric family H ε>0 = p 2 + x 2 (ix) ε , and of many other hamiltonians that were studied later [5], is that they are invariant respect to Parity-Time Reversal (PT) symmetry. These type of hamiltonians are a particular case of quasi-hermitian operators [6]. They have proved to be very useful in the understanding of physical problems with manifiest PT symmetry, i.e. microwave cavities [7], atomic diffusion [8], electronic circuits [9], optical waveguide arrays [10], quantum critical phenomena [11].
In the study of a parametric family of hamiltonians, it is usual to observe regions with different symmetry. The points, in terms of the parameters of the model, at which the symmetry is broken are called exceptional points [12,13]. Thus, exceptional points [12][13][14] divide the family of hamiltonians in two regions: the region where the eigenvalues are real (region of unbroken symmetry), and the region where only a finite number of eigenvalues are real and the rest appear in conjugated pairs (region of broken symmetry). The existence of exceptional points has been visualized in various laboratory experiments [15][16][17]. The recent literature is abundant concerning the study of the region where there is not rupture in symmetry, both theoretically [18] and with examples of physical interest [3]. Recently, the Authors of [19] have studied the region with broken symmetry, for the family of Hamiltonians H ε . Their findings have clarified the existence of divergences in different perturbative developments [20][21][22].
The formalization of the time evolution of the observables of physical systems, which are described by nonhermitian hamiltonians, is related to the introduction of well definite scalar products [23][24][25]. The literature devoted to time evolution of physical systems, which are modeled by non-hermitian hamiltonians, is predominately focussed in the determination of the survival of a particular component of the initial state as it evolves in time. To our knowledge, the time evolution of these type of physical observables have been mostly addressed by means of perturbative expansions [26] or by phenological approaches, i.e master equation for the density matrix [27]. Less has been said about the time evolution of physical observables from first principles, i.e. microscopic hamiltonians [1,28,29], except for a very recent work [30]. In this work we propose the construction of the time evolution of a given initial state, in the presence of an interaction modeled by a non-hermitian hamiltonian. We explore different regimes in terms of the coupling constants of the system. We construct metric operators and scalar products for each of the possible scenarios. We apply the formalism to a non-hermitian One Axe Twisting (OAT) hamiltonain. We explore the behavior of the steady state of the system, in the regime of real spectrum, and in the regime where the spectrum of H includes complex eigenvalues. Also, we discuss the behaviour of the system under decoherence in both regimes.
The work is organized as follows. The details of the general formalism are presented in Section II. The results of the calculations are presented and discussed in Section III. Our conclusions are drawn in Section IV.

II. FORMALISM
Consider a non-hermitian hamiltonian, H. We can write the action H on an orthonormal basis of a Hilbert space, A k . From the diagonalization of the matrix of reperesentation of H in the basis A k , we obtain the set of eigenfunctions of H, A H = {| ϕ j } j=1...Nmax , which are given by In same way, the set of eigenfunctions of ..Nmax , are given by As it has been pointed before, we aim to describe the mean value of a physical observable when a given initial state evolves in time under the action of a non-hermitian hamiltonian. We shall represent the physical observable by the linear hermitian operator o. In order to work with the basis formed by the eigenstates of H, A H , to calculate the expectation value of o, we have to equipped the linear vector space with an scalar product. To do so we look for a metric operator S, i.e. an operator which is self-adjoint and positive definite. Given the metric operator S, we shall define a new scalar product .|. S . The Hilbert space H equipped with the scalar product .|. S is the new physical linear space H S = (H, .|. S ). Because of its properties, the metric operator S can always be diagonalized, that is with [S] AS = D, being D a real and positive definite diagonal matrix.
Here, R is the transformation matrix from the basis A k to the basis of eigenstates of S, A S . That is, the coordinates of a vector f ∈ H transform, under the change of basis, as while the matrix representation of a linear operator, o, transform as We shall denote vectors of the basis A S by | ≈ v , the vectors of the basis A H by | v , and the vectors of the basis A k , and by |v , and the corresponding coordinates as ≈ c k , c k and c k , respectively.
As it has been discussed in [32], to fix the metric uniquely, such there is no ambiguities in the interpretation of physical observables, we shall assume that matrix representation of the hermitian o on the basis A S , transforms as and that the coordinates of the vectors transform as So that A particular case of non-hermitian hamiltonians are the so called quasi-hermitian hamiltonians. If H is quasihermitian hamiltonian, its spectrum consists of real or of complex pair conjugates eigenvalues, and it coincides with the spectrum of H † . In this case, we can construct a metric operator S, so that it obeys the relation SH = H † S. Along this work, when we work with a quasi-hermitian hamiltonian H, we shall assume that it belongs to a parametric family of hamiltonians H δ . This parameter, or possibly parameters, is in direct relation with the coupling constants of the physical problem under consideration. We say that the parameter δ is in the symmetry zone if H δ is diagonalizable and has real punctual spectrum. Those values of δ for which H δ has pair conjugate complex spectrum will cover the zone of broken symmetry. We shall call exceptional points to those values of δ in the boundary between the zone of symmetry and of broken symmetry. On these points the Hamiltonian H δ is not diagonalizable.
In this work, when dealing with the construction of metric operators in the case of quasi-hermitian hamiltonians with broken symmetry, we shall make use of the formalism of Krein Spaces.
For the sake of completeness, we shall briefly review the theory of Krein Spaces. Let us consider a vector space F over the field C of complex numbers, and let Q(x, y) be a sesquilinear hermitian form in F , Q : F × F → C. By a sesquilinear hermitian form, we understand that Q(x, y) is linear in the second argument, that is given x, y 1 , y 2 ∈ F and λ 1 , λ 2 ∈ C, Q(x, λ 1 y 1 + λ 2 y 2 ) = λ 1 Q(x, y 1 ) + λ 2 Q(x, y 2 ), and that Q(y, x) = Q(x, y) * . Clearly, Q(x, y) is semi-linear (or anti-linear) in its first argument.
Though, an hermitian form Q(x, y) defines a Q-metric on F , [x, y] = Q(x, y) (x, y, ∈ F ), this metric can be indefinite, i.e there exist x 0 and y 0 elements in F such that [x 0 , x 0 ] > 0 and [y 0 , y 0 ] < 0. Respect to this indefinite metric, we can classify the vectors x ∈ F as positive, negative or neutral depending on whether [x, x] > 0, [x, x] < 0 or [x, x] = 0, respectively.
As usual, two vectors x, y ∈ F are said to be Qorthogonal, x[⊥]y, if [x, y] = 0. The Q-orthogonality of sets M, N (⊂ F ) is naturally defined by requiring that x[⊥]y for all x ∈ M and y ∈ N .
Let F be a linear vector space with a Q-metric [x, y] = Q(x, y). Consider the subsets F + and F − of F , constructed as If F admits the decomposition where [+] denote the Q-orthogonal direct sum, we say that (11) is a canonical decomposition of the space F . Notice that F + and F − are pre-Hilbert spaces: , admits the canonical decomposition of Eq. (11), with both F + and F − complete subspaces, i.e Hilbert spaces relatives to the norms [33].
The canonical decomposition of Eq.(11) enables the construction of a positive definite scalar product, x|y , into the whole Krein space. It reads for all x, y ∈ F , being x = x + + x − and y = y + + y − , with x + , y + ∈ F + and x − , y − ∈ F − . Equipped with the scalar product of Eq. (12), the space F is complete, i.e it is a Hilbert space relative to the norm x = x|x 1/2 of Eq. (12).
In what follows we shall construct the metric operator S for the different classes of non-hermitian hamiltonians.

Case I: quasi-hermitian hamiltonian: Real Spectrum.
As is well known from literature [4,6], if the nonhermitian hamiltonian H has real spectrum, it is isospectral to an hermitian hamiltonian h, h = Υ r HΥ r −1 . This allows us to define a symmetry operator S ψ = Υ † r Υ r , so that S ψ | φ j = |ψ j . In terms of the eigenvectors of H † it takes the form and it obeys S ψ H = H † S ψ . The symmetry operator S ψ is a self-adjoint and positive definite operator, so that we can define an scalar product on H of the form Notice that in this case A S coincides with A H , and that D is the identity matrix. The Hilbert space H equipped with the scalar product .|. S ψ is the new physical Hilbert space H S ψ := (H, .|. S ψ ).

Case II: quasi-hermitian hamiltonian: Complex non-degenerate pair conjugate Spectrum.
If the spectrum of H includes complex non-degenerate pair conjugate eigenvalues, the operator S ψ of Eq. (13) is not longer a symmetry operator, S ψ H = H † S ψ . The symmetry self-adjoint operator, which enables us to recovery the property SH = H † S, takes the form However, S is not positive definite. Thus, the scalar product [x, y] = (x, Sy) is indefinite. To solve this difficulty, we shall use the formalism of Krein Spaces. Let us begin by saying that as S is a self-adjoint operator, its eigenvalues can be classified according to their sign, semipositive or negative. We shall assume that S has M semi-positive eigenvalues,{λ +i } (λ +i ≥ 0), and N negative ones, {λ −i }. From the eigenvalues of S, we shall determine a decomposition of H as a direct sum H + ⊕H − , where H + is spanned by eigenfunctions corresponding to {λ +i } positive eigenvalues and H − is spanned by eigenfunctions corresponding to {λ −i } negative eigenvalues, respectively.
We proceed as follows. If S is a diagonalizable operator, S can be written as S = P DP −1 , where D is the diagonal matrix containing the eigenvalues We shall define, in H, the operators S + = P D + P −1 and S − = P D − P −1 . Both S + and S − are self-adjoint operators, being S + positive definite and S − negative definite. This means that x|S + x ≥ 0 and x|S − x ≤ 0 for all x ∈ H.
In the frame of Krein Spaces, because of the canonical decomposition, H can be written as H = H + ⊕ H − . Notice that S + | H − = 0 and S − | H + = 0. Then, we can introduce the metric operator S K = S + − S − , which is self-adjoint and positive definite. Consequently, we shall define the inner product .|. SK in H as for all x, y ∈ H being x ± , y ± its components in the canonical decomposition.
To preserve the mean value of a observable o, we proceed as has been detailed at beginning of the Section II, Eqs. (5-10), with [S K ] AS = D = D + − D − , and R = P.
When H is not longer diagonalizable, we have to make use of the Jordan Decomposition technique. In this case, by a suitable change of basis, H can be written as H = P J P −1 . As it is known, generalized eigenvectors and eigenvectors constitute the columns of matrix P . These vectors form a basis of Hilbert space, H, of dimension N max . In the same way, H † =P J †P −1 , with P = P †−1 . Let |ψ k being the k th column ofP . Vectors {|ψ k } 1≤k≤Nmax form a non-orthonormal basis of H † . As P −1P = I, the set {|v k } 1≤k≤Nmax , where |v k is the k th column of (P −1 ) T , forms a basis of H † , which is biorthonormal to {|ψ k } 1≤k≤Nmax , i.e v k |ψ j = δ k,j . Let us construct a new self-adjoint operator of symmetry as It is straightforward to prove S J H = H † S J . As before, S J is a non-positive definite operator, so [f, g] SJ = f |S J g is an indefinite scalar product for H. As S J is a self-adjoint operator, we can follow the steps of II 2. After diagonalization of S J , it reads S J = RD J R −1 . As before, D J = D J+ + D J− . At this point, we are in condition to introduce the metric operator S KJ = S +J − S −J , which is self-adjoint and positive definite. Also, S KJ = Υ † KJ Υ KJ . Consequently, we shall define the inner product .|. SKJ in H as As before, we preserve the mean value of a observable o, by following the steps presented in the Section II, Eqs.

Case IV: non-pseudo-hermitian hamiltonian.
If the spectrum of H contains complex eigenvalues, which are not complex pair conjugates, we can define inner product by introducing the operator Notice that S g has the same form of S ψ of Eq.(13), but nowĒ j = E * j . Also, as H is no longer a pseudo-hermitian operator, it results that S g H = H † S g .
The operator S g of Eq. (22) is self-adjoint and positive definite, so that we can introduce a operator Υ g such that S g = Υ † g Υ g . We are in condition to introduce an scalar product on H of the form The Hilbert space H equipped with the scalar product .|. Sg is the new physical Hilbert space H Sg = (H, .|. Sg ). As in Case I of this Section, II 1, the basis A S coincides with A H and D is the identity matrix.
We can summarize the previous results as follows. We have constructed, depending on the characteristics of the spectrum of H, a self-adjoint positive definite operator, S. The metric operator S allows to define an scalar product, so that mean values of physical observables can be computed.
A. Time Evolution.
We shall construct the time evolution of a general initial state, |I . In the basis A k , it reads In terms of the basis formed by the eigenvectors of H the initial state can be written as with Υ the transformation matrix from basis A k to basis A H . We shall assume that the initial state is normalized, that is I|I = 1. The initial state of Eq.(25) evolves in time as If H can be diagonalized, c α (t) is given by c α (t) = e −i Eαt c α . In the case of exceptional points, the hamiltonian H can be decomposed, in terms of the Jordan matrix J, as H = Υe −iJt Υ −1 , and the form of the coordinates c α (t) will depend upon the particular hamiltonian under consideration.
In terms of the eigenvectors of the symmetry operator S, the initial state reads with Υ ′ being the transformation matrix from the basis A H to the basis A S . We are know in condition of evaluate the mean value of an operator o as a function of time as As it has been stated before, in order to evaluate we follow the prescription given at the Section II, Eqs. (5-10).

III. RESULTS AN DISCUSSION.
Let us apply the previous results to the study of the evolution of a given initial state, under the action of a collective system of spins interacting through a nonhermitian One Axes Twisting (OAT) hamiltonian of the form [34] In writing the collective Hamiltonian of Eq. (29), H OAT is the OAT hamiltonian introduced by Kitagawa and Ueda in [34]. While, the term H d models the effects of decoherence of the system. The collective pseudo-spin operator of the system, S = ( S x , S y , S z ), is governed by the cyclic commutation relations [ S i , S j ] = i ǫ ijk S k , where the suffixes i, j, k stand for the components of the spin in three orthogonal directions and ǫ ijk is the Levi-Civita symbol.
We shall assume that the initial state is prepared as a coherent spin-state (CSS) [35] given by with z(θ 0 , φ 0 ) = e −iφ0 tan(θ 0 /2). The angles (θ 0 , φ 0 ) define the direction n 0 = (sin θ 0 cos φ 0 , sin θ 0 sin φ 0 , cos θ 0 ), such that S · n 0 |I = −S|I [35]. An observable of interest related to the hamiltonian of Eq. (29) is the spin squeezing parameter. Spin-squeezedstates are quantum-correlated states with reduced fluctuations in one of the components of the total spin. Following the work of Kitagawa and Ueda [34], we shall define a set of orthogonal axes {n x ′ , n y ′ , n z ′ }, such that n z ′ is the unitary vector pointing along the direction of the total spin < S > . We shall fix the direction n x ′ looking for the minimum value of ∆ 2 S x ′ , so that the Heisenberg Uncertainty Relation reads Consequently, we define the squeezing parameters [34] as The state is squeezed in the x ′ -direction if ζ 2 x ′ < 1 and ζ 2 y ′ > 1. If the minimum value of the Heisenberg Uncertainty Relation, Eq. (31), is achieved and ζ 2 x ′ < 1, the state is called Intelligent Spin State [36][37][38][39].

Quasi-hermitian OAT hamiltonian.
In Figure 1, we show the results concerning the number of real eigenvalues of the hamiltonian H of Eq. (29), as a function of the ratio κ/λ, for systems with different number of particles, N = 2S. The systems with an even number of particles have always, at least, one real eigenvalue, due to the fact that the number of eigenstates of the hamiltonian is 2S + 1. On the other hand, as the ratio κ/λ is increased, the systems with odd number of particles has no real eigenvalues. In Figure 2, we display the results obtained for the Squeezing Parameters of Eq. (32) as a function of the time, in units of [dB]. We have considered a system with N = 10 particles. Insets (a) and (c) show the results obtained for the coupling constants ratio κ/λ = 0.01, while Insets (b) and (d) correspond to κ/λ = 1.5. In Insets (a) and (b), we show the results obtained for an initial CSS with (θ 0 , φ 0 ) = (π/4, 0). For Insets (c) and (d) we have taken (θ 0 , φ 0 ) = (π/8, 0) . It is clear from the Figure that the time evolution of the initial state in the region of real spectrum H is quite different to that in the region with complex spectrum of H. For small values of the ratio κ/λ, the model show a patron of revivals, even when decoherence is present in the system, κ = 0. If the ratio κ/λ is greater than 1 (See Figure 1), the number of real eigenvalues reduces to one, and the initial state evolves into a stationary state which behaves as an Intelligent State. As complementary information, in Figure  3, the mean value of the collective spin components of the system are displayed. Figure 3 confirms the pattern of revival of the physical observables in the region of unbroken symmetry, and the effect of decoherence, that is the existence of a pointer state, in the region of broken symmetry.

Quasi-hermitian hamiltonian: Exceptional points.
The hamiltonian of Eq. (29), has exceptional points [12][13][14]. At this points, some eigenstates of the system become degenerate and the hamiltonian is not, in general, diagonalizable.
As an example, let us consider a system with N = 4 particles. In this case, there are two values of the coupling ratio κ/λ at which the system has exceptional points. In Figure 4, we plot the eigenstates of the system, as a function of the coupling ratio κ/λ. In Inset (a) we show the behaviour of the real part of the eigenvalues, while in Insets (b), we present the imaginary part of each eigenvalues. Clearly, the exceptional points take place whenever κ/λ = 0.0739815 or κ/λ = 0.375. For values of κ/λ < 0.0739815, the hamiltonian has real eigenvalues, while for κ/λ > 0.0739815 complex pair conjugate eigenvalues are present. At the exceptional points the hamiltonian can be decomposed as H = P JP −1 .
Let us consider the exceptional point κ/λ = 0.0739815, Notice that, as H = H T , H † = P * J * P −1 * . At this exceptional point, the hamiltonian has real eigenvalues, E 1 = E 2 = 2.24, E 3 = 0.514, E 4 = 0.515 and E 5 = 1.99. The symmetry operator, S J , of Eq. (19), takes the form At this exceptional point the hamiltonian has three real eigenvalues, E 1 = E 2 = 1.25, E 3 = 0.754, and two complex eigenvalues, E 4 = 2.12−1.34i and E 5 = 2.12+1.34i. The symmetry operator, S J , of Eq. (19), takes the form so that, when writing |I(t) of Eq. (26), the coefficients c α (t) are given by In Figure 5, we present the numerical results we have obtained for the squeezing parameters and for the mean value of the components of spin, as a function of the time. The Squeezing Parameters of Eq. (32), as a function of the time, in units of [dB], are presented in Insets (a) and (b). The Mean Values of the components of the spin, S k , as a function of the time, are displayed in Insets (c) and (d). We have considered an initial CSS with (θ 0 , φ 0 ) = (π/4, 0). In Insets (a) and (c), we show the results obtained for the exceptional point κ/λ = 3.75, and in Insets (c) and (d) we plot the results obtained for the exceptional point κ/λ = 0.0739815. We have proceeded in the way proposed in II 3. At the exceptional point κ/λ = 0.0739815 the hamiltonian has real eigenvalues, this fact is reflected in the oscillatory behaviour of the observables of the system, though there is not a periodical behaviour because of the structure of the coordinate c − 1(t) of Eq. (42). At the other exceptional point, κ/λ = 3.75, due to the presence of complex eigenvalues, the behavior of physical observables, is similar to the case with broken symmetry. That is, the system evolves to a squeezed stationary state, which minimizes the corresponding uncertainty relations.

General non-hermitian hamiltonian.
As an example of a general non-hermitian hamiltonian let us consider another generalization of the OAT hamiltonian [3,40]. That is The hamiltonian of Eq. (43) consists of a OAT term, H OAT , plus a Lipkin-type, H L , term. In addition, we shall assume that the particles of the system have a finite lifetime, which is given by the line-width γ [3,40]. This hamiltonian has complex non-pair-conjugate eigenvalues if ǫ = 0. In Figure 6, we plot both the squeezing parameters, in units of [dB], and the mean value of the components of spin, as a function of the time. As an example, we have fixed the number of particles to N = 15, and the In Insets (a) and (b), we show the results obtained for an initial CSS with (θ 0 , φ 0 ) = (π/4, 0). For Insets (c) and (d) we have taken (θ 0 , φ 0 ) = (π/8, 0) Also, the initial coherent state evolves in time to a stationary squeezed state.
The numerical results we have presented suggest that we can moderate the effect of decoherence by adopting coupling constants in regime of real spectrum, or that we can obtained an stationary squeezed state, by working in the parameter region of broken symmetry of the model.

IV. CONCLUSIONS
In this work, we have studied the time evolution of discrete non-hermitian hamiltonians. In doing so, we have constructed metric operators and the corresponding scalar products. In the case of quasi-hermitian hamiltonians, we have analyzed the regime of real and of complex pair conjugate spectrum. Also, we have studied the time evolution of quasi-hermtian hamiltonians at exceptional points. We make use of the theory of Krein Spaces to define scalar products when dealing with quasi-hermitian hamiltonians with complex spectrum. As an example, we study the stationary behavior of non-hermitian One Axis Twisting hamiltonians. We discussed the effect of decoherence in the different coupling schemes. As it is expected, we observe that the results depend drastically on the characteristic of the spectrum of the hamiltonian. If the spectrum of the hamiltonian is real, the observables of the system show a patron of revivals as function of time.
On the other hand, if the spectrum of the hamiltonian includes complex eigenvalues, the effect of decoherence is dominant and the system evolves into a pointer state. Work is in progress concerning the extension of the formalism to non-hermitian hamiltonians with unbound spectrum.

ACKNOWLEDGMENTS
This work was partially supported by the National Research Council of Argentine (PIP 282, CONICET) and by the Agencia Nacional de Promocion Cientifica (PICT 001103, ANPCYT) of Argentina.