Content uploaded by Mahmoud Abdel-Aty

Author content

All content in this area was uploaded by Mahmoud Abdel-Aty on Jan 26, 2017

Content may be subject to copyright.

arXiv:0805.3576v2 [quant-ph] 11 Jan 2009

Entanglement sudden birth of two trapped ions interacting with a

time-dependent laser ﬁeld

M. Abdel-Aty1and T. Yu2

1Mathematics Department, College of Science, Bahrain University, 32038 Kingdom of Bahrain

2Rochester Theory Center and Department of Physics and Astronomy, University of

Rochester, New York 14627, USA

Journal of Physics B: At. Mol. Opt. Phys. 41 (2008) 235503

We explore and develop the mathematics of the two multi-level ions. In particular,

we describe some new features of quantum entanglement in two three-level trapped ions

conﬁned in a one-dimensional harmonic potential, allowing the instantaneous position of

the center-of-mass motion of the ions to be explicitly time-dependent. By solving the

exact dynamics of the system, we show how survivability of the quantum entanglement

is determined by a speciﬁc choice of the initial state settings.

1 Introduction

The ability to laser cool ions within the vicinity of the motional ground state is a key factor

in the realization of eﬃcient quantum communication [1]. Laser-cooled ions conﬁned in

an electromagnetic trap are good candidates for various quantum information processing

processes such as quantum control and quantum computing [2]. Various interactions

models and higher order non-linear models can be implemented in this system by simply

choosing appropriate applied laser tunings [3, 4]. For more than one ion, as required in

any realistic quantum logic schemes, individual addressing imposes small trap frequencies,

whereas sideband cooling imposes high trap frequencies [5]. Experimentally, the sideband

cooling of two ions to the ground state has been achieved in a Paul trap that operates

in the Lamb-Dicke limit [6]. Recent advances in the dynamics of trapped ions [7]) have

demonstrated that a macroscopic observer can eﬀectively control dynamics as well as

perform a complete measurement of states of microscopic quantum systems.

On the other hand, entanglement is the quintessential property of quantum mechanics

that sets it apart from any classical physical theory, and it is essential to quantify it in

order to assess the performance of applications of quantum information processing [8].

With the reliance in the processing of quantum information on a cold trapped ion, a long-

living entanglement in the ion-ﬁeld interaction with pair cat states has been observed [9].

1

Also, experimental preparation and measurement of the motional state of a trapped ion,

which has been initially laser cooled to the zero-point of motion, has been reported in

[10]. It is well known that the loss of entanglement cannot be restored by local operations

and classical communications, which is one of the main obstacles to build up a quantum

computer [11]. Therefore it becomes an important subject to study entanglement dynam-

ics of qubits [12, 13, 14, 15, 16, 17]. Quite recently, it has been shown that entanglement

of two-qubit systems can be terminated abruptly in a ﬁnite time in the presence of either

quantum or classical noises [12, 17]. This non-smooth ﬁnite-time decay has been referred

as entanglement sudden death (ESD) [12, 18, 19]. At least two experimental veriﬁcations

of ESD have been reported so far [20, 21]. Interestingly, as the entanglement can suddenly

disappear, it can be suddenly generated, a process called entanglement sudden birth [22].

These recent results together with the fundamental interest of the subject, are the

motivation for the present work: we study here the entanglement evolution of two three-

level trapped ions interacting with a laser ﬁeld taking into account the time-dependent

modulated function. We focus our attention on the entanglement decay and generation

due to the presence of a time-dependent modulated function in the two interacting ions and

intrinsic decoherence. We show that entanglement of two ions in the Lamb-Dicke regime

presents some novel features with respect to single-trapped ion. In particular, we show

how sudden feature in the entanglement(sudden birth and sudden death of entanglement)

occurs for the two-ion system coupled to a laser ﬁeld.

The paper is organized as follows. In section 2, we introduce the model and deﬁne

the diﬀerent parameters which will be used throughout the paper. We devote section 3 to

an alternative approach to ﬁnd an exact dynamics of the coupled system in the presence

of the time-dependent modulated function. Section 4 is devoted to study some examples

of entanglement evolutions which lead to diﬀerent types of decay for entangled systems,

and show how diﬀerent initial state settings and time-dependent interaction aﬀect the

decay of the entanglement. Then we conclude the paper with some remarks on how the

presented results could be generalized to multi-ions systems.

2 Model

A useful model of controlled entanglement evolution consists of two three-level ions that

are irradiated by laser beams whose wavevectors lie in the radial plane of the trap. This

2

laser conﬁguration excites the vibrational motion in the radial plane only (1Dionic mo-

tion) [23, 24]. Therefore, the physical system on which we focus is two three-level har-

monically trapped ions with their center-of-mass motion quantized. We denote by aand

a†the annihilation and creation operators and υis the vibrational frequency related to

the center-of-mass harmonic motion along the direction ˆx. Without the rotating wave

approximation, the trapped ions Hamiltonian for the system of interest may be written

as

ˆ

H=ˆ

Hcm +ˆ

Hion +ˆ

Hint,(1)

where

ˆ

Hcm =~υˆa†ˆa,

ˆ

Hion =

2

X

i=1 X

j=a,b,c

~ωiS(i)

jj ,

ˆ

Hint(t) = ~ℑ(ˆx, t)ˆ

S(1)

ab +~ℑ∗(ˆx, t)ˆ

S(1)

ba +~ℑ(ˆx, t)ˆ

S(2)

ab +~ℑ∗(ˆx, t)ˆ

S(2)

ba

+~ℑ(ˆx, t)ˆ

S(1)

ac +~ℑ∗(ˆx, t)ˆ

S(1)

ca +~ℑ(ˆx, t)ˆ

S(2)

ac +~ℑ∗(ˆx, t)ˆ

S(2)

ca .(2)

We denote by ˆ

S(i)

lm the atomic ﬂip operator for the |mii→ |liitransition between the two

electronic states, where ˆ

S(i)

lm =|liiihm|,(l, m =a, b, c). Suppose the ions are irradiated by

a laser ﬁeld of the form ℑ(br, t) = ~−1ǫhi|d.℘|jiexp[−i(kbr−Ωt)],where ǫis the amplitude

of the laser ﬁeld with frequency Ω and polarization vector ℘. The transition in the

three-level ions is characterized by the dipole moment dand kis the wave vector of the

laser ﬁeld. Therefore if we express the center of mass position in terms of the creation

and annihilation operators of the one-dimensional trap namely ˆx= ∆x(a†+a),where

∆x= (~/2υm)1/2=η/k is the widths of the dimensional potential ground states, in the

xdirection (ηis called Lamb–Dicke parameter describing the localization of the spatial

extension of the center-of-mass), and mis the mass of the ions.

By using a special form of Baker-Hausdorﬀ theorem [3], the operator exp[iη(a†+a)]

may be written as a product of operators i.e. exp(iη(a†+a)) = exp η2

2[a†, a]exp iηa†exp (iηa).

The physical processes implied by the various terms of the operator

exp iη a†+a= exp −η2

2∞

X

n=0

(iη)na†n

n!

∞

X

m=0

(iη)mam

m!,(3)

may be divided into three categories (i) the terms for n > m correspond to an increase in

energy linked with the motional state of center of mass of the ion by (n−m) quanta, (ii)

3

the terms with n < m represent destruction of (m−n) quanta of energy thus reducing

the amount of energy linked with the center of mass motion and (iii) (n=m), represents

the diagonal contributions. When we take Lamb-Dicke limit and apply the rotating wave

approximation discarding the rapidly oscillating terms, the interaction Hamiltonian (2)

takes a simple form

ˆ

Hint =

2

X

i=1 ~λ(i)

1(t)E(a†a)ˆ

S(i)

12 a†+~λ(i)∗

1(t)E∗(a†a)ˆ

S(i)

21 a

+~λ(i)

2(t)E(a†a)ˆ

S(i)

13 a†+~λ(i)∗

2(t)E∗(a†a)ˆ

S(i)

31 a,(4)

where λ(i)

j(t) is a new coupling parameter adjusted to be time dependent. The other

contributions which rapidly oscillate with frequency νhave been disregarded. Note that

in the Lamb-Dicke regime only processes with ﬁrst and second order of ηare considered,

while in the general case, the nonlinear coupling function is derived by expanding the

operator-valued mode function as

Ek(a†a) = −ǫ

2exp −η2

2∞

X

n=0

(iη)2n+k

n!(n+k)!a†nan.(5)

Since Ek(a†a) depends only on the quantum number a†a, in the basis of its eigenstates,

a†a|ni=n|ni, (n= 0,1,2, ...), these operators are diagonal, with their diagonal elements

hn|Ek(a†a)|niis given by E(j)

k(n) = −0.5ǫ(n+k)!)−1n!Lk

n(η2) exp (−η2/2) where Lk

n(η2)

are the associated Laguerre polynomials.

3 Solution

3.1 Master equation

To study the time evolution of the quantum entanglement, we will start by ﬁnding the

general solution of the two-ion system with the Hamiltonian ( 4). The intrinsic decoher-

ence approach [25], is based on the assumption that on suﬃciently short time scales the

system does not evolve continuously under unitary evolution but rather in a stochastic

sequence of identical unitary transformations. Under the Markov approximation [25], the

master equation governing the time evolution for the two-ion system under the intrinsic

decoherence is given by

d

dtˆρ(t) = −i

~[ˆ

H, ˆρ]−γ

2~2[ˆ

H, [ˆ

H, ˆρ]],(6)

4

where γis the intrinsic decoherence parameter. The ﬁrst term on the right-hand side

of equation (6) generates a coherent unitary evolution of the density matrix, while the

second term represents the decoherence eﬀect on the system and generates an incoherent

dynamics of the coupled system. In order to obtain an exact solution for the density

operator ˆρ(t) of the master equation (6), three auxiliary superoperators b

J, b

Sand b

Lcan

be introduced as (~= 1)

exp b

Jτ ˆρ(t) = ∞

X

k=0

(γτ )k

k!ˆ

Hkˆρ(t)ˆ

Hk,

exp b

Sτ ˆρ(t) = exp −iˆ

Hτ ˆρ(t) exp iˆ

Hτ ,(7)

exp b

Lτˆρ(t) = exp −τγ

2ˆ

H2ˆρ(t) exp −τγ

2ˆ

H2,

where b

Jˆρ(t) = γˆ

Hˆρ(t)ˆ

H, b

Sˆρ(t) = −i[ˆ

H, ˆρ(t)] and b

Lˆρ(t) = −(γ/2)( ˆ

H2ˆρ(t) + ˆρ(t)ˆ

H2).

Then, it is straightforward to write down the formal solution of the master equation (6)

as follows

ˆρ(t) = exp( b

Jt) exp( b

St) exp( b

Lt)ˆρ(0)

=∞

X

k=0

ˆ

Mk(t)ˆρ(0) ˆ

M†

k(t).(8)

We denote by ˆρ(0) the density operator of the initial state of the system and ˆ

Mk=

(γt)k/2

√k!ˆ

Hkexp −iˆ

Htexp −γt

2ˆ

H2, where the so-called Kraus operators ˆ

Mksatisfy ∞

P

k=0

ˆ

Mk(t)ˆ

M†

k(t) =

ˆ

Ifor all t.

Eq. (8) can also be written as

ˆρ(t) = e−iˆ

Ht exp −γt

2ˆ

H2ne(ˆ

SMt)ˆρ(0)oexp −γt

2ˆ

H2eiˆ

Ht ,(9)

where, we deﬁne the superoperator ˆ

SMˆρ(0) = γˆ

Hˆρ(0) ˆ

H. The reduced density matrix of

the ions is given by taking the partial trace over the ﬁeld system, i.e. ρa(t)≡trFρ(t) =

ρij,lk(t)|ijihlk|.We use the notation |iji=|i1i⊗|j2i,(i, j =a, b, c),where |a1(2)i,|b1(2)iand

|c1(2)iare the basis states of the ﬁrst (second) ion and ρij,lk(t) = hij|ρa(t)|lkicorresponds

the diagonal (ij =lk) and oﬀ-diagonal (ij 6=lk) elements of the density matrix ρa(t).

It is worth referring here to the fact that, Eq. (9) represents the general solution of

the master equation (6) that can be used to study the eﬀect of the intrinsic decoherence

when the modulated function λ(i)

j(t) is taken to be time-independent. However, to study

5

the time-dependent case as seen below, it will be much more convenient to use the exact

wave function.

3.2 Wave function

At any time the state vector |Ψ(t)iin the interaction picture can be written as

|Ψ(t)i=

9

X

i=1

∞

X

n=0

Ai(n, t)|ψii.(10)

where |ψ1i=|n;a1a2i,|ψ2i=|n+ 1; a1b2i,|ψ3i=|n+ 1; a1c2i,|ψ4i=|n+ 1; b1a2i,|ψ5i=

|n+ 2; b1b2i,|ψ6i=|n+ 2; b1c2i,|ψ7i=|n+ 1; c1a2i,|ψ8i=|n+ 2; c1b2i,|ψ9i=

|n+ 2; c1c2i,where |nicorresponds to the initial state of the vibrational motion. We

denote by Ai(n, t) the probability amplitude for the ion to be in the state |ψii. In general,

the dynamics of the probability amplitudes Ai(n, t) is given by the Schr¨odinger equation,

i∂Aj(n, t)

∂t =

9

X

l=1

Cjl Al(n, t),(11)

where Cjk =ψjˆ

Hint |ψki.These equations are exact for any two three-level system.

In order to consider the most general case, we solve equation (11), by assuming a new

variable G(n, t) as G(n, t) =

9

P

i=1

xiAi(n, t),where x1= 1,which means that

idG(n, t)

dt =γ(t)β1 A1(n, t) +

9

X

i=2

β′

iAi(n, t)!,(12)

where λ(i)

j(t) = γ(t)λjand βiare given by β1=x2C21 +x3C31 +x4C41 +x7C71,and β′

i=

βi/β1,where β2=C11 +x5C51 +x8C81, β3=C13 +x6C63 +x9C93, β4=C14 +x5C54 +x6C64 ,

β5=x2C25 +x4C45, β6=x3C36 +x4C46, β7=C17 +x8C87 +x9C97, β8=x2C28,

β9=x3C39 +x7C79,

Let us emphasize that in addition to the general form of equation (12 ), the present

method is also suitable for any initial conditions. It is instructive to examine the formation

of a general solution of the two three-level systems. Therefore, we use equation (12) and

seek G(n, t) such that i˙

G(n, t) = zζ(t)G(n, t).This holds if β1=zand βi=xi.Some

simple algebra gives rise to an equation that has nine eigenvalues such that the zi[26] are

determined. Correspondingly, there are also nine corresponding eigenfunctions

Gj(n, t) = Gj(0) exp −izjZt

0

ζ(τ)dτ,(13)

6

where Gj(n, t) =

9

P

k=1

Mjk Ak(n, t),and Mji =bpTˆexi +

9

P

i=1

ˆxT

iˆexi.We denote by ˆexi mutually

orthogonal unit vectors, given by ˆexi= (δi1, δi2, δi3, δi4, δi5, δi6, δi7, δi8, δi9), δij = 1 if i=j

and δij = 0 if i6=jand bp= (1,1,1,1,1,1,1,1,1),ˆxi= (xi1, xi2, xi3, xi4, xi5, xi6, xi7,

xi8, xi9).Then, one can express the unperturbed state amplitude Ai(n, t) in terms of the

dressed state amplitude Gj(n, t) as follows

Ai(n, t) =

9

X

j=1

M−1

ij Gj(n, t) =

9

X

j=1

M−1

ij Gj(0) exp −izjZt

0

ζ(τ)dτ.(14)

We have thus completely determined the exact solution of a two three-level system in the

presence of the time-dependent modulated function.

4 Entanglement evolution

Good measures of entanglement are invariant under local unitary operations. They are

also required to be entanglement monotones, that is, they must be non-increasing under

local quantum operations combined with classical communication [27, 28, 29, 30, 31, 32].

computable entanglement measures for a generic multiple level system do not exist. For

two-qubit systems, concurrence can be used to compute entanglement for both pure and

mixed states [33]. Rungta et al [34] deﬁned the so-called I-concurrence in terms of a

universal-inverter which is a generalization to higher dimensions of two spin ﬂip operation,

therefore, the concurrence in arbitrary dimensions takes the form

Iψ(t) = phψ|SN1⊗SN1(|ψihψ|)|ψi.(15)

Another generalization is proposed by Audenaert et al [35] by deﬁning a concurrence

vector in terms of a speciﬁc set of antilinear operators. As a complete characterization of

entanglement of a bipartite state in arbitrary dimensions may require a quantity which,

even for pure states, does not reduce to single number [36], Fan et al. deﬁned the concept

of a concurrence hierarchy as N−1 invariants of a group of local unitary for N−level

systems [37].

Now, we suppose that the initial state of the principle system is given by

|ψ(0)i=cos θ|a1, b2i+ sin θeiφ |b1, a2i⊗∞

X

n=0

qn|ni,(16)

7

012345

lt0

1

2

3

q

0

0.2

0.4

0.6

01234

lt

012 3 4 5

lt

0

0.5

1

1.5

2

2.5

3

q

Figure 1: The evolution of the concurrence Iψ(t) as a function of the scaled time λ1tand

θ. The parameters are n= 5, λ2/λ1= 0.01, and ϕ= 0.

where θ∈[0,2π], φ ∈[0, π] and qn=hn|ψ(0)i.

In absence of intrinsic decoherence, the concurrence (15) will be calculated using Eq.

(10), while to study the eﬀect of γwe use equation (9). In ﬁgures 1 and 2 the dependence

on θand the scaled time λ1t, in the Lamb-Dicke regime, of the concurrence Iψ(t) is

illustrated. In the typical experiments at NIST [39], 9Be+ions are stored in a RF Paul

trap with a secular frequency along bxof ν/2π≃11.2 MHz, providing a spread of the

ground state wave function of ∆x≃7 nm, with a Lamb-Dicke parameter of η≃0.202.

With these data we ﬁnd ǫ≃0.01, so they can be considered as small parameters. We

see that the concurrence Iψ(t) exhibits some peaks whose amplitude decreases as the

interaction time increases. The main consequence is that we can select a given value of the

superposition parameter θ, or rather a speciﬁc instant time to obtain strong entanglement

or disentanglement.

The entanglement sudden birth phenomenon is observed in ﬁgure 1. Entanglement

is not present at earlier times, and suddenly at some ﬁnite time an entanglement starts

to build up. Also, from ﬁgure 1, it is clearly seen that the value of ﬁrst local maximum

signiﬁcantly exceeds the second local maximum when the two ions start from a superpo-

sition state. Recall that the entanglement attains the zero value (i.e., disentanglement)

when the trapped ions start from either |a1, b2ior |b1, a2istates, while strong entangle-

8

ment occurs when the inversion is equal to zero, i.e, the two ions start from a maximum

entangled state, such as |ψAB(0)i= (|a1, b2i+|b1, a2i)/√2. In other words, for the initial

entangled state, there are some intervals of the interaction time where the entanglement

reaches its local maximum and drops to zero (entanglement sudden death).

012345

lt0

1

2

3

q

0

0.2

0.4

0.6

01234

lt

012 3 4 5

lt

0

0.5

1

1.5

2

2.5

3

q

Figure 2: The same as ﬁgure 1 but n= 15.

To gain more insight into the general behavior of the quantum entanglement evolution,

we plot in ﬁgure 2 the time evolution of the generalized concurrence Iψ(t) for a larger value

of the ﬁeld intensity, n. It can be seen that the dynamics is strongly modiﬁed where a

smooth decay of the entanglement is observed. In contrast to the dynamics with the small

n, where the sudden death of entanglement-like feature has been observed after the ﬁrst

local maximum of the entanglement (see ﬁgures 1 and 2), here the entanglement smoothly

decays and vanishes as the time increased further. Note that the eﬀect of the superposition

parameter θon the entanglement distributions in both ﬁgure 1 and 2 is similar and shows

symmetry around θ=π

2while the local maxima correspond to θ=nπ

4,(n= 1,3,5, ...).

Obviously from these ﬁgures, if θ=nπ

2,(n= 0,1,2, ...) i.e., the system starts from a

separable state, the entanglement is always zero. On the other hand, if the system starts

from an entangled state and because of the existence of further system parameters, the

entanglement decay occurs with small or large values of n. Moreover, comparison of

ﬁgures 1-2 shows that the entanglement decay takes a longer time to reach zero in the

case where large nis considered. We hence come to understand that considering a large

9

intensity of the initial state of the ﬁeld, can be used positively in preventing entanglement

sudden death or delaying the disentanglement of the two ions. A physical explanation

of why is the intensity of the ﬁeld playing such a role in the disentangling process of

the two ions, is that the strong ﬁeld providing some sort of shielding mechanism to the

decoherence eﬀectively induced by the trace over the motional degree of freedom.

00.5 1

1.5

2

2.5

lt

0

0.25

0.5

0.75

1

gl

0

0.25

0.5

0.75

1

00.5 1

1.5

2

lt

0 0.5 1 1.5 2 2.5

lt

0

0.2

0.4

0.6

0.8

1

gl

Figure 3: The concurrence as a function of the scaled time and decoherence parameter.

The other parameters are the same as ﬁgure 1.

Quantum coherence and entanglement typically decay as the result of the inﬂuence of

decoherence and much eﬀort has been directed to extend the coherence time of systems

of interest. However, it has been shown that under particular circumstances where there

is even only a partial loss of coherence of each ion, entanglement can be suddenly and

completely lost [12]. This has motivated us to consider the question of how decoherence

eﬀects the scale of entanglement in the system under consideration. Once the decoherence

taken into account, i.e., γ6= 0, it is very clear that the decoherence plays a usual role

in destroying the entanglement. In this case and for diﬀerent values of the decoherence

parameter γ, we can see from ﬁgure (3) that after switching on the interaction the entan-

glement function increases to reach its maximum showing strong entanglement. However

its value decreases after a short period of the interaction time to reach its minimum.

The function starts to increase its value again however with lower local maximum values

showing a strong decay as time goes on. Also, from numerical results we note that with

10

the increase of the parameter γ, a rapid decay of the entanglement (entanglement sudden

death) is shown [15, 38].

-10 -5

0

5

10

lt0

1

2

3

q

0

0.2

0.4

0.6

-10 -5

0

5

lt

-10 -5 0510

lt

0

0.5

1

1.5

2

2.5

3

q

Figure 4: The same as ﬁgure 1 but the modulated function ζ(t) = sech(t/2τ).

The above results and connections are very intriguing, and naturally lead us to ask

what is the role played by the modulated function in obtaining these associated phenomena

of the entanglement. In order to answer this question, we consider the modulated function

ζ(t) to be time-dependent of the form ζ(t) = sech(t/2τ) [40, 41].In this form the coupling

increases from a very small value at large negative times to a peak at time t= 0, to

decrease exponentially at large times. Thus, depending on the value of τand the initial

time t0, various limits such as adiabatically or rapidly increasing (for t0< λt ≤0) or

decreasing (for 0 ≤λt < t0) coupling can be conveniently studied. This allows us to

investigate, analytically, the eﬀect of transients in various diﬀerent limits of the eﬀect

of switching the interaction on and oﬀ in the ion-ﬁeld system. The vanishing of the

interaction at large positive times leads to the leveling out of the inversion.

It should be noted that the time dependence speciﬁed in ζ(t) is one of a class of

generalized interactions that may oﬀer analytical solutions. It is evident from ﬁgure 4

that the entanglement sudden birth phenomenon is more clearly observed for the time-

dependent interaction. In this case, the entanglement starts suddenly to build up at later

times compared with the time-independent case. At this point the entanglement from zero

evolves to its local maximum value and then oscillates with lower local maximum followed

11

by smooth decay (see ﬁgure 4). Although increasing the ﬁeld intensity leads to strong

entanglement (maximum value of entanglement), however the local maximum values of

the entanglement also vary and occur for some short periods of the interaction time. This

indicates that in a regime where coherent state is considered, the underlying states are

highly entangled. Consequently, the presence of the time-dependent modulated function

increases the number of oscillations and delays the disentanglement. All these results

conﬁrm the possibility of a practical observation of time-dependence of the modulated

function eﬀects for prolonging time for the disentanglement. Our results here show the

important role played by the modulated function in the entanglement dynamics which is

crucial for the onset of either entanglement sudden death or sudden birth in the trapped

ion systems considered in this paper. In addition, our results suggest that the analytical

results presented here, could be attained for diﬀerent conﬁgurations of any two trapped

ions systems [42].

The remaining task is to identify and compare the results presented above for the

entanglement with another accepted entanglement measure such as the quantum relative

entropy. Eisert and Plenio [43] have raised the question of the ordering of entanglement

measures. It has been proven that all good asymptotic entanglement measures are either

identical or fail to uniformly give consistent orderings of density matrices [44]. One of the

best understood cases is entanglement measure deﬁned in terms of the quantum relative

entropy. More explicitly, for the entangled states ˆρ(t) the quantum relative entropy is

deﬁned by the following formula as the distance between the entangled state ˆρ(t) and

disentangled state trAˆρ(t)⊗trBˆρ(t)∈S(H1⊗ H2) [45, 46]

IρρA

t, ρB

t=trˆρ(t)(log ˆρ(t)−log(trAˆρ(t)⊗trBˆρ(t))),(17)

where ρA

t=trB(ˆρ(t)) and ρB

t=trA(ˆρ(t)) ,A(B) refers to the ﬁrst (second) ion.Note

that if the entangled state ˆρ(t) is a pure state, S(ˆρ(t)) = 0 and then S(trAˆρ(t)) =

S(trBˆρ(t)),which means that we have IρρA

t, ρB

t= 2S(trBˆρ(t)). One, possibly not very

surprising, principal observation is that the numerical calculations corresponding to the

same parameters, give nearly the same behavior with diﬀerent scales. This means that the

entanglement measured by either quantum relative entropy or concurrence measures gives

rise to qualitatively the same results. We must stress, however, that no single measure

alone is enough to quantify the entanglement in a multilevel system.

12

5 Conclusions

In summary, we have derived an intuitive extension of the standard quantum model of two

three-level trapped ions interacting with a laser ﬁeld to include the time-dependent mod-

ulated function and intrinsic decoherence. This study reveals that the time-dependent

modulated function can be used for generating either entanglement sudden death or sud-

den birth depending on a proper manipulation of the initial state setting. We note that

the existence of entanglement sudden death reveals a fact that the non-interacting and

non-communicating two ions can abruptly lose their entanglement. Also, it will be very

interesting to extend these results to the case of mixed states in the presence of the de-

coherence. We hope the presented results can be useful for the ongoing theoretical and

experimental eﬀorts in multi-levels particles interaction.

Acknowledgment

T. Y. acknowledges grant support from US National Science Foundation (PHY-0758016).

We are grateful to Prof. A.-S. F. Obada for helpful discussions.

References

References

[1] D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King, and D. M.

Meekhof, J. Res. Natl. Stand. Tech. 103, 259 (1998).

[2] J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995).

[3] C.A. Blockley, D.F. Walls and H. Kisken, Europhys. Lett. 17, 509 (1992).

[4] W. Vogel and R.L.de Matos Filho, Phys. Rev. A 52, 4214 (1995); W. Vogel and

D.-G. Welsch, ibid. 40, 7113 (1989); A. Steane, C.F. Roos, D. Stevens, A. Mundt, D.

Leibfried, F. Schmidt-Kaler, and R. Blatt, ibid. 62, 042305 (2000); L.F. Wei, Y.-X.

Liu, and F. Nori, Phys. Rev. A 70, 063801 (2004)

[5] G. Morigi, J. Eschner, J. I. Cirac, and P. Zoller, Phys. Rev. A 59, 3797 (1999)

13

[6] B. E. King, C. S. Wood, C. J. Myatt, Q. A. Turchette, D. Leibfried, W. M. Itano,

C. Monroe, and D. J. Wineland, Phys. Rev. Lett. 81, 1525 (1998).

[7] D. Leibfried, R. Blatt, C. Monroe and D. Wineland, Rev. Mod. Phys. 75, 281 (2003).

[8] S. Hill and W. K. Wootters, Phys. Rev. Lett. 78, 5022 (1997); M. A. Nielsen, Phys.

Rev. Lett. 83, 436 (1999).

[9] M. Abdel-Aty, Appl. Phys. B: Laser and Opt. 84, 471 (2006).

[10] D. Leibfried, D. M. Meekhof, B.E. King, C. Monroe, W.M. Itano and D.J. Wineland,

J. Mod. Opt. 44, 2485 (1997).

[11] C. H. Bennett, H. J. Bernstein, S. Popescu and B. Schumacher, Phys. Rev. A 53,

2046 (1996).

[12] T. Yu and J. H. Eberly, Opt. Commun. 264, 393 (2006).

[13] M. Yonac, T. Yu and J. H. Eberly, J. Phys. B: At. Mol. Opt. Phys. 39, S621(2006).

[14] C. Pineda and T. H. Seligman, Phys. Rev. A 73, 012305 (2006).

[15] T. Yu and J. H. Eberly, Phys. Rev. B 66, 193306 (2002).

[16] T. Yu and J. H. Eberly, Phys. Rev. B 68, 165322 (2003).

[17] T. Yu and J. H. Eberly, Phys. Rev. Lett. 93, 140404 (2004).

[18] T. Yu and J. H. Eberly, Phys. Rev. Lett. 97, 140403 (2006).

[19] T. Yu and J.H. Eberly, Quant. Inf. Comp. 7, 459 (2007).

[20] M.P. Almeida et al, Science 316, 579 (2007). See also J.H. Eberly and T. Yu, Science

316, 555 (2007).

[21] J. Laurat et al., Phys. Rev. Lett. 99, 180504 (2007).

[22] Z. Ficek and R. Tanas, Phys. Rev. A 77, 054301 (2006); C. E. Lopez, G. Romero,

F. Lastra, E. Solano and J. C. Retamal, Phys. Rev. Lett. 101, 080503 (2008).

[23] A. Messina, S. Maniscalco and A. Napoli, J. Mod. Opt. 50, 1 (2003), M. Abdel-Aty,

Prog. Quant. Electron., 31, 1 ( 2007).

14

[24] S. S. Sharma and N. K. Sharma, J. Phys. B 35, 1643 (2002).

[25] G.J. Milburn, Phys. Rev. A 44, 5401 (1991); S. Schneider and G.J. Milburn, Phys.

Rev. A 57, 3748 (1998); S. Schneider and G.J. Milburn, Phys. Rev. A 59, 3766

(1999).

[26] J. H. Mc-Guire, K. K. Shakov and K. Y. Rakhimov, J. Phys. B 36, 3145 (2003).

[27] V. Vedral, M. B. Plenio and P. L. Knight, ”The Physics of Quantum Information”,

edited by D Bouwmeester, A Ekert and A Zeilinger, Springer (2000); V. Vedral, M.

B. Plenio, K. Jacobs, and P. L. Knight, Phys. Rev. A 56, 4452 (1997); G. Vidal, J.

Mod. Opt. 47, 355 (2000).

[28] S. J. D. Phoenix and P. L. Knight, Ann. Phys. (N. Y) 186, 381 (1988).

[29] S. J. D. Phoenix and P. L. Knight, Phys. Rev. A 44, 6023 (1991).

[30] S. J. D. Phoenix and P. L. Knight, Phys. Rev. Lett. 66, 2833 (1991).

[31] D. W. Berry and B. C. Sanders, J. Phys. A 36, 12255 (2003).

[32] S. Bose, I. Fuentes-Guridi, P. L. Knight, and V. Vedral, Phys. Rev. Lett. 87, 050401

(2001); S. Scheel, J. Eisert, P. L. Knight, M. and B. Plenio, J. Mod. Opt. 50, 881

(2003); E. Ciancio and P. Zanardi, Phys. Lett. A 360, 49 (2006).

[33] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).

[34] P. Rungta, V. Buzek, C. M. Caves, M. Hillery and G. J. Milburn, Phys. Rev. A 64,

042315 (2001); S. J. Akhtarshenas, J. Phys. A: Math. Gen. 38, 6777 (2005).

[35] K. Audenaert, F. Verstraete and De Moor, Phys. Rev. A 64, 052304 (2001).

[36] D. Meyer and N. Wallach, in The Mathematics of Quantum Computation, edited by

R. K. Brylinski and G. Chen (CRC Press, Boca Raton, 2002), p. 77.

[37] H. Fan, K. Matsumoto and H. Imati, J. Phys. A: Math. Gen. 36, 4151 (2003).

[38] A.-S. F. Obada, and M. Abdel-Aty, Phys. Rev. B 75, 195310 (2007); M. Abdel-Aty,

and H. Moya-Cessa, Phys. Lett. A 369, 372 (2007).

15

[39] D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano and D. J. Winel,

Phys. Rev. Lett. 77, 4281 (1996); Q. A. Turchette, et. al., Phys. Rev. A 61, 063418

(2000).

[40] S. V. Prants and L. S. Yacoupova, J. Mod. Opt. 39, 961 (1992); A. Joshi and S.V.

Lawande, Phys. Rev. A 48, 276 (1993); A. Joshi and S.V. Lawande, Phys. Lett. A

184, 390 (1994).

[41] A. Dasgupta, J. Opt. B: Quantum Semiclass. Opt. 1, 14 (1999).

[42] E. Solano, R. L. de Matos Filho and N. Zagury, Phys. Rev. A 59, 2539 (1999)

[43] J. Eisert and M. B. Plenio, J. Mod. Opt. 46, 145 (1999)

[44] S. Virmani and M. B. Plenio, Phys. Lett. A 268, 31 (2000)

[45] S. Furuichi and M. Abdel-Aty, J. Phys. A: Math. & Gen. 34 6851 (2001)

[46] G. Lindblad, Commun. Math. Phys., 33, 111 (1973); E. Leib and M. B. Ruskai,

Phys. Rev. Lett. 30, 434 (1973); J. Math. Phys. 14, 1938 (1973).

16