A. Abidi, A. Trabelsi, and S. Krichene

ARTICLE

Coupled harmonic oscillators and their application in the

dynamics of entanglement and the nonadiabatic Berry phases

A. Abidi, A. Trabelsi, and S. Krichene

Abstract:In the dynamic description of physical systems, the two coupled harmonic oscillators’ time-dependent mass, angular frequency, and coupling parameter are recognized as a good working example. We present in this work an analyti- cal treatment with a numerical evaluation of the entanglement and the nonadiabatic Berry phases in the vacuum state. On the basis of an exact resolution of the wave function solution of the time-dependent Schrödinger equation (TDSE) using the Heisenberg picture approach, we derive the wave function of the two coupled harmonic oscillators. At the logarithmic scale, we derive the entanglement entropies and the temperature. We discuss the existence of the cyclical initial state (CIS) based on an instant Hamiltonian and we obtain the corresponding nonadiabatic Berry phases through a periodT. Moreover, we extend the result to the case ofNcoupled harmonic oscillators. We use the numerical calculation to follow the dynamic evolution of the entanglement in comparison to the time dependance of the nonadiabatic Berry phases and the time dependance of the temperature. For two coupled harmonic oscillators with time-independent mass and angular frequency, the nonadiabatic Berry phases present very slight oscillations with the equivalent period as the period of the entanglement.

A second model is composed of two coupled harmonic oscillators with angular frequency, which change initially as well as later. Herein, the entanglement and the temperature exhibit the same oscillatory behavior with exponential increase in temperature.

Key words: instantaneous Hamiltonian, dynamics of entanglement, nonadiabatic Berry phases, cyclical initial state, coupled harmonic oscillators.

Résumé :Dans la description dynamique des systèmes physiques, on reconnait comme un bon exemple le système de deux oscilla- teurs harmoniques couplés, avec masse, fréquence angulaire et paramètre de couplage qui dépendent du temps. Nous en présen- tons ici un traitement analytique, avec évaluation numérique de l’intrication et des phases non adiabatiques de Berry à l’état du vide. Sur la base d’une résolution exacte de la fonction d’onde solution de l’équation de Schrödinger dépendante du temps (ESDT/

TDSE) dans la représentation de Heisenberg, nous dérivons la fonction d’onde des deux oscillateurs harmonique couplés. À l’échelle logarithmique, nous obtenons les entropies d’intrication et la température. Nous discutons l’existence d’un état initial cyclique (EIC/CIS) basé sur un hamiltonien instantané et nous obtenons les phases non adiabatiques de Berry correspondantes le long d’une périodeT. De plus, la méthode est étendue à un système deNoscillateurs harmoniques couplés. Nous utilisons le calcul numérique pour suivre l’évolution dynamique de l’intrication face à la dépendance dans le temps des phases non adiabatiques de Berry et de la température. Pour deux oscillateurs harmoniques couplés avec masses et fréquences angulaires indépendantes du temps, les phases de Berry non adiabatiques présentent une légère oscillation avec la période équivalente vue comme la période d’intrica- tion. Un deuxième modèle est composé de deux oscillateurs harmoniques couplés avec la fréquence angulaire changeant au début aussi bien que plus tard. Ici, l’intrication et la température montrent le même comportement oscillatoire avec une croissance exponentielle de la température. [Traduit par la Rédaction]

Mots-clés : hamiltonien instantané, dynamique de l’intrication, phases non adiabatiques de Berry, état initial cyclique, oscillateurs harmoniques couplés.

1. Introduction

Schrödinger’s cat [1], Einstein–Podolsky–Rosen paradox [2], and the Bell inequalities [3] are three quantum mechanical fea- tures connected to entanglement. This concept was introduced by Schrödinger in 1935 [4]. On one hand, many significant appli- cations have been developed on the basis of quantum entangle- ment, for instance the superdense coding of quantum teleportation [5], quantum cryptography [6,7], quantum computing [8,9], etc. On

the other hand, in recent years, various entanglement measure- ments have proven to be effective tools for illustrating and under- standing the composition of matter [10]. Rényi and von Neumann entropies are considered as groundbreaking quantum methods for quantifying these entanglement procedures; they were introduced to characterize the degree of unpredictability of information for bipartite systems, which are no longer in a pure state [11–13]. In general, these entropies are defined by a time-dependent or time-

Received 22 July 2020. Accepted 12 May 2021.

A. Abidi.Tunis University, National Higher School of Engineers of Tunis, Tunis, Tunisia; Research Unit of Nuclear and High Energy Physics, Faculty of Science of Tunis, University of Tunis El Manar, Tunis 2092, Tunisia.

A. Trabelsi.Research Unit of Nuclear and High Energy Physics, Faculty of Science of Tunis, University of Tunis El Manar, Tunis 2092, Tunisia; National Center for Nuclear Sciences and Technologies, Technopole of Sidi, Thabet 2020, Tunisia.

S. Krichene.Tunis University, Preparatory Institute for Engineering Studies of Tunis, Nabeul, Tunisia.

Corresponding author: A. Abidi (email:abidiiahlem0@gmail.com).

Copyright remains with the author(s) or their institution(s). Permission for reuse (free in most cases) can be obtained fromcopyright.com.

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independent Gaussian densities matrix [14–16]. Other entropies are based on the generalized Tsallis entropies that employ the Wigner function to be more fitting in the non-commutative phase space [17,18].

Alternatively, the evolution of time-dependent quantum sys- tems has increasingly stimulated considerable interest in various fields of physics, particularly quantum entanglement, in relation to the temporal analysis of entanglement entropies in non-equi- librium quantum systems [19]. If the time-dependent evolution is periodic, a quantum system is indefinitely in the unprocessed state after each period T, and it acquires a phase factor as it evolves. In this case we are referring to the Berry phases. Berry’s adiabatic formulation for a periodic Hamiltonian extends to the case of nonadiabatic evolution through the appropriate choice of initial state, in particular, the nonadiabatic Berry phases, which we favor in this paper. We focus on the time-dependent Hamilto- nian, particularly periodic with aT-period when its eigenvectors are periodic to a phase with period T; the phase by which the state vector of a physical system is multiplied [20]. In this regard, we could think of the initial states of the system. However, the system appears as a closed circuit [21,22] and these phases accu- mulate when the parameters of the Hamiltonian vary instantane- ously [23]. We are studying these issues in this work with instantaneous Hamiltonian for two coupled harmonic oscillators HðtÞ ¼ 1

2MðtÞ½P²₁ðtÞ þP²₂ðtÞ þ1

2MðtÞv²₁ðtÞx²₁ðtÞ þ1

2MðtÞv²₂ðtÞx²₂ðtÞ þ1

2KðtÞx1ðtÞx2ðtÞ In this context, we discuss the resolution of the wave function solution of the time-dependent Schrödinger equation (TDSE) obtained using the Heisenberg picture approach, passing in brief by the diagonalization of the Hamiltonian and deriving the wave function of two coupled harmonic oscillators. In Sect. 3, to char- acterize the dynamics of entanglement in the vacuum state, we derive the Rényi and the von Neumann entropies using the den- sity matrix and wefind the temperature. Section 4 discusses the wave function, the instantaneous Hamiltonian, and the initial cyclic state by the time variable. In addition, we obtain the nona- diabatic Berry phases corresponding to the two coupled harmonic oscillators and we close this work by deriving the nonadiabatic Berry phases forNcoupled harmonic oscillators.

2. Wave function solution of TDSE

To analyse the two applications, entanglement and nonadia- batic Berry phases, our concern is the system of two coupled har- monic oscillators with the Hamiltonian

HðtÞ ¼ 1

2MðtÞ½P²₁ðtÞ þP²₂ðtÞ þ1

2MðtÞv²₁ðtÞx²₁ðtÞ þ1

2MðtÞv²₂ðtÞx²₂ðtÞ þ1

2KðtÞx1ðtÞx2ðtÞ (2.1) where [xl,Pm] =idlm,h¼1, andxl,Pl,M,vl, andKare the time- dependent canonical coordinates, momenta, mass, angular fre- quency, and coupling parameter, respectively, withl= 1, 2.

We define a unit operation through a rotation angle (a/2) in the double Hilbert space such that the new canonical coordinates and momenta are

ðy1;y2Þ ¼ y1¼cosa

2x1sina 2x2

y2¼sina

2x1þcosa 2x2

8>

<

>: (2.2)

and

ðP1;P2Þ ¼ P1¼cosa

2P1sina 2P2

P2¼sina

2P1þcosa 2P2

8>

<

>: (2.3)

Note that yj andPjverify the canonical commutation relation, and thus we havePj=–i(@/@yj) (j= 1, 2).

By setting tana¼ KðtÞ

MðtÞ½v²₂ðtÞ v²₁ðtÞ (2.4)

the HamiltonianH(t) takes the diagonal form HðtÞ ¼ 1

2MðtÞP²1ðtÞ þP²2ðtÞ þ 1

2B²₁ðtÞy²₁ðtÞ þB²₂ðtÞy²₂ðtÞ

(2.5) where

B²₁ðtÞ ¼v²₁ðtÞ KðtÞ 2MðtÞtana

2 (2.6)

B²₂ðtÞ ¼v²₂ðtÞ þ KðtÞ 2MðtÞtana

2 (2.7)

Thus, the new time-dependent expressions B²₁ðtÞ;B²₂ðtÞ are obtained by a direct trigonometric calculation. They are the most appropriate forms to be taken into account for the analysis of the results, and they can be rewritten as

B²₁ðtÞ ¼1

2 MðtÞv²₂ðtÞ þv²₁ðtÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v²₂ðtÞ v²₁ðtÞ

2

M²ðtÞ þK²ðtÞ

q

(2.8)

B²₂ðtÞ ¼1

2 MðtÞv²₂ðtÞ þv²₁ðtÞ

þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v²₂ðtÞ v²₁ðtÞ

2

M²ðtÞ þK²ðtÞ

q

(2.9) In the following, we setv2>v1.

Using the Hamiltonian in (2.5), the corresponding TDSE is shown as follows:

i@

@tcn;mðy1;y2;tÞ ¼ 1 2MðtÞ @²

@y²₁ þ @²

@y²₂

! (

þ1

2½B²₁ðtÞy²₁ðtÞ þB²₂ðtÞy²₂ðtÞ )

c_n;mðy1;y2;tÞ (2.10) wheren,m= 0, 1, 2,. . .are the standard harmonic oscillator quan- tum numbers.

Going back to the techniques presented in Ref. [24], the wave function solution of the TDSE in(2.10)using the Heisenberg pic- ture approach is given as

c_n;mðy1;y2;tÞ ¼ 1 2^nþmn!m!

1=2

ˆh1ðtÞˆh2ðtÞ p²g1ðtÞg2ðtÞ

1=4

expfi½anðtÞ þamðtÞgexp ˆh1ðtÞ

2g₁ðtÞy²₁ˆh2ðtÞ 2g₂ðtÞy²₂

exp i g01ðtÞ 2g₁ðtÞy²₁ i g02ðtÞ

2g2ðtÞy²₂

Hn

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˆh1ðtÞ g1ðtÞ s

y1

2 4

3 5Hm

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˆh2ðtÞ g2ðtÞ s

y2

2 4

3

5 (2.11)

with

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anðtÞ ¼ ð^t

0

ˆh1ðt1Þ½nþ ð1=2Þ

Mðt1Þg1ðt1Þ dt1 amðtÞ ¼ ð^t

0

ˆh2ðt1Þ½mþ ð1=2Þ Mðt1Þg2ðt1Þ dt1

(2.12) being the corresponding phases at timet, and

ˆhiðtÞ ¼ ½gþiðtÞgiðtÞ g²_0iðtÞ¹⁼² (2.13) is a constant of motion. From the Lie algebra,g+i(t),g_–1(t),g0i(t) ver- ify the linear system of the differential equations

_

g_iðtÞ ¼ 2

MðtÞg0iðtÞ (2.14)

_

g_0iðtÞ ¼MðtÞB²_ig_iðtÞ 1

MðtÞg_þiðtÞ (2.15)

_

g_þiðtÞ ¼2MðtÞB²_ig0iðtÞ (2.16)

With (i= 1, 2), the general solution of the system has the form of the classical equationg_–i(t) in the form

g_iðtÞ ¼c1iðtÞf_1i²ðtÞ þc2iðtÞf1iðtÞf2iðtÞ þc3iðtÞf_2i²ðtÞ (2.17) andf1i,f2iare the two linearly independent solutions for the equa- tion of motion

€f_1;2iðtÞ þMðtÞ_

MðtÞf__1;2iðtÞ þB²_iðtÞf1;2iðtÞ ¼0 (2.18) wherec1i,c2i,c3iare arbitrary constants andg0i(t),g+i(t) are obtained by direct differentiation of(2.14)and(2.16). At initial time, we have

g_iðt0Þ ¼ 1

Mðt0Þ g0iðt0Þ ¼0 g_þiðt0Þ ¼MB²_iðt0Þ and ˆhiðt0Þ ¼Biðt0Þ

In terms of the original coordinates (x1,x2), the wave function in(2.11)can be rewritten

c_n;mðx1;x2;tÞ ¼ 1 2^nþmn!m!

1=2

ˆh1ð Þˆt h2ð Þt p²g1ð Þgt 2ð Þt

" #₁₌₄

expi½anð Þ þt amð Þt

exp ˆh1ð Þt

2g1ð Þt x1cosa

2x2sina 2

2

ˆh2ð Þt

2g2ð Þt x1sina

2þx2cosa 2

2

" #

exp i g01ð Þt

2g1ð Þt x1cosa

2x2sina 2

2

i g02ð Þt

2g2ð Þt x1sina

2þx2cosa 2

2

" #

Hn

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˆh1ð Þt g1ð Þt s

x1cosa

2x2sina 2

2 4

3 5 Hm

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˆh2ð Þt g2ð Þt s

x1sina

2þx2cosa 2

2 4

3

5 (2.19)

Note that the general solution can be written as a linear combination ofcn,m

cðy1;y2;tÞ ¼X

n

X

m

Cn;mc_n;mðy1;y2;tÞ (2.20)

In the next section, the focus is on the analysis of the time evolution of the density matrix, the purity function, and the entanglement entropies using the exact full time-dependent Schrödinger wave function [25–27]. The dynamical treatment presented here is concealed in the expressiong_–i(t).

3. Rényi entropy

In relation to the reduced density matrix of the system, the Rényi entropy is expressed at orderkas

SkðtÞ ¼ ð1kÞ¹lnTr½rredðx;x⁰;tÞ^k (3.1)

So, to discuss the dynamics of entanglement wefirst derive the density matrix and assess the corresponding Rényi and von Neumann entro- pies. From the wave function in(2.19)and its conjugate, the exact density matrix is

r_n;mx1;x2;x⁰₁;x⁰₂:t

¼c_n;mðx1;x2:tÞc_n;mx⁰₁;x⁰₂:t

¼ 1

2^nþmn!m!

ˆh1ð Þˆt h2ð Þt p²g1ð Þgt 2ð Þt

" #₁₌₂

Hn

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˆh1ð Þt g1ð Þt s

x1cosa

2x2sina 2

2 4

3 5

Hm

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˆh1ð Þt g1ð Þt s

x1sina

2þx2cosa 2

2 4

3 5Hn

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˆh1ð Þt g1ð Þt s

x⁰₁cosa

2x⁰₂sina 2

2 4

3 5Hm

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˆh1ð Þt g1ð Þt s

x⁰₁sina

2þx⁰₂cosa 2

2 4

3 5

exp x²₁

2 g₁cos²a

2g₂sin²a 2

x²₂

2 g₁sin²a

2þg₂cos²a 2

þx1x2sina 2cosa

2ðg₁g₂Þ

" #

exp x⁰²₁

2 g₁cos²a

2þg₂sin²a 2

x⁰²₂

2 g₁sin²a

2þg₂cos²a 2

þx⁰₁x⁰₂sina 2cosa

2g₁g₂

" #

(3.2) where

g_j¼ˆhjðtÞ

gjðtÞ þig0jðtÞ

gjðtÞ (3.3)

The vacuum state is illustrated byn=m= 0 and the following density matrix:

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r_ð0;0Þðx1;x2;x⁰₁;x⁰₂:tÞ ¼ ˆh1ðtÞˆh2ðtÞ p²g1ðtÞg2ðtÞ

1=2

exp x²₁

2 g1cos²a

2 þg2sin²a 2

x²₂

2 g1sin²a

2þg2cos²a 2

þx1x2sina 2cosa

2ðg1g2Þ

" #

exp x⁰²₁

2 g₁cos²a

2þg₂sin²a 2

x⁰²₂

2 g₁sin²a

2 þg₂cos²a 2

þx⁰₁x⁰₂sina 2cosa

2ðg₁g₂Þ

" #

(3.4)

Thefirst harmonic oscillator is defined by the variables (x1;x⁰₁) and the associated reduced density matrix

r_1ð0;0Þðx1;x⁰₁:tÞ ¼ ð

dx2r_ð0;0Þðx1;x2;x⁰₁;x2:tÞ (3.5) According to expression(3.4), the reduced density matrix is r_1ð0;0Þðx1;x⁰₁:tÞ ¼ ˆh1ðtÞˆh2ðtÞ

p²g1ðtÞg2ðtÞ 1=2

exp½ðl1þl3Þ þil2x²₁ ½ðl1þl3Þ il2x⁰²₁ þl3x1x⁰₁g (3.6) where

l1¼ 1 2a

ˆh1ðtÞ g₁ðtÞ

ˆh2ðtÞ g₂ðtÞ

l2¼ 1 2a

ˆh1ðtÞ g1ðtÞ

g02ðtÞ g2ðtÞ

sin²a

2þ ˆh2

g2ðtÞ g01ðtÞ

g1ðtÞ

cos²a 2

( )

l3¼ 1 4a sin²a

2cos²a 2

ˆh1ðtÞ

g₁ðtÞ ˆh2ðtÞ g₂ðtÞ

2

þ g01ðtÞ

g₁ðtÞ g02ðtÞ g₂ðtÞ 2

( )

a¼ ˆh1ðtÞ g₁ðtÞ

sin²a

2þ ˆh2ðtÞ g₂ðtÞ

cos²a

2

(3.7) The same procedures are done for the second harmonic oscillator.

The reduced density matrix of the second system with the variables (x2;x⁰₂) is given as

r_2ð0;0Þðx2;x⁰₂;tÞ ¼ ð

dx1r_ð0;0Þðx1;x2;x1;x⁰₂:tÞ

¼ ˆh1ðtÞˆh2ðtÞ p²g1ðtÞg2ðtÞ

₁₌₂

expf½ðt1þt3Þ þit2x²₂ ½ðt1þt3Þ it2x⁰²₂þt3x2x⁰₂g (3.8) where

t1¼ 1 2b

ˆh1ðtÞ g1ðtÞ

ˆh2

g2ðtÞ

t2¼ 1 2b

ˆh1ðtÞ g₁ðtÞ

g02ðtÞ g₂ðtÞ

sin²a

2þ ˆh2ðtÞ g₂ðtÞ

g01ðtÞ g₁ðtÞ

cos²a 2

( )

t3¼ 1 4b sin²a

2cos²a 2

ˆh1ðtÞ

g1ðtÞ ˆh2ðtÞ g2ðtÞ

2

þ g01ðtÞ

g1ðtÞ g02ðtÞ g2ðtÞ

2

( )

(3.9) Using(3.6)and(3.8), it is easy to confirm that

Tr½r_1ð0;0Þ ¼Tr½r_2ð0;0Þ ¼1 (3.10) To characterize the system information, we move into the pu- rity function. Its explicit expression for thefirst system reads

P1¼Trf½r_1ð0;0Þðx1;x⁰₁;tÞ²g ¼ l1

l1þ2l3

1=2

¼ 1

§ ˆh1ðtÞ

g1ðtÞ

ˆh2ðtÞ g2ðtÞ

1=2 (3.11)

where

§¼abþsin²a 2cos²a

2 g01ðtÞ

g1ðtÞ g02ðtÞ g2ðtÞ

2

b¼ ˆh1ðtÞ g₁ðtÞ

cos²a

2þ ˆh2ðtÞ g₂ðtÞ

sin²a

2

(3.12)

The second system has the purity function P2¼Trn½r_2ð0;0Þðx2;x⁰₂;tÞ²o

¼ t1

t1þ2t3

1=2

¼ 1

§ ˆh1ðtÞ

g₁ðtÞ

ˆh2ðtÞ g₂ðtÞ

1=2 (3.13)

The two particles 1 and 2 have the same purity, which can be writtenP1=P2=P. Let us consider the equationP= Trr², which represents the density of the probability for the system in the vacuum state. Additionally, it characterizes the mixedness of the stater. It is clear that whena= 0, the purity becomesP= 1 and r(x,x⁰,t) is a pure state. Also, ifa=6p/2 the purity func- tion is

P¼2

ˆh1ðtÞ g1ðtÞ

ˆh2ðtÞ g2ðtÞ

ˆh1ðtÞ

g₁ðtÞ þˆh2ðtÞ g₂ðtÞ

2

þ g01ðtÞ

g₁ðtÞg02ðtÞ g₂ðtÞ

2

8>

>>

<

>>

>:

9>

>>

=

>>

>;

1=2

(3.14)

andr(x,x⁰;t) characterizes a mixed state. To calculate the Rényi entropy, we solve the expression of the eigenvalue corresponding to the reduced density matrix and given by

ð

þ1

1

r_redðx;x⁰;tÞFnðx⁰:tÞ ¼PnðtÞFnðx;tÞ (3.15)

Herenis an integer noted in theFn(x⁰:t) eigenfunction with the eigenvaluePn(t).

The solution of the eigenfunction of the corresponding eigen- value equation is presented in the form

PnðtÞ ¼ ½1%ðtÞ%ⁿðtÞ Fnðx;tÞ ¼ 1

ffiffiffiffiffiffiffiffiffi 2ⁿn!

p k

p

1=4

Hnð ffiffiffiffi pk

xÞexp k

2x²þil2x²

(3.16)

where

%ðtÞ ¼l3 ðl1þl2Þ þk 2

1

k¼2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðl1þl3Þ²l²₃ q

(3.17) Delving into a certain calculation procedure, the Rényi entropy can be rewritten in the form

SkðtÞ ¼ ð1kÞ¹ln½1%ðtÞ^k

½1%^kðtÞ (3.18)

When k! 1, the expression of the Rényi entropy in (3.18) is reduced to the von Neumann entropy where the result is

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S1ðtÞ ¼ ln½1%ðtÞ %ðtÞ½1%ðtÞ¹ln%ðtÞ (3.19) In the ground state boosted byn= 0 using expression(3.16), the density matrix in(3.15)becomes the thermal density matrix at thermal equilibrium with the specific frequency of a harmonic oscillatork. It is useful to define the canonical temperature for this small system, as stressed in Refs. [28,29]. Following Han et al.

and Kim and al. [30,31], we have T ¼k ln 1

%ðtÞ

1

(3.20) These results provide all the ingredients to study the dynamics of entanglement in the context of the analysis of the role of coupling parameterK(t) and the mixing anglea. Whena= 0 and K= 0,%tends to zero. The Rényi and the von Neumann entropies disappear. So, we are dealing with a system of two unbound harmonic oscillators [32]. Next, we are interested in an immedi- ate Hamiltonian that it has a closed form, in the particular case where its parameters evolve periodically in time, where we can discuss the nonadiabatic Berry phases.

4. Nonadiabatic Berry phases

To calculate the nonadiabatic Berry phases, the system of two coupled harmonic oscillators evolves periodically with a Hamiltonian [33–35]

HðtþTÞ ¼HðtÞ (4.1)

Using the su(2) group of the Lie algebraic approach, the Hamiltonian in(2.5)can be re-expressed [36] as

HðtÞ ¼X

l¼1;2

X

j¼0;6

hjlðtÞXjlðtÞ

(4.2)

where

h0lðtÞ ¼g²_0lðtÞ þM²ðtÞB²_lðtÞg²_lðtÞ þˆ²_hlðtÞ

g0lðtÞMðtÞˆhlðtÞ (4.3)

h₆lðtÞ ¼g²_0lðtÞ þM²ðtÞB²_lðtÞg²_lðtÞ þˆ²_hlðtÞ62ig0lðtÞˆhlðtÞ

2g0lðtÞMðtÞˆhlðtÞ (4.4)

X0lðtÞ ¼A^þ_l AlþAlA^þ_l

2 X_lðtÞ ¼A²_lðtÞ

2 X_þlðtÞ ¼A^þ2_l ðtÞ 2

(4.5) andA^þ_l andAlare the time-dependent creation and annihilation operators, respectively. They are given as

A^þ_lðtÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiˆ_hlðtÞ 2glðtÞ r

i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

2ˆhlðtÞglðtÞg0lðtÞ 2 s

4

3 5ylðtÞ i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gl

ðtÞ2ˆhlðtÞ r

PjðtÞ (4.6)

AlðtÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiˆhlðtÞ

2glðtÞ r

þi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

2ˆhlðtÞglðtÞg0lðtÞ 2 s

4

3 5y_lðtÞ þi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g_lðtÞ 2ˆhlðtÞ

s

PjðtÞ (4.7) with [yl(t),Pm(t)] =idlm.

4.1. Dynamic and geometric phases

Using the wave function defined in(2.11)and the Hamiltonian (2.5), the initial state is characterized by the wave functionc(t0) and the TDSE

i @

@t0

c_n;mðy1;y2:t0Þ ¼Hðt0Þc_n;mðy1;y2:t0Þ (4.8) where the energies of the decoupled system are given by

En¼B1 nþ1 2

Em¼B2 mþ1 2

(4.9) After a periodTthe system acquires the eigenstatec(t0+T), if we choose

cðt0þTÞ ¼e^ixn;mðTÞcðt0Þ (4.10) Note that e^ixn;mðTÞcðt0Þis not an eigenstate of the HamiltonianH(t0);

we refer to it as the cyclical initial state. From the geometrical phase xn,m(T) in(4.10), the nonadiabatic Berry phases are defined as

b ¼xd (4.11)

Here d is the generalisation of the dynamical phase and is expressed by the instantaneous expectation value of the Hamiltonian (4.2)defined as

d ¼ ð^T

0

hc_n;mðtÞjHðtÞjc_n;mðtÞidt (4.12)

The nonadiabatic Berry phases become

b_n;mðTÞ ¼x_n;mðTÞ þ ð^T

0

hc_n;mðtÞjHðtÞjc_n;mðtÞidt (4.13)

By studying Ref. [24], it is not difficult tofind the expression of the nonadiabatic Berry phases corresponding to the two coupled harmonic oscillators wheng_–i(t) isT-periodic, with the result

b_n;mðTÞ ¼ nþ 1 2

ð^T

0

g²₀₁ðtÞ MðtÞg₁ðtÞˆh1ðtÞdt mþ1

2

ð^T

0

g²₀₂ðtÞ

MðtÞg₂ðtÞˆh2ðtÞdt (4.14) The ground state is characterized by the nonadiabatic Berry phases

b_0;0ðTÞ ¼ 1 2 ð^T

0

g²₀₁ðtÞ

MðtÞg1ðtÞˆh1ðtÞdt 1 2 ð^T

0

g²₀₂ðtÞ MðtÞg2ðtÞˆh2ðtÞdt

(4.15) Ifa= 0 alsoK= 0, expression(4.15)is presented as the nonadia- batic Berry phases for two decoupled harmonic oscillators.

5. Nonadiabatic Berry phases in N dimensions

Ncoupled harmonic oscillators are described by the time- dependent Hamiltonian

H^NðtÞ ¼1 2

1 MðtÞ

X^N

j¼1

P²_jðtÞ þMðtÞX^N

j¼1

v²_jðtÞx²_jðtÞ þX^N

j¼1

Kj;jþ1ðtÞxjðtÞxjþ1ðtÞ 2

4

3 5

(5.1) Also in the form

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H^NðtÞ ¼ 1 2MðtÞ

X^N

j¼1

P²_jðtÞ þMðtÞX^N

j¼1

X^N

l¼1

xjðtÞJjlðtÞxlðtÞ (5.2)

whereJjl(t) is a symmetric matrix.

In terms of the orthogonal transformationD, we define a diago- nal matrixJ_jj^DðtÞ ¼D¹JjlðtÞDand a new coordinate system

Y¼D¹X¼ ðy1;y2;y3;. . .;yNÞ^N (5.3)

Thus, the Hamiltonian forNdecoupled harmonic oscillators can be expressed as

H^0NðtÞ ¼1 2

X^N

j¼1

1

MðtÞP²_jðtÞ þy²_jðtÞJ^D_jjðtÞ

(5.4)

From expression(4.2), the HamiltonianH^0NðtÞtakes the form H^0NðtÞ ¼X^N

j¼1

X

k¼0;6

hkjðtÞXkjðtÞ

(5.5) where

h0iðtÞ ¼g²_0jðtÞ þM²ðtÞB²_jðtÞg²_jðtÞ þˆ²_hjðtÞ

g0jðtÞMðtÞˆhjðtÞ (5.6)

h₆iðtÞ ¼g²_0jðtÞ þM²ðtÞB²_jðtÞg_j²ðtÞ þˆ²_hjðtÞ62ig0jðtÞˆhjðtÞ

2g0jðtÞMðtÞˆhjðtÞ (5.7) and

A^þ_j ðtÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˆhjðtÞ 2g_jðtÞ s

i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

2ˆhjðtÞg_jðtÞg0jðtÞ 2 s

4

3 5y_jðtÞ i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g_jðtÞ 2ˆhjðtÞ s

PjðtÞ (5.8)

AjðtÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˆhjðtÞ 2gjðtÞ s

þi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

2ˆhjðtÞgjðtÞg0jðtÞ 2 s

4

3 5yjðtÞ þi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g_jðtÞ 2ˆhjðtÞ s

PjðtÞ (5.9) The time-dependent wave function for the system becomes Y^N

j¼1

c_njðyj;tÞ ¼ 2 PN

j¼1njY^N

j¼1

nj! 0

@

1 A

1=2

Y^N

j¼1

ˆhjðtÞ pgjðtÞ 2

4

3 5

1=4

exp iX^N

j¼1

anjðtÞ

2 4

3 5exp 1

2 X^N

j¼1

ˆhjðtÞ gjðtÞy²_j i

2 X^N

j¼1

g0jðtÞ gjðtÞy²_j 2

4

3 5

Y^N

j¼1

Hn_j

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˆhjðtÞ

gj y²_j 2s

4

3

5 (5.10)

Going back to the original variables, expression(5.10)becomes Y^N

j¼1

c_njðxj;tÞ ¼ 2 PN

j¼1njY^N

j¼1

nj! 0

@

1 A

1=2

detC p 1=4

exp iX^N

j¼1

anjðtÞ 2

4

3 5 exp X^TCX

2 i X^TgX 2

Y^N

j¼1

Hnj½X^TC¹⁼²X (5.11) where

g¼Dg^DD¹ C¼DC^DD¹ X¼ ðx1;x2;. . .;xNÞ^T (5.12)

andC^D,g^D are the diagonal matrices of the elementsˆhjðtÞ=gj

andg0j/g_–j, respectively [14,37].

Using(5.5)and(5.11), the dynamical phase ofNdecoupled har- monic oscillators is given by

d^N¼ ð^T

0

*Y^N

j¼1

c_njðxj;tÞ H^0N

Y^N

j¼1

c_njðxj;tÞ +

dt (5.13)

Hence, the corresponding nonadiabatic Berry phases as Y^N

j¼1

b_njðTÞ ¼Y^N

j¼1

x_njðTÞ þð^T

0

*Y^N

j¼1

c_njðyj;tÞ H^0N

Y^N

j¼1

c_njðyj;tÞ +

dt

Y^N

j¼1

bn_jðTÞ ¼ X^N

j¼1

njþ1 2

ð^T

0

g²_0jðtÞ Mg²_jðtÞˆhjðtÞdt

(5.14) with

M¼KðtÞ

tana^ðv²_jþ1v²_jÞ (5.15)

Of the vacuum state, we can write Y^N

j¼1

b₀ðTÞ ¼ 1 2

X^N

j¼1

ð^T

0

g²_0jðtÞ

Mg²_jðtÞˆhjðtÞdt (5.16)

6. Entanglement

Now, we have illustrated our present objective in the different results of the entanglement, the nonadiabatic Berry phases, and the temperature for two important cases. Thefirst is visualized by two coupled harmonic oscillators with time-independent mass and angular frequency. The coupling parameterKis time dependent. We can define three parameters,c₁= 1/M,c₂= 0,c₃= S⁴/M, where S is the Schrödinger-picture operator [38], which takes on exponential form

S1ðtÞ ¼S0e^iB1t S2ðtÞ ¼S0e^iB2t (6.1) whereB₁,B₂are given by(2.6)and(2.7).

The two independent solutions in(2.17)are

f1kðtÞ ¼cosB1kðtÞ (6.2)

f2kðtÞ ¼sinB2kðtÞ (6.3)

(k= 1, 2). Using(2.17),(2.14), and(2.16)we get g_kðtÞ ¼ 1

Mðcos²BktþS⁴_ksin²BktÞ (6.4) g0kðtÞ ¼ ðS⁴_k1ÞBkcosBktsinBkt (6.5) gþkðtÞ ¼MB²_kðcos²BktþS⁴_ksin²BktÞ (6.6)

The constant of motion in(2.13)becomes ˆhk¼ ˆh1¼B1S²₁

ˆh2¼B2S²₂

(6.7) By settingv1= 0.8,v2= 1.6, andS0= 2, we plot inFigs. 1aand1b the time dependence of von Neumann entropy and nonadiabatic

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Berry phases for three values of the coupling parameterK,K= 3 (blue dashed line),K= 7 (black dashed line), andK= 13 (red solid line). In Fig. 1c, we plot the time dependence of nonadiabatic Berry phases forK= 3 and we vary the number of the harmonic oscillators forN= 2 (blue dashed line),N= 4 (black dashed line), andN= 8 (red solid line).

The time dependence of von Neumann entropy begins with a maximum oft= 0.488, it has a double oscillatory behavior resulting in the expression of the constant of motionˆhkin (6.7), more precisely the oscillatory behavior presented in expressions (2.6) and (2.7). As the order of entanglement decreases with parameter K, it is expected that at K = 0 the two coupled harmonic oscillators become a separable system. The time dependence of nonadiabatic Berry phases exhibits exponential behavior with very slight oscillations of the same period as the periods of the time-dependent von Neumann entropy oscillations and those represented byˆhk. In the regions 0≤t≤0.2 and 0.488≤t≤0.748, the growth of the time-dependent nonadiabatic Berry phases is propor- tional to the value ofK. This trend is reversed in the regions 0.2≤t≤0.488 andt≥0.748. AsNincreases, the time depend- ence of nonadiabatic Berry phases increases rapidly, and the harmonic oscillator chain shown in the Hamiltonian (5.4) and the nonadiabatic Berry phases(5.16)can be described by a single massive harmonicfield that produces a single nonadia- batic Berry phases.

The second is the two coupled harmonic oscillators where the unit mass and the frequency are changed linearly for some initial and late time [36,39]. We havec1=c3= 1 andc2= 0

f1kðtÞ ¼ f1kiðtÞ ¼cosBkit t<0 f1kfðtÞ ¼cosBkft t>0

(6.8) whereiis the initial time, andfis late time.

f2kðtÞ ¼

f2kiðtÞ ¼sinBkit t<0 f2kfðtÞ ¼ Bkit

Bkft

sinBkft t>0 8<

: (6.9)

B²_kðtÞ ¼ B²_ki t<0

B²_kf ¼B²_kið1þb₀Þ t>0 (

(6.10) From(2.17),(2.14), and(2.16),

g_kðtÞ ¼

g_ki¼1 t<0 g_kf¼ Bki

Bkf 2

þ 1 Bki

Bkf

" 2#

cos²Bkft t>0 8>

<

>:

(6.11)

g0kðtÞ ¼

g0ki¼0 t<0 g0kf¼ B²_kfB²_ki

Bkf

sinBkftcosBkft t>0 8<

: (6.12)

g_þkðtÞ ¼ g_þki¼B²_ki t<0

g_þkf¼B²_kf ðB²_kfB²_kiÞcos²Bkft t>0 (

(6.13)

and expression(2.13)is ˆhk¼ ˆhki¼Bki t<0

ˆhkf ¼Bki t>0

(6.14) Whent<0, the constant of motion becomes

ˆhki¼ ˆh1i¼B1i

ˆh2i¼B2i

(6.15) and fort>0, we get

Fig. 1. Results of (a) von Neumann entropyS(t), (b) nonadiabatic Berry phasesb(T) for different values ofKand (c) the nonadiabatic Berry phasesb₁(T) when we vary the number of harmonic oscillators. [Colour online.]

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ˆhkf¼ ˆh1f¼B1i

ˆh2f¼B2i

(6.16) InFig. 2a, we plot the time dependence of von Neumann en- tropyS1(t) fora=p/6 (blue solid line),a=p/8 (red dashed line), anda=p/12 (black dashed line). InFig. 2b, we plot the time de- pendence of Rényi entropyS(t) fork= 2 (blue solid line),k= 4 (red dashed line), andk= 100 (black dashed line). InFig. 2c, we plot the time dependence of temperatureT1(t) fora=p/4 (blue solid line), a=p/8 (red dashed line), anda=p/24 (black dashed line).

In the region wheret≤0, the time dependence of von Neumann entropy exhibits the same oscillatory behavior as the time depend- ence of Rényi entropy. The oscillations have the same amplitude and they widen when it tends to zero; they are contributions from the two independent solutions presented by(6.8)and(6.9).

By changing the region wheret≥0, the oscillations change forms.

This change appears from the effect of the two independent solu- tions, which have a periodic variation. The time dependence of temperature exhibits the same behavior except that the oscilla- tions show exponential growth. Whenaincreases, the dynam- ics of entanglement and the time dependence of temperature increase while the Rényi entropy exhibits a variation inversely proportional to the value ofk. Furthermore, entanglement of the massive particles particularly in the coupled harmonic oscil- lators is studied in Refs. [40,41] based on the Schmidt decompo- sition. They study the case when the coupling parameterKand the frequencies are constant. That is, forKconsidered small in relation to the frequencies, the quantum entanglement can be significant for different quantum states, particularly the vac- uum state. In this case, the exact solution is useful. However, in this article we specify only the treatment of the vacuum state.

7. Conclusion

We use the Heisenberg picture approach to examine the dy- namics of entanglement and the nonadiabatic Berry phases by solving the wave function solution of TDSE of two coupled har- monic oscillators. We then generalize the result of the nonadia- batic Berry phases forNcoupled harmonic oscillators. To take full advantage of this resolution, we use a Hamiltonian whose terms show a time evolution. The dynamics of entanglement is discussed of the vacuum state with the Rényi and the von Neu- mann entropies and we deduce the temperature. After following the dynamics of entanglement with respect to the time evolution of the nonadiabatic Berry phases and the temperature, wefind a double oscillatory behavior of the entanglement with periods equal to the periods of the nonadiabatic Berry phases. While the nonadiabatic Berry phases for N coupled harmonic oscillators appear as the Berry phases of a single massive harmonic oscilla- tor when the mass and the angular frequency are time independ- ent. For two coupled harmonic oscillators whose frequency varies with negative and positive time intervals, the entangle- ment exhibits the same oscillatory behavior as temperature with an exponential increase for temperature. The wave function of the studied system and the cyclical initial state are analyzed with the instantaneous hamiltonian in the initial time (t0).

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