Feb 27, 2009 - February 27, 2009 quant-ph/0608177v2. Abstract. We introduce creation and annihilation operators of pseudo-Hermitian fermions for two-level ...
arXiv:quant-ph/0608177v2 27 Feb 2009
quant-ph/0608177v2
Fermionic coherent states for pseudo-Hermitian two-level systems O. Cherbal(1) , M. Drir(1) , M. Maamache (1)
(2)
, and D.A. Trifonov(3)
Physical Faculty, Theoretical Physics Lab, USTHB,
B.P. 32 El-alia, Bab Ezzouar, 16111 Algiers, Algeria. (2)
Laboratoire de Physique Quantique et Syst`emes Dynamiques,
Department of Physics, Setif University, Setif 19000, Algeria. (3)
Institute of nuclear research, 72 Tzarigradsko chauss´ee, 1784 Sofia, Bulgaria.
February 27, 2009
Abstract We introduce creation and annihilation operators of pseudo-Hermitian fermions for two-level systems described by pseudo-Hermitian Hamiltonian with real eigenvalues. This allows the generalization of the fermionic coherent states approach to such systems. Pseudo-fermionic coherent states are constructed as eigenstates of two pseudo-fermion annihilation operators. These coherent states form a bi-normal and bi-overcomplete system, and their evolution governed by the pseudo-Hermitian Hamiltonian is temporally stable. In terms of the introduced pseudo-fermion operators the two-level system’ Hamiltonian takes a factorized form similar to that of a harmonic oscillator.
1
Introduction
In the last years a great deal of interest has been devoted to the study of nonHermitian Hamiltonians with real spectrum (see [1]- [17] and references therein). Bender and Boettcher were the first to touch on this issue [7], by introducing the notion of PT-symmetry for one-dimensional non-Hermitian Hamiltonian Hν = p2 + x2 (ix)ν , (ν ≥ 0), that possesses real, positive and discrete spectrum. In [15, 16] the Bessis - Zinn Justin conjecture about the reality of the spectrum of the P T -symmetric Hamiltonian −d2 /dx2 − (ix)2M for M ≥ 1 has been proven. A criterion for the reality of the spectrum of non-Hermitian P T -symmetric Hamiltonians is provided in [17]. 1
By definition a P T -symmetric Hamiltonians H satisfies the relation [H, P T ] = 0 ,
(1)
where P and T are the operators of parity and time-reversal transformations respectively. These are defined according to P xP = −x,
P pP = T pT = −p,
T i1T = −i1 ,
(2)
where x, p, 1 are the position, momentum, and identity operators respectively, √ acting on the Hilbert space, and i := −1. Later, Mostafazadeh [8]-[12] introduced the notion of pseudo-Hermiticity in order to establish the mathematical relation with the notion of PT symmetry. He explored the basic structure responsible for the reality of the spectrum of non-Hermitian Hamiltonians, and established that all the PT-symmetric nonHermitian Hamiltonians are pseudo-Hermitian. He also has shown that any diagonalizable operators with discrete spectra, is pseudo-Hermitian if and only if its eigenvalues are either real or grouped in complex-conjugate pairs (with the same multiplicity). Moreover this result has been generalized to the class all PT-symmetric standard Hamiltonians having R as their configuration space, and to the class of possibly nondiagonalizable Hamiltonians that admit a blockdiagonalization with finite-dimensional diagonal blocks. In fact many of the later developments in the field are anticipated in the paper by Scholtz, Geyer and Hahne [18] (see also [19]). By definition [8], a Hamiltonian H is called pseudo-Hermitian if it satisfies the relation H † = ηHη −1 , (3) where η is a linear, Hermitian, and invertible operator. One can also express the definition (3) in the form H # = H, where H # = η −1 H † η is the η-pseudo adjoint of H [8]. An interesting area where the pseudo-Hermiticity is applied is in the study of non-Hermitian two-level systems [11, 14]. These simple Hamiltonian systems model accurately many physical systems in condensed mater, atomic physics, and quantum optics [20] - [26]. The latter field provides a beautiful implementation of the coherent states formalism [28] - [31]. Rabi oscillations in the non-Hermitian system of a two-level atom in electromagnetic field have been recently examined in ref. [14]. In the preceding paper [32], we have shown how the exact evolution and nonadiabatic Hannay’s angle of Grassmannian classical mechanics of spin one half in a varying external magnetic field, is associates with the evolution of Grassmannian invariant-angle coherent states. In this paper we extend the fermionic coherent states approach [33, 34, 35, 37] to two-level non-Hermitian Hamiltonians which are pseudo-Hermitian (pHermitian). The underlying number system is Grassmann algebra [38, 39]. The set of coherent states (CS) for pseudo-fermionic (shortly p-fermionic) system 2
turned out to consist of two subsets of states, which are bi-normalized and biovercomplete (shortly bi-normal CS). The paper is organized as follows. In section 2, we study a non-Hermitian two-level systems (a two-level atom interacting with electromagnetic field)and its pseudo-Hermitian properties. Then we introduce the creation and annihilation operators for the two-level p-Hermitian system with real energy spectrum, such that it’s Hamiltonian ascribes a form similar to that of the free harmonic oscillator: H = Ω(b# b − 1/2), where b and b# are pseudo-fermionic (p-fermionic) lowering and raising operators. In section 3, we construct the p-fermionic CS as eigenstates of two annihilation operators, the eigenvalues being complex Grassmann variables. The set of such eigenstates form a bi-normal and bi-overcomplete system. Then, in section 4, we study the time evolution of the constructed p-fermionic CS for the corresponding two-level p-Hermitian system. This evolution is shown to be temporally stable. The paper ends with concluding remarks.
2
Two-level systems and pseudo-Hermitian fermions
We consider a two-level atom interacting with an electromagnetic field. The general state of the two-level atomic system is |ψi = Ca′ (t)|+i + Cb′ (t)|−i , where C∓ are the amplitudes of being in |∓i. They are time dependent due to atom-field interaction. In the interaction picture (dipole interaction and phenomenologically described decay), and in the rotating wave approximation, the evolution of the system is described by the equation [24, 25] ′ ′ 1 ∂ Ca (t) −iγ a ω∗ Ca (t) = . (4) i Cb′ (t) ω −iγ b Cb′ (t) ∂t 2 The real constants γ a , γ b are the decay rates for the upper and lower level respectively. The quantity ω characterizes the radiation-atom interaction matrix element between the levels ( ω∗ is the complex conjugate ). The basic vectors of the upper (lower ) level are | +i and | −i . We remove the average effect of the decay terms by means of a nonunitary transformation in the state space, 1 (γ + γ b ). (5) 4 a The probability amplitudes in the new representation are Ci (t) = exp(Γt)Ci′ (t), i = a, b, and the non-Hermitian Hamiltonian takes the following matrix form 1 −iδ ω ∗ , (6) H= ω iδ 2 |ψi → U (t)|ψi,
U (t) = eΓt , Γ =
3
where δ = (γ a − γ b )/2. The trace of H, eq.(4), is vanishing, and the determinant of H is real, det H = (δ 2 − |ω|2 )/4. Therefore it is η-pseudo-Hermitian (η-p-Hermitian) [10]. Its matrix is a particular case of a more general 2 × 2 traceless matrix studied in [11], where the complete biorthonormal system {|ψi i, |φi i} for H and the operator η are explicitly constructed. We reproduce this system and η (up to certain common factors) in our specific notations. The eigenvalues Ei of H, i = 1, 2, and the related complete biorthonormal system are given by (we consider the nondegenerate case of Ei 6= 0)
E1 = −
Ω , 2
√ Ω+iδ |ω|
−ω ∗
E2 =
|φ1 i =
where Ω =
−ω ∗
√
Ω∗ −iδ |ω|
(7)
ω∗
1 |ψ 2 i = √ √ 2Ω
1 |ψ 1 i = √ , √ 2Ω Ω − iδ
Ω , 2
1 √ √ , ∗ 2Ω Ω∗ + iδ
|φ2 i = √
q 2 |ω| − δ 2 .
√
Ω−iδ |ω|
Ω + iδ
ω∗
√ Ω∗ +iδ |ω|
,
1 √ , ∗ 2Ω Ω∗ − iδ
(8)
(9)
For both real and complex eigenvalues (i.e. real and complex Ω) the Hamiltonian (6) satisfies the p-Hermiticity relation (3) with η given by ! iδω ∗ 1 |ω|2 η= . (10) iδω − |ω| 1 2 As noted in [14], the real eigenvalues correspond to the case where the dipole interaction is large relative to the damping effect. In this case the ordinary Rabi frequency is replaced by the ’pseudo Rabi frequency’ which in our notations are |ω|/2 and Ω/2 correspondingly. In the case of Ω = 0 (i.e. |ω|2 = δ 2 ), the amplitudes Ca′ ,Cb′ are given [14] by (1 − δt) exp(−Γt) and (iωt/2) exp(−Γt) correspondingly, where Γ = (γ a + γ b )/4. Due to the exponential decay factor exp(−Γt), any divergence [27] does not occur in our system. In the case of |ω|2 < δ 2 the eigenvalues of H are pure imaginary, but the Hamiltonian is still pseudo-Hermitian [14]. The P T -symmetry of our Hamiltonian (6) is considered in [14] following the method of Bender, Brody and Jones [3]. The Parity operator P of the two-level system can be defined as [3] 0 1 P = , 1 0 4
so that P 2 = 1, P = P −1 . The generalized parity operator for the two-level systems is defined [12, 13] as P = |φ1 ihφ1 | − |φ2 ihφ2 |, which in our case results to ! ω∗ 0 − |ω| P = . ω 0 − |ω| Different definitions have been introduced by Mostafazadeh [12] and Ahmed [13] for the antilinear time-reversal operator T . As in the paper [14], we use the representation introduced by Bender et al [3], namely, T = K0 , where K0 is the complex conjugation operator. One has K02 = 1, (P T )2 = 1, and one finds that P T commutes with H: P K0 HK0−1 P −1 = H. If one uses the generalized parity operator P = |φ1 ihφ1 |−|φ2 ihφ2 |, then one can take T = U K0 , where U is a 2 × 2 unitary diagonal matrix, U22 = U11 = ω/|ω|. The generalized charge conjugation operator C for two-level system is given by the expression [12] C = |ψ 1 ihφ1 | − |ψ 2 ihφ2 | , which in our case reads 1 2 iδ −ω∗ C= = − H. 2Ω −ω −iδ Ω From the last equation we deduce that C commutes with the Hamiltonian H: [C, H] = 0. This invariance property eliminates negative inner products [14]. Furthermore, unless otherwise stated, we consider the small damping effect case of our system, i.e. ω2 > δ 2 , that is Ω real (and if real it is positive). One can verify that, in this regime (i.e. |ω|2 > δ 2 ) this η, eq. (10), can be represented in terms of |φi i as η=
Ω Ω (|φ1 ihφ1 | + |φ2 ihφ2 |) ≡ η . |ω| |ω| +
(11)
The operator η is positive definite. The construction η + = |φ1 ihφ1 |+|φ2 ihφ2 | was introduced by Mostafazadeh [40]. Now, let us introduce the annihilation operator b associated to the Hamiltonian H given in eq. (6), ! −ω ∗ (Ω+iδ) − |ω| 1 |ω| b= . (12) ω(Ω−iδ) 2Ω |ω| |ω| Its adjoint operator reads (Ω is real) 1 b+ = 2Ω
− |ω|
−ω(Ω−iδ) |ω|
and its η-p-Hermitian adjoint b# , defined by 5
ω ∗ (Ω+iδ) |ω|
|ω|
!
,
(13)
b# = η −1 b+ η ,
(14)
takes the form 1 b = 2Ω #
− |ω|
−ω(Ω+iδ) |ω|
ω ∗ (Ω−iδ) |ω|
|ω|
!
.
(15)
Next we examine the properties of the operators b# and b. First of all is that b# and b realize a pseudo-Hermitian generalization of the fermion algebra, namely # b2 = b#2 = 0, b, b = bb# + b# b = 1 . (16)
b# and b could be called the creation and annihilation operators of p-Hermitian fermions [40]. One can verify that they raise and lower the eigenvalues of H by a quantity Ω = 2E, i.e. they act on the states |ψ i i as follows b |ψ 1 i = 0,
b |ψ 2 i = |ψ 1 i ,
b# |ψ 2 i = 0,
b# |ψ 1 i = |ψ 2 i ,
(17) (18) #
The operator b annihilates the lowest eigenstates |ψ 1 i, and b brings this state onto the upper eigenstates |ψ 2 i. Introducing the quadratic operator N = b# b, the p-fermionic number operator, we find the following natural anticommutation relations {N, b} = b
N, b# = b#
.
(19)
In terms of the operators b and b# the Hamiltonian H is factorized (up to an additive C-number term) to a form, similar to that of the free (boson) harmonic oscillator, 1 # . (20) H =Ω b b− 2 Taking the Hermitian conjugate of both sides of (20) we reaffirm the pHermiticity of H (according to definition (3)): 1 H + = Ω(b+ ηbη−1 − ) 2 1 = Ωηη −1 (b+ ηbη −1 − )ηη −1 2 = ηHη −1 . The above relations confirm that b and b# are lowering and raising operators for the two-level p-Hermitian system (with real eigenvalues) and can be regarded as p-fermionic annihilation and creation operators. This is consistent with the limit δ = 0, corresponding to a Hermitian Hamiltonian, when η = 1 6
and b# = b+ , i.e. the p-Hermitian generalization of the fermion algebra reduces to the usual fermion algebra. Quantum system with Hamiltonian of the form (20) should be referred to as p-fermionic oscillator.
3
Pseudo-fermionic coherent states
Having introduced the p-fermion lowering and raising operators we now embark on the construction of the p-fermionic coherent states (CS) for our system described by p-Hermitian Hamiltonian H given in eq. (6). We shall follow as closely as possible the scheme of fermionic CS developed in papers [33, 34, 35, 37], generalizing it to the p-fermion case. For the reader convenience we begin with a brief reminder of the properties of complex Grassmann variables [33, 35, 36, 37], denoted here as ξ and ξ ∗ . The complex Grassmannian variables ξ i and their complex conjugates ξ ∗i satisfy the anticommutation relations: ξi, ξj = ξiξj + ξj ξi = 0 , ∗ ∗ ∗ ξi , ξj = 0 . ξ i , ξ j = 0,
(21) (22)
The ξ i ’s anticommute with b and b# ,
{ξ i , b} = 0, {ξ ∗i , b} = 0 , and have the following properties ξ|ψ 1 i = |ψ 1 iξ ,
ξ i , b# = 0 ,
ξ|ψ 2 i = −|ψ 2 i ,
ξ|φ1 i = |φ1 iξ ,
ξ|φ2 i = −|φ2 iξ .
(23)
(24) (25)
The pseudo-Hermitian conjugation reverses the order of all fermionic quantities, both the operators and the Grassmann variables: (b# ξ i + ξ ∗i b)# = ξ ∗i b + b# ξ i .
(26)
The Grassmann integration and differentiation over the complex Grassmann variables are given by Z Z Z Z dξ 1 = 0, dξξ = 1, dξ ∗ 1 = 0, dξ ∗ ξ ∗ = 1 , (27) d 1 = 0, dξ
d d ξ = 1, 1 = 0, dξ dξ ∗
d ∗ ξ = 1. dξ ∗
(28)
The Grassmann integration of any function is equivalent to the left differentiation Z ∂ f (ξ) . (29) dξ f (ξ) = ∂ξ 7
We define the displacement operators D(ξ) for any set of complex Grassmannian variables ξ in the following way: D(ξ) = exp b# ξ − ξ ∗ b
1 ∗ = 1 + b# ξ − ξ ∗ b + b# b − ξ ξ. 2
(30) (31)
The pseudo-Hermitian adjoint D# is given by D# (ξ) = exp ξ ∗ b − b# ξ
= 1 + ξ ∗ b − b# ξ + b# b −
1 2
(32) ξ∗ξ .
(33)
These two operators satisfy the following displacement relation, D# (ξ)bD(ξ) = b + ξ . Using the explicate formulas of D and D# , and the anticommutation relations between operators b, b# and Grassmann variable ξ we establish that D(ξ) are pseudo-unitary: D# (ξ)D(ξ) = 1 = D(ξ)D# (ξ). We now define the p-fermionic CS |ξi as eigenstates of the annihilation operator b, b |ξi = ξ |ξi .
(34)
|ξi = D(ξ) |ψ 1 i .
(35)
hξ| = hψ 1 | | D† (ξ) 1 = exp − ξ ∗ ξ (hψ 1 | +ξ ∗ hψ 2 |) , 2
(37)
The eigenvalue ξ is a complex Grassmannian variable. The Hermitian adjoint of the CS (the bra-vector) is hξ| and it is left eigenstate of b† , hξ|b† = hξ|ξ ∗ . In order to meet the alternative relation η hξ|b# = η hξ|ξ ∗ one has to define η hξ| ≡ (|ξi)# := hξ|η. Similarly to the cases of Glauber bosonic CS [28] and of fermionic CS [33] our p-fermion eigenstates |ξi can be constructed from the lowest (ground) eigenstate |ψ 1 i of the Hamiltonian H, acting on it by the pseudo-unitary operator D(ξ): By using the formula (31) for the displacement operator, we may write the state |ξi in the form 1 ∗ (36) |ξi = exp − ξ ξ (|ψ 1 i − ξ |ψ 2 i) . 2 The Hermitian adjoint of the CS is
8
(38)
and the inner product hξ|ξi is hξ|ξi = hψ 1 |ψ 1 i + (hψ 2 |ψ 2 i − hψ 1 |ψ 1 i)ξ ∗ ξ − 2iIm(ξhψ 1 |ψ 2 i) ,
(39)
so that the |ξi are not normalized. Now we have to examine for (over)completeness the set {|ξi}. One can straightforwardly check (using the rules (21) - (27)) that neither the integral (against the measure dξ ∗ dξ) of the Hermitian |ξihξ| nor the integral of p-Hermitian |ξi η hξ| (unnormalized) projectors result in the identity operator: Z Z dξ ∗ dξ|ξihξ| = 6 1, dξ ∗ dξ|ξi η hξ| = 6 1. (40)
The way out of this impasse is suggesting by the known transition from ’orthonormal system’ of eigenstates of Hermitian H to the ’biorthonormal system’ of states of p-Hermitian H. With this idea in mind we introduce another f of the operator ˜b, that continuous family of states namely the eigenstates |ξi annihilates the dual state |φ1 i, ˜2
f = ξ |ξi, f ˜b|ξi
˜b = ηbη −1 .
(41)
Operator ˜b is nilpotent, b = 0 and anticommutes with b . Representing b† = η˜b† η −1 we see that b† is η ′ -p-Hermitian adjoint to ˜b, η ′ = η −1 . Denoting this pseudo-conjugation by #′ we obtain the pair of p-fermionic operators ˜b and ˜b#′ , †
˜b ˜b#′ + ˜b#′˜b = 1,
˜b2 = (˜b#′ )2 = 0 .
(42)
In view of the p-fermionic algebra (42) we introduce new displacement operators e e #′ (ξ) ˜b D(ξ) e D(ξ) = exp(˜b#′ ξ − ξ ∗˜b), D = ˜b + ξ , and construct eigenstates of ˜b according to the above described scheme (see eqs. (35), (36)), f = D(ξ)|φ e |ξi 1i 1 ∗ = exp − ξ ξ (|φ1 i − ξ|φ2 i) . 2 f ξi e takes the form The scalar product between hξ| while
f ξi e = hφ |φ i + (hφ |φ i − hφ |φ i)ξ ∗ ξ − 2iIm(ξhφ |φ i), hξ| 1 1 2 2 1 1 1 2
or more generally,
(43) (44)
(45)
f = |ω| hξ|η|ξi = 1, hξ|ξi Ω
1 ∗ ∗ ∗ e = hψ |D† (ξ)D(ζ)|φ e hξ|ζi 1 1 i = ξ ζ + (2 − ξ ξ)(2 − ζ ζ). 4 9
(46)
f the resolution of the identity By means of the two type of states |ξi and |ξi is realized in the following way, Z Z ∗ f f 1 = dξ dξ |ξihξ| = dξ ∗ dξ |ξihξ|. (47)
f in The equations (47) can be easily verified using the expansions of |ξi and |ξi terms of |ψ i i and |φi i (eqs. (36) and (44)) and the rules of permutation and integration (21) - (27). We have obtained that the system of one-mode p-fermionic CS consists of e two subsets, namely {|ξi} and {|ξi}. In view of (45) and (46) this continuous system should be called bi-normalized and bi-overcomplete, or shortly system e of bi-normal CS. Similarly the two sets of pseudo-unitary operators D(ξ), D(ξ) should be called bi-unitary: e † (ξ) = 1 = D e † (ξ)D(ξ). D(ξ)D
e Note that D(ξ) is η-pseudo-unitary, while D(ξ) is η ′ -pseudo-unitary with η ′ = −1 η .
4
Time evolution of p-fermionic coherent states
A given parametric set of states is said to be realizable for a physical system if the time evolution |ψ; ti of any initial state |ψi from the set, governed by the Hamiltonian, leaves the state in the set [42]. In other word |ψ; ti, for any t, obeys the defining criteria of the set. In such a case one shortly says that the time evolution (of the parametric set of states) is stable [42]. In Hermitian mechanics this means that the time dependence of the states is included, up to a phase factor, in the state parameters. For example the time evolution |α; ti of Glauber CS |αi [28] is stable with respect to the harmonic oscillator evolution operator exp(−iHt), H = ω(a† a + 1/2): |α; ti = e−iωt/2 |α(t)i,
a|α; ti = α(t)|α(t)i,
α(t) = αe−iωt .
(48)
In the case of our p-fermionic CS {|ξi, e|ξi} the set’ parameter is the complex Grassmann variable ξ, the eigenvalue of the p-fermionic lowering operators b or gti remain ˜b. The time evolution is stable if the evolved states |ξ; ti and |ξ; ˜ eigenstates of the operators b and b respectively, gti = ξ(t)|ξ; g ˜b|ξ; ti .
b|ξ; ti = ξ(t)|ξ; ti,
(49)
This implies that the time evolved CS {|ξ; ti and e|ξ; ti} should form bi-normal and bi-overcomlete system. Let us first consider the time evolution of an initial CS |ξi. Clearly we have |ξ; ti = exp(−iHt)|ξi, |ξ; 0i ≡ |ξi. Using the form (36) of |ξi and the facts that |ψ 1,2 i are eigenstates of H (with eigenvalues E1,2 ) we get 10
1 |ξ; ti = e−iE1 t 1 − ξ ∗ ξ |ψ 1 i − e−iE2 t ξ|ψ 2 i . 2
(50)
Taking into account that E1 = −E and E2 = E we put ξ(t) = e−i2Et ξ and rewrite the last equation in the form 1 (51) |ξ; ti = eiEt 1 − ξ(t)∗ ξ(t) |ψ 1 i − ξ(t)|ψ 2 i = eiEt |ξ(t)i . 2 which manifests the stability of the time evolution of CS |ξi. Note that the overall time dependent factor exp(iEt) is a phase factor since Ei are real. gti of an initial In a similar manner we establish, that the time evolution |ξ; f ˜ |ξi, is stable (remains eigenstate of b): gti = exp(−iH † t)|ξi f |ξ; 1 = exp(−iE1 t) 1 − ξ ∗ ξ |φ1 i − exp(−iE2 t)ξ|φ2 i 2 1 ∗ = exp(iEt) 1 − ξ(t) ξ(t) |φ1 i − ξ(t)|φ2 i = exp(iEt)|ξ(t)i . 2
(52)
The results (51) and (52) reveal the bi-normality and bi-overcompleteness of the gti} of the p-fermionic oscillator system family of time evolved states {|ξ; ti, |ξ; g (20): one has ht; ξ |ξ; ti = 1, and Z Z ∗ g gtiht; ξ| . 1 = dξ dξ|ξ; tiht; ξ| = dξ ∗ dξ |ξ; (53)
gti differ from CS We observe that here the time evolved states |ξ; ti and |ξ; ] in phase factors only. In more general cases the overall factors |ξ(t)i and |ξ(t)i e (t) ascribed in the stable evolution of bi-normal and bi-overcomplete N (t) and N system of states could not be phase factors, but their product should equal unity, e (t) = 1. N ∗ (t)N Finally we have to note that a complementary bi-normal and bi-overcomplete system of states can be constructed, in a symmetrical manner using the opera# , that annihilate the ’upper-level states’ |ψ i and |φ i. tors b# and bf 2 2
5
Concluding Remarks
In this paper, we have generalized the fermionic coherent states (CS) for two-level systems described by pseudo-Hermitian Hamiltonian with real spectrum. Unlike the standard bosonic and fermionic cases the system of pseudofermionic (p-fermionic) CS consists of two subsets, which are bi-normalized and bi-overcomplete. In this sense the system of p-fermionic CS can be regarded as 11
a continuous analogue of the bi-orthonormal system of discrete eigenstates of p-Hermitian H. The two subsets are built up as eigenstates of the p-fermion annihilation operators b and ˜b = ηbη−1 , where η is the Hermitian operator that ensures the p-Hermiticity of the Hamiltonian, H = η −1 H † η. In terms of b and b# = η −1 b† η the Hamiltonian is factorized to the form of p-fermionic oscillator, eq. (20). The evolution of the p-fermionic CS governed by the p-Hermitian two-level Hamiltonian (6) is shown to be time stable - the evolved states remain eigenstates of the p-fermionic annihilation operators, preserving the bi-normality and bi-overcompleteness of the system at later time. In the Hermitian limit of η = 1 (that is δ = 0 in (6) our p-fermionic CS and all related formulas recover standard fermionic CS of refs. [33, 34]. Time evolution of fermionic CS for a Pauli spin in a slowly varying magnetic field was examined by Abe [35] (see also the comment [36]).
Acknowledgments One of the authors (O.C.) acknowledges Drs M. Le Bellac and J.P. Provost for their useful comments and valuable discussions.
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