Nonlinear quantum mechanics in a q-deformed Hilbert space

Physics Letters A 383 (2019) 2729–2738

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Physics Letters A www.elsevier.com/locate/pla

Nonlinear quantum mechanics in a q-deformed Hilbert space Bruno G. da Costa a,∗ , Ernesto P. Borges b a b

Instituto Federal de Educação, Ciência e Tecnologia do Sertão Pernambucano, 56316-686 Petrolina-PE, Brazil Instituto de Física, Universidade Federal da Bahia, Rua Barao de Jeremoabo, 40170-115 Salvador-BA, Brazil

a r t i c l e

i n f o

Article history: Received 1 May 2019 Received in revised form 31 May 2019 Accepted 31 May 2019 Available online 4 June 2019 Communicated by M.G.A. Paris Keywords: Nonlinear Schrödinger equation Quantum decoherence Nonextensive thermostatistics

a b s t r a c t We introduce a generalized temporal evolution operator that belongs to a q-deformed Hilbert space. This q-unitary operator is used to revisit a nonlinear form of the Schrödinger equation related to the nonextensive thermostatistical framework through a q-deformed Dirac formalism. We also obtain generalized versions of the density matrix operator, the Liouville-von Neumann equation and the Bloch vector, that can be used in the description of decoherence phenomena. Deformed rotation matrices are used for the problem of the spin 21 precession in a uniform magnetic field. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Microscopic quantum systems are not perfectly isolated, such as occurs in a simple measurement process of an observable, what leads to the phenomenon of quantum decoherence. It can be understood as the loss of information of the initial state of a physical ˆ (t ) = |ψ(t )ψ(t )| plays a fundamental role in the description of system due to interaction with the environment [1]. The density matrix  quantum decoherence, for instance, in a two-level system. More specifically, the quantum decoherence includes (but is not limited to) both contributions of dissipation and dephasing [2]. Dissipation refers to processes in which populations of quantum states, represented by the main diagonal elements of the density matrix are modified by interactions with environment, while dephasing refers to random processes between the relative phases of quantum states, represented by the anti-diagonal elements of the density matrix. Both are associated with the non-unitary dynamics of the system [3]. Besides that, the quantum decoherence is related to memory effects and long-range interactions [4], features of non-Markovian processes in which nonextensive statistical mechanics [5–7] plays an alternative approach in its description. Investigations of possible formulations for a deformed time evolution operator associated to nonextensivity has been developed previously. Tirnakli et al. [8] have proposed a non-unitary deformed time evolution operator which depends on a deformation parameter q, and as an application, this operator was used to analyze the effects of nonextensivity on a beam of orthohydrogen molecules through a SternGerlach apparatus. From an analogy between the nonlinear Fokker-Planck equation associated to anomalous diffusion and the nonlinear Schrödinger equation for a free particle, Lavagno has introduced a deformed time evolution operator which dependents explicitly either on the initial state or on the final state [9]. This behavior is similar to the anomalous diffusion related to memory effects. Vidiella-Barranco et al. [10] have proposed a generalization of the Liouville-von Neumann equation for the density operator within the context of nonextensive statistical mechanics: the phenomenon of decoherence emerges naturally due to a non-unitary time evolution of ˆ q (t ) embodied by the nonextensive parameter q. More recently, Santos et al. [11] have studied the Walecka many-body field theory by using of the deformed time evolution operator with the Tsallis nonextensive framework. Several experimental studies have revealed the usefulness of the nonextensive formalism for the description of quantum systems, for instance, Larmor spin precession of trapped particles in the presence of magnetic-field gradients and electric fields [12], spin-glass relaxation in CuMn and AuFe [13], single ions in radio frequency traps interacting with a classical buffer gas [14], momentum distribution of cold atoms in dissipative optical lattices [15], driven-dissipative two-dimensional dusty plasma [16], transverse momentum distributions of high-energy pp and p p collisions at LHC experiments [17]. Furthermore, generalizations of equations of quantum physics have been

*

Corresponding author. E-mail addresses: [email protected] (B.G. da Costa), [email protected] (E.P. Borges).

https://doi.org/10.1016/j.physleta.2019.05.056 0375-9601/© 2019 Elsevier B.V. All rights reserved.

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proposed and analyzed within the context of this formalism [18–33]. One of these formulations deals with the nonrelativistic fundamental equation in quantum mechanics, proposed by Nobre, Rego Monteiro and Tsallis [18], — we refer to it as NRT nonlinear Schrödinger equation. According to their approach, a physical system is described by two coupled fields ((r , t ) and (r , t )) and their respective conjugated complexes [19]. The equations have nonlinear terms characterized by powers that depend on a parameter q in such a way that the standard forms of the equations are recovered for q → 1. The solutions present properties similar to solitons, and are written in terms of the q-exponential function which plays a central role in nonextensive statistical mechanics. In this work, we present new developments for the formalism proposed in [18]. We consider a generalization for the standard time evolution operator in a deformed Hilbert space. This operator is associated with the nonlinear Schrödinger equation written in a q-deformed ket notation, and it leads to decoherence phenomena. The paper is organized as follows: we first present a brief review of generalized trigonometric functions in Sec. 2. A q-deformed Hilbert space is introduced in Sec. 3. From a generalized time evolution operator in this deformed space, we obtain a formulation for the generalized nonlinear Schrödinger equation in the Dirac formalism. We show that NRT nonlinear Schrödinger equation can be used with just one field q (x, t ) and its deformed complex conjugate, instead of two coupled fields. A possible connection with the nonextensive statistical mechanics is presented from a Liouville-von Neumann equation for a deformed density matrix. Sec. 4 exemplifies the formalism with a deformed rotation operator. Lastly, conclusions are drawn in Sec. 5. 2. Properties of generalized trigonometric functions The mathematical basis of the nonextensive statistical mechanics is developed on the q-exponential function [5–7] defined by expq (u ) ≡ 1/(1−q)

[1 + (1 − q)u ]+ for u ∈ R, with [ A ]+ = A, if A > 0, and [ A ]+ = 0, if A ≤ 0. The q-exponential function satisfies expq (a ⊕q b) = 1 expq (a) expq (b) and expq (a q b) = expq (a)/ expq (b), where a ⊕q b ≡ a + b + (1 − q)ab and a q b ≡ 1+(a1−−bq)b (b = q− ) are named the 1 q-addition and q-difference operators, respectively [34,35]. For a pure imaginary iu, expq (iu ) is defined by principal value of expq (iu ) ≡ [1 + (1 − q)iu ]1/(1−q) ,

(1)

with exp1 (iu ) = exp(iu ). Generalized trigonometric functions cosq (u ) and sinq (u ) has been defined in [36] such that

expq (±iu ) = cosq (u ) ± i sinq (u ).

(2)

From the expansion in Taylor series

expq (iu ) = 1 +

∞ 

(iu )ν , ν!

Q ν −1 (q)

ν =1

with Q ν (q) =

ν 

(3)

[1 + (1 − q)k], it follows

k=1

∞ 

cosq (u ) = 1 +

Q 2ν −1 (q)

ν =1

(−1)ν u 2ν , (2ν )!

(4a)

and

sinq (u ) =

∞ 

Q 2ν (q)

ν =0

(−1)ν u 2ν +1 . (2ν + 1)!

(4b)

These generalized trigonometric functions can be written as

cosq (u ) = ρq (u ) cos[ϕq (u )]

(5a)

sinq (u ) = ρq (u ) sin[ϕq (u )],

(5b)

and

where

ρq2 (u ) = expq (iu ) expq (−iu ) = [1 + (1 − q)2 u 2 ]1/(1−q) ,

(6)

and

ϕq (u ) =

1 1−q

atan[(1 − q)u ].

(7)

The Pythagorean theorem is rewritten as

cosq2 (u ) + sinq2 (u ) = ρq2 (u )

(8)

with ρ1 (u ) = 1. Functions cosq (u ) and sinq (u ) oscillate non-periodically through the deformed phase ϕq (u ). The function ρq (u ) modulates the generalized trigonometric functions, and the vector v q = (cosq (u ), sinq (u )), for q = 1 describes a spiral. As it is done in Refs. [36,37],

B.G. da Costa, E.P. Borges / Physics Letters A 383 (2019) 2729–2738

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Fig. 1. Generalized trigonometric functions cosq (u ) and sinq (u ) for (a) q = 1.1 and (b) q = 0.90. (c) Parametric representation (cosq (u ), sinq (u )) is shown for these same values of q and u > 0. The usual case (q = 1) is shown for comparison (dash-dotted line).

the definitions (2) can be extended for complex numbers. Herein, we explore these functions in the real domain. Fig. 1 shows the functions (a) cosq (u ), (b) sinq (u ), and (c) the diagram (cosq (u ), sinq (u )). From Eq. (5), we obtain the following identities

cosq (u ) cos2−q (u ) = cos2 [ϕq (u )],

(9a)

sinq (u ) sin2−q (u ) = sin2 [ϕq (u )],

(9b)

and

sinq (u ) cos2−q (u ) = sin2−q (u ) cosq (u ) =

1 2

sin[2ϕq (u )].

(9c)

Therefore

1 = expq (iu ) exp2−q (−iu )

= cosq (u ) cos2−q (u ) + sinq (u ) sin2−q (u ) = v q · v 2−q ,

(10)

and

exp2−q (iu ) expq (iu ) = e 2i ϕq (u ) .

(11)

While the functions sinq (u ) and cosq (u ) approaches 0 for q > 1 as u → ±∞, the respective generalized cofunctions cos2−q (u ) and sin2−q (u ) diverge (see Fig. 1 (a) and (b)). Eq. (10) is related to this behavior. Fig. 2 shows the products given by equations (9) for q = 1.1.

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Fig. 2. Product of deformed trigonometric functions: (a) cosq (u ) cos2−q (u ) and sinq (u ) sin2−q (u ), (b) sinq (u ) cos2−q (u ) and − sin2−q (u ) cosq (u ) for q = 1.1.

3. Time evolution operator in q-deformed nonlinear quantum mechanics 3.1. q-Deformed Hilbert space Following Eq. (10), we propose a q-deformed Hilbert space for which the inner product satisfy the following properties

 w q | v q  =  v 2−q | w 2−q ∗ ,

(12a)

 w q |λq,1 v q,1 + λq,2 v q,2  = λq,1  w q | v q,1  + λq,2  w q | v q,2 ,

(12b)

λq,1 w q,1 + λq,2 w q,2 | v q  = λ∗2−q,1  w q,1 | v q  + λ∗2−q,2  w q,2 | v q . For a given ket vector | v q  = as

|| v q ||2 =  v q | v q  =





k cq,k |k,

({|k} is an orthonormal basis), there is a q-dual bra  v q | =

c 2∗−q,k cq,k .



(12c) ∗

k k|c 2−q,k .

The q-norm is defined

(13)

k

ˆ q | v q , and the matrix elements of which are  j |

ˆ q |k = q, jk . ˆ q that transforms | v q  into | w q , i.e. | w q  =

Consider an operator

ˆ ˆq =

ˆ ˆ are the matrix elements of the q-deformed conjugate operator. If

2−q , then q is a q-Hermitian operator. If q preserves the q-deformed inner product, i.e.

∗

†

2−q,kj

ˆ†

ˆ ˆ

2−q q = 1,

(14)

ˆ q is a q-unitary operator. then

3.2. NRT nonlinear Schrödinger equation in Dirac notation Let the deformed time evolution operator Uˆ q (t , t 0 ) defined by [8]



Uˆ q (t , t 0 ) = expq −

ˆ (t − t 0 ) iH h¯



(15)

,

ˆ a necessary condition for describing quantum decoherence and |ψq (t ) = Uˆ q (t , t 0 )|ψq (t 0 ). The operator Uˆ q (t , t 0 ) is not unitary ( Uˆ q Uˆ q = 1), phenomena [3]. The index q may be viewed as a parameter associated to a proper measure of the information loss of the system. The operator Uˆ q (t , t 0 ) satisfies the following nonlinear differential equation †

ˆ Uˆ q (t , t 0 ), ih¯ Dq,t Uˆ q (t , t 0 ) = H where Dq,t is a deformed derivative operator defined by Dq,u f (u ) = [ f (u )]1−q df d [ f (u )]1−q du {[ f (u )]1−q du },

df (u ) du

[18]. The second derivative must be used

f (u ) = and the product rule is Dq,u [ f (u ) g (u )] = g (u )Dq,u f (u ) + f (u )Dq,u g (u ). In particular, Dq,u expq (u ) = expq (u ). Multiplying both sides of (16) by the initial ket state |ψq (t 0 ) leads to as

Dq2,u

(16)

ˆ |ψq (t ). ih¯ Dq,t |ψq (t ) = H

(17)

B.G. da Costa, E.P. Borges / Physics Letters A 383 (2019) 2729–2738

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This equation corresponds to a nonlinear time evolution equation for the ket vector |ψq (t ), which can be used to extend the results proposed in [18] to systems that do not have classical analog, as particles of spin s in an external magnetic field. Thus, by means the basis {|n} of eigenstates of the Hamiltonian Hˆ , whose respective eigenenergies are E n , the effect of the deformed time evolution operator is to lead an initial state |ψq (t 0 ) to



|ψq (t ) = expq − =



ˆ (t − t 0 ) iH 

expq −

h¯

−i E n (t −t 0 )/¯h

|ψq (t 0 )

i E n (t − t 0 )



h¯

n

Let cq,n (t ) ≡ eq



|nn|ψq (t 0 ).

(18)

cn,q (t 0 ) where cq,n (t 0 ) = n|ψq (t 0 ). The norm of the ket vector |ψq (t ) =



n cq,n (t )|n

is not preserved for q = 1

due to non-unitarity of Uˆ q (t , t 0 ). However, according to Eq. (13), its q-norm is constant. We present a possible connection with the NRT nonlinear Schrödinger equation. By using the wave function r |ψq (t ) ≡ q (r , t )/0 (q), where 0 (q) is a normalizing constant,



ih¯ Dq,t

   q (r , t ) q (r , t ) = Hˆ . 0 (q) 0 (q)

(19)

ˆq = Equation (19) corresponds to the definition of an energy operator [18]. A deformed Hamiltonian operator H deformed operator momentum defined by [18]



pˆ q

   q (r , t ) q (r , t ) = −ih¯ Dq,r , 0 (q) 0 (q)

yields

 ih¯ Dq,t

i.e.

ih¯

1 2m

pˆ q2 + Vˆ , with pˆ q a

(20)

     q (r , t ) q (r , t ) q (r , t ) h¯ 2 2 =− + V (r ) , Dq,r 0 (q) 2m 0 (q) 0 (q)

(21)

 2−q     q (r , t ) q ∂ q (r , t ) 1 h¯ 2 2 q (r , t )  =− ∇ + V (r ) , ∂ t 0 (q) 2 − q 2m 0 (q) 0 (q)

(22)

which corresponds to the nonlinear Schrödinger equation proposed in [18]. There is, thus, an equivalence between the nonlinear wave equation (22) and its matrix form given by Eq. (19). Specifically for a particle under a constant potential V (r ) = V 0 , the solution of Eq. (22) is a q-plane wave (r , t )/0 (q) = expq [i (k · r −

 one obtains E = p 2 /2m + V 0 ωt )], that satisfies the dispersion relation h¯ ω = h¯ 2k2 /2m + V 0 . From de Broglie’s postulate, E = h¯ ω and p = h¯ k, [23]. The q-dual complex conjugate of the nonlinear Schrödinger equation yields



−ih¯ D2−q,t

∗2−q (r , t )



∗0 (2 − q)

=−

h¯ 2 2m



D22−q,r

∗2−q (r , t )

∗0 (2 − q)





+ V (r )

∗2−q (r , t ) 0 (2 − q)

 .

(23)

From the wave function for a free particle submitted to constant potential V 0 , and property (10), an additional field q (r , t )/0 (q) was introduced by [19]:

 ∗ q   2−q (r , t ) q (r , t ) q (r , t ) −q = , = 0 (q) ∗0 (2 − q) 0 (q)

(24)

so Eq. (23) can be rewritten as

 1−q         r, t) q (r , t ) q−1 q (r , t ) ∂ q (r , t ) h¯ 2 q (r , t ) 2 q (  = − qV 0 . ∇ ih¯ ∂ t 0 (q) 2m 0 (q) 0 (q) 0 (q) 0 (q)

(25)

3.3. q-Deformed Liouville-von Neumann equation A possible connection with nonextensive thermostatistics may be found by multiplying Eq. (18) by r | on the left, so the wave function is



q (r , t ) = with

dr  K q (r , t ; r  , t 0 )q (r  , t 0 )

   i E n (t − t 0 )  r |nn|r  expq − h¯ n   ˆ (t − t 0 ) iH = r | expq − |r  , h¯

(26)

K q (r , t ; r  , t 0 ) ≡

(27)

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B.G. da Costa, E.P. Borges / Physics Letters A 383 (2019) 2729–2738

i.e., the deformed propagator is formally the same, but with the q-exponential function. For the sake of simplicity, r  = r and t 0 = 0, thus



G q (t ) ≡

dr  K q (r  , t ; r  , 0)

 i Ent |r  |n|2 expq − h¯ n

  i Ent , expq − = G q (0) h¯ n

=

dr 



(28)

G q (t ) is a sum over all states. Similarly to the standard case, the deformed partition function is obtained by replacing it /¯h by a parameter β (see [38]):

Zq =

G q (h¯ β/i ) G q (0)

=



expq (−β E n ) .

(29)

n

ˆ q (t ) = |ψq (t )ψq (t )|, with Consider the q-deformed density matrix operator for a pure state  expected value of an observable in the generalized Hilbert space is given by

ˆ q = ψq |

ˆ |ψq  = 



ˆ ). q,nm mn = tr(ˆ q

q,nm = c2∗−q,m (t )cq,n (t ), so that the (30)

n,m

From Eq. (17) it follows

ˆ q,nm = ih¯ c 2∗−q,m (t )[Dq,t cq,n (t )] + ih¯ [D2−q,t c 2∗−q,m (t )]cq,n (t ) ih¯ Dq,t  = E n c 2∗−q,m (t )cq,n (t ) − E m c 2∗−q,m (t )cq,n (t ) = −ˆ q,nm Hˆ mm + Hˆ nn ˆ q,nm ,

(31)

[ Hˆ , ˆ q ]

(32)

thus,

Dq,t ˆ q =

1 ih¯

which corresponds to a deformed von Neumann’s equation. 1/(1−q) ˆ q ≡ [ Hˆ , ˆ q ], the formal solution to (32) is given by ˆ q (t ) = [1 + (1 − q)Lˆt ]+ ˆ q (0) originally proposed by VidiellaDefining i h¯ Lˆ q  Barranco et al. [10] to describe decoherence in quantum mechanics within the Tsallis statistics formalism. The solution was obtained ˆ q (t ) = expq (Lˆt )ˆ q (0). through an ansatz  4. q-Deformed nonlinear quantum dynamic for a spin 4.1. q-Deformed rotation operator for spin

1 2

system

1 2

We revisit the problem of a spin 12 particle in an external magnetic field with the q-deformed time evolution operator. A deformed rotation operator in the q-deformed Hilbert space is defined by



ˆ q,ˆn (φ) ≡ expq − R

i J · φ

(33)

h¯

ˆ φ is the rotation angle, nˆ is a unit vector, and J is the generator of rotations. The components of J satisfy the commutation with φ ≡ φn, relations of angular momentum [ J j , J k ] = i  jkl h¯ J l ( jkl is the Levi-Civita symbol). 1 , 2 Sˆ j = h2¯

For a system of spin Pauli matrices, with

S can be written as



ˆ q,ˆn (φ) = expq − R =I+

the components of the generator of rotations S at the basis of eigenstates of Sˆ z ({|±}) are described by the

σ j ( j = 1, 2, 3 for x, y , z). Using the Taylor series (3), the q-deformed rotation operator associated with the spin

i S · φ

h¯

∞  (−1)ν Q 2ν −1 (q) 1 ν =1

(2ν )!

2

σ · φ

2ν −i

∞  (−1)ν Q 2ν (q) 1 ν =0

(2ν + 1)!

2

σ · φ

 2 ν +1 ,

(34)

and using Eq. (4a) and (4b), we arrive at

ˆ q,ˆn (φ) = I cosq (φ/2) − i (σ · nˆ ) sinq (φ/2), R

(35)

that is q-unitary in the sense of Eq. (14). Explicitly, the q-deformed rotation operators around the x, y, and z axes by an angle φ are given by

B.G. da Costa, E.P. Borges / Physics Letters A 383 (2019) 2729–2738



ˆ q,x (φ) = R ˆ q, y (φ) = R



cosq (φ/2) −i sinq (φ/2) −i sinq (φ/2) cosq (φ/2) cosq (φ/2) − sinq (φ/2) sinq (φ/2) cosq (φ/2)



 (36a)

,



(36b)

,

−i φ/2

eq

ˆ q,z (φ) = R

2735

0

(36c)

.

i φ/2

0

eq

ˆ q,z (φ)|ψq . The effect of the deformed rotation operator Let us focus on the effects of a deformed rotation about the z-axis, |ψq  = R

on the state |ψq  = aq |+ + bq |− (NB: ||ψq ||2 = ψq |ψq  = a∗2−q aq + b∗2−q bq ) leads to





|ψq  = expq − −i φ/2

= aq e q

i Sˆ z φ





aq |+ + bq |−

h¯

i φ/2

|+ + bq eq 

|−

 = ρq (φ/2) aq e −i ϕq (φ/2) |+ + bq e i ϕq (φ/2) |− , with

(37)

ρq2 (φ/2) = ρ q+1 (φ) and ϕq (φ/2) = 12 ϕ q+1 (φ). (s)

2

2

 (s) 2 ||

Denoting ||ψq ||2 = |aq |2 + |bq |2 as the standard norm of the ket vector |ψq , ||ψq

(s)

 (s)

= ρ 2q+1 (φ/2)||ψq ||2 , that is, ||ψq

|| is depen-

2

dent of the rotation angle φ , and therefore the norm of |ψq  is not preserved under this q-deformed rotation. On the other hand, the q-norm of the rotated ket vector is preserved. In fact,







 iφ iφ iφ iφ + b∗2−q bq expq ||ψq ||2 = a∗2−q aq expq − exp2−q exp2−q − 2

2

2

2

2

= ||ψq || ,

(38)

ˆ q,z (φ). in accordance with the q-unitarity of R ˆ q,z (φ) under the expectation values of the spin component along the x-axis is  Sˆ x q = ψq | Sˆ x |ψq  −→ The effect of the operator R ψq | Sˆ x |ψq , where

ˆ ˆ ˆ ψq | Sˆ x |ψq  = ψq |R 2−q, z (φ) S x Rq, z (φ)|ψq . †

(39)

From Eq. (36c) and using the relation (11), the q-rotated operator Sˆ x can be written as

ˆ ˆ ˆ† R z,2−q (φ) S x R z,q (φ) i φ/2

 −i φ/2 0 h¯ e 2−q 0 1 eq = −i φ/2 2

0

1

e 2−q

0

0 i φ/2

0

eq

1 1 = e 2i ϕq (φ/2) ( Sˆ x − i Sˆ y ) + e−2i ϕq (φ/2) ( Sˆ x + i Sˆ y ) 2

2

= Sˆ x cos[ϕ q+1 (φ)] − Sˆ x sin[ϕ q+1 (φ)], 2

(40)

2

and consequently we get that

 Sˆ x q −→  Sˆ x q cos[ϕ q+1 (φ)] −  Sˆ y q sin[ϕ q+1 (φ)]. 2

(41)

2

The other components follow analogously:

 Sˆ y q −→  Sˆ x q sin[ϕ q+1 (φ)] +  Sˆ y q cos[ϕ q+1 (φ)], 2

(42)

2

and

 Sˆ z q −→  Sˆ z q .

(43)

Through the use of a deformed rotation matrix R, we can see that  Sˆ i q −→

⎛

⎞

cos[ϕ q+1 (φ)] − sin[ϕ q+1 (φ)] 0 2 2 ⎜ ⎟ cos[ϕ q+1 (φ)] 0 ⎠ . R = ⎝ sin[ϕ q+1 (φ)] 2

0

2

0

j

R i j  Sˆ j q , with

(44)

1

Thus, the spin S revolves in an q-deformed angle similar the usual case.



ϕ q+1 (φ) = 2ϕq (φ/2), whereas the ket vector revolves half of this angle, i.e. ϕq (φ/2), 2

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If the q-hermiticity is not considered, the rotation matrix is given in terms of the generalized trigonometric functions. A rotation of angle φ around the z-axis leads to

⎛

− sin q+1 (φ) 2 ⎜  cos q+1 (φ) R = ρ q+1 (φ)R = ⎝ sin q+1 (φ) 2 2 cos q+1 (φ)

0

2

2

0

0

⎞ ⎟ ⎠.

(45)

ρ q+1 (φ)

0

2

This rotation collapses for q > 1 and diverges for q < 1. 4.2. q-Deformed Bloch vector The q-normalized ket vector for two-level quantum system can be represented by a deformed qubit state −i δ/2

|ψq  = eq =

i δ/2

cosq (γ /2)|0 + eq

−i δ/2

eq

cosq (γ /2)

sinq (γ /2)|1 (46)

,

i δ/2 eq sinq (

γ /2 )

with |0 = |−, |1 = |+, and the parameters (δ , qubit state (46) is

γ ) define a point in unit three-dimensional spherical surface. The density matrix for a

ˆ q = |ψq ψq | ⎛

=⎝ =

i δ/2 −i δ/2

e 2−q eq

i δ/2 i δ/2

e 2−q eq

−i δ/2 −i δ/2

cosq (γ /2) cos2−q (γ /2) e 2−q eq

sinq (γ /2) cos2−q (γ /2)

−i δ/2 i δ/2

e 2−q eq

⎞ sin2−q (γ /2) cosq (γ /2)

⎠

sinq (γ /2) sin2−q (γ /2)

cos2 [ϕq (γ /2)]

e −2i ϕq (δ/2) sin[ϕq (γ /2)] cos[ϕq (γ /2)]

e 2i ϕq (δ/2) sin[ϕq (γ /2)] cos[ϕq (γ /2)]

sin2 [ϕq (γ /2)]

(47)

.

ˆ q ) = 1. The diagonal elements of ˆ q , referred to as populations, give probabilities of occupation in the base states, and which ensures tr( The off-diagonal elements are complex numbers with a deformed factor phase, and they correspond to the quantum coherence. From (s) (s) ˆ q . The the standard density matrix ˆ q in the usual Hilbert space for the q-deformed qubit (46), one obtains ˆ q = ρ q+1 (δ)ρ q+1 (γ ) 2

2

q = (ηq,x , ηq, y , ηq,z ) corresponds to a geometrical representation of the qubit state |ψq  with the Stokes paramq-deformed Bloch vector η eters given by

⎧ ⎪ ⎨ ηq,x = sin[ϕ q+2 1 (γ )] cos[ϕ q+2 1 (δ)], ηq, y = sin[ϕ q+1 (γ )] sin[ϕ q+1 (δ)], 2 ⎪ ⎩ η = cos[ϕ q+2 1 (γ )]. q, z

(48)

2

The density matrix (47) can be written as

ˆ q =

1



2

1 + ηq,z ηq,x − i ηq, y

ηq,x + i ηq, y 1 − ηq,z



1

= ( I + ηq · σ ).

(49)

2

ˆ q,ˆn (φ) on ˆ q is ˆ  = |ψ  ψ  | = The effect of R q q q formula [39]

+ ηq · σ ), where ηq is the q-rotated Bloch vector expressed by Euler-Rodrigues

1 (I 2

ηq = cos[ϕ q+1 (φ)]ηq + sin[ϕ q+1 (φ)](ˆn × ηq ) + (1 − cos[ϕ q+1 (φ)])(ˆn · ηq )ˆn. 2

2

(50)

2

Thus, the effect of the rotation operator (35) is to revolve the q-deformed Bloch vector by an angle

ˆ ϕ q+1 (φ) around the axis n. 2

4.3. q-Deformed precession

ˆ = − e S · B = ω0 Sˆ z , and B = B zˆ ; ω0 ≡ |e | B /mc is the Larmor freFor the problem of a two-level system in uniform magnetic field H mc quency. The deformed time evolution operator Eq. (15) (with t 0 = 0) is formally the same as a deformed rotation with φ = ω0 t (Eq. (33)), thus, from Eq. (37), the time evolution of the state is given by

|ψq (t ) =





i

i



ρ q+1 (ω0t ) e− 2 q (ω0 t ) |++|ψq + e 2 q (ω0 t ) |−−|ψq  ,

(51)

2

with

q (ω0 t ) = ϕ 1+q (ω0 t ) = 2

atan [(1 − q)ω0 t /2]

(1 − q)/2

.

The time evolution of the expected values of the components of S satisfy

(52)

B.G. da Costa, E.P. Borges / Physics Letters A 383 (2019) 2729–2738

2737

Fig. 3. Expected values  Sˆ x (t )q and  Sˆ y (t )q (normalized by S 0 ) for q = 1.2: (a) linear scale (0 < ω0 t /2π < 10), (b) semi-log scale (101 < ω0 t /2π < 103 ) (initial conditions:

 Sˆ x (0)q = S 0 and  Sˆ y (0)q = 0). (c) d Sˆ x (t )q /dt and d Sˆ y (t )q /dt (normalized by ω0 S 0 ). Dotted curves: ±[1 + 14 (1 − q)(ω0 t )2 ]−1 .

 Sˆ x (t )q =  Sˆ x (0)q cos[q (ω0 t )] −  Sˆ y (0)q sen[q (ω0 t )],

(53a)

 Sˆ y (t )q =  Sˆ x (0)q sen[q (ω0 t )] +  Sˆ y (0)q cos[q (ω0 t )],

(53b)

 Sˆ z (t )q =  Sˆ z (0)q ,

(53c)

and

d S (t )q dt

 =

1 1 + 14 (1 − q)2 (ω0 t )2



ω 0 ×  S (t )q

 0 ×  S (t )q = [ρq (ω0 t /2)]q−1 ω  q−1 = ρ (s) (t )/ρ (s) (0) ω 0 ×  S (t )q ,

(54)

 0 = ω0 zˆ , where ρ (s) (t ) is the trace of the standard density matrix ˆ (s) for a pure state with  S (t )q =  Sˆ x (t )q xˆ +  Sˆ y (t )q yˆ +  Sˆ z (t )q zˆ and ω given by (51). The spin of the particle precesses with a deformed phase given by (52) modulated by [ρq (ω0 t /2)]q−1 (Lorentzian function). The phase of state vector is half of the phase spin precession, but the deformed precession is not periodic. As t → ∞, d S q /dt → 0, and consequently the components x and y reach stationary values lim  Sˆ x (t )q =  Sˆ x (0)q cos s −  Sˆ y (0)q sin s ,

(55a)

lim  Sˆ y (t )q =  Sˆ x (0)q sin s +  Sˆ y (0)q cos s ,

(55b)

t →∞ t →∞

with s = π /(1 − q). Fig. 3 shows the expected values  Sˆ x (t )q and  Sˆ y (t )q , as well as their temporal variation rates d Sˆ x (t )q /dt and d Sˆ y (t )q /dt,  Sˆ x (t )q and for q = 1.2.  Sˆ y (t )q oscillate within the range [−1, 1], and the stationary state is reached for t  2π /ω0 , in accordance to Eq. (55).

2738

B.G. da Costa, E.P. Borges / Physics Letters A 383 (2019) 2729–2738

5. Conclusions The unitary operators are related to the conservation of the norm of a ket vector, and consequently to the conservation of probability. Quantum decoherence for open systems has been described in the literature with non-unitary time evolution operators, non-Hermitian Hamiltonians, pole theory, among others [40]. We have introduced a possible q-deformed Hilbert space within the framework of an algebraic structure that emerges from nonextensive statistical mechanics. The usual unitary operator is generalized into a deformed q-unitarity relation given by Eq. (14). The q-unitary operators conserve a q-norm associated with q-deformed inner product, Eq. (12). Furthermore, the operator Uˆ q (t , t 0 ) leads to a q-deformed nonlinear Schrödinger equation by means of a convenient Dirac’s notation. The use of two coupled fields, q (r , t ) and q (r , t ) and their conjugated complexes q∗ (r , t ) and q∗ (r , t ), as proposed in [19], can be replaced solely by the field q (r , t ) and its q-conjugated complex ∗2−q (r , t ) considering the properties of a q-deformed Hilbert space, where the auxiliary field q (r , t ) is equivalent to ∗2−q (r , t ). We also introduce generalized versions of the density matrix operator, Liouville-von Neumann

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