the tangle of the lambda-type three levels atoms and the two-mode cavity field

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Physica A 358 (2005) 313–327

0378-4371/$ -

doi:10.1016/j

�CorrespoE-mail a

zwzhou@ust

www.elsevier.com/locate/physa

The tangle of the Lambda-type three levels atomsand the two-mode cavity field

Ting Huanga,�, Xiu-Min Lina,b, Zheng-Wei Zhoua,Zhuo-Liang Caoc, Guang-Can Guoa

aKey Laboratory of Quantum Information, Department of Physics, University of Science and Technology of

China, Hefei 230026, People’s Republic of ChinabSchool of Physics and Optoelectronics Technology, Fujian Normal University, Fuzhou 350007,

People’s Republic of ChinacDepartment of Physics, Anhui University, Hefei 230039, China

Received 6 February 2005

Available online 23 June 2005

Abstract

A system comprising Lambda-type three level atoms and the two-mode Fock cavity field is

considered in this paper. Under the adiabatical approximation and the large detuning

condition, the effective Hamiltonian of the system in the interaction picture can be given out.

If the two identical three level atoms pass through the cavity in turn, the entangled state atoms

can be generated. When the interaction time is taken to an appropriate value, the maximally

entangled states are created. On the same time, the tangles for the multipartite of the system

are studied in detail.

r 2005 Elsevier B.V. All rights reserved.

Keywords: Entangled atoms; Tangle

see front matter r 2005 Elsevier B.V. All rights reserved.

.physa.2005.04.037

nding author.

ddresses: [email protected] (T. Huang), [email protected] (X.-M. Lin),

c.edu.cn (Z.-W. Zhou).

www.elsevier.com/locate/physa

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T. Huang et al. / Physica A 358 (2005) 313–327314

1. Introduction

As a potential resource for communication and information processing, quantumentanglement has been the subject of much study in recent years. It is the quantummechanical property that Schrodinger pointed out many years ago as ‘‘thecharacteristic trait of quantum mechanics’’ [1]. A pure state of a pair of quantumsystems is called entangled if it is unfactorizable, for example, for the singlet state oftwo spin-1

2particles,

1ffiffiffi2

p ðj "#i þ j #"iÞ .

A mixed state is entangled if it cannot be represented as a mixture of factorizablepure states. As with other resources such as free energy and information, thereshould be a quantitative theory of entanglement giving detailed rules about how itcan and cannot be manipulated; such a theory has begun to develop. Theentanglement must be quantified. In recent years, a number of entanglementmeasures for bipartite states have been introduced and analyzed [2–8]. Measures ofentanglement associated with spin squeezed states have been studied by Stockton etal. [9] under the assumption that all of the atoms in the ensemble are symmetricallycoupled to the ‘‘quantum bus’’.However, totally quantifying entanglement in general cases is very difficult, which

is an unsolved problem [10]. Unlike classical correlations, quantum entanglementcannot be freely shared among many parties. Entanglement in tripartite system hasbeen studied by Coffman et al. [11] for the case of three qubits. They show us thatsuch quantum entanglement cannot be arbitrarily distributed amongst thesubsystems; and the existence of three-body entanglement restricts the distributionof the bipartite entanglement of the remainder after tracing over any one of thequbits. Entanglement sharing in the two-atom Tavis–Cummings model has beenstudied by Tessier et al. [12]. They consider how the multipartite entanglement isgenerated in the context of a system consist of two two-level atoms resonantlycoupled to a single mode of the electromagnetic field. The restriction on the sharingof entanglement takes a particularly elegant form in terms of a measure ofentanglement called the ‘‘tangle’’, which is closely related to the entanglement offormation.There are many schemes for the generation of the max entangled state. Shi-Biao

Zheng has proposed a scheme for the generation of Greenberger–Home–Zeilinger(GHZ) states for three atoms. It is based on resonant atom–field interactions [13].Moreover, he proposed another scheme of the generation of GHZ states. Thisscheme is easily generalized for preparing n-atom entangled states [14].In this paper, we let a Lambda-type three level atom pass through a two-mode

cavity field. The correlations between the atom and two-mode cavity field and thatbetween one mode and the remainder of the system can be quantified by the tangle.After the first atom leaves the cavity, another identical atom passes through thecavity, too. Via the procedure, the entangled atoms can be generated. When theinteraction time is taken to an appropriate value, the maximally entangled atoms can

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T. Huang et al. / Physica A 358 (2005) 313–327 315

be created. Here, the ground hyperfine levels of the Lambda-type three-level atoms areused as qubit, which is more stable than the qubit composed of the excited state andground state of a two-level state because the former is not related to the spontaneousemission of the atom. On the same time, it is convenient to let the atoms pass throughthe cavity. It is more operable on experiments than the situation that the atom movesin the cavity. There is no need to measure the states of the atoms in the cavity. Thetangles between the multipartite of the system are analyzed at length. This has theimplications for the research of quantum control of ensembles. If the quantified cavityfield is measured, the degree of correlation between the field and the atom ensembledetermines the degree of backaction on the atom ensemble. We can then quantify thedegree to which one can perform quantum control on the atom ensemble.The remainder of the paper is organized as follows. A system comprises a Lambda-

type three-level atom and two-mode cavity is introduced in detail in Section 2. Theeffective Hamiltonian can be achieved under the limit of large detuning in this section.In Section 3, we draw our attention to the situation that one atom passes through thetwo-mode cavity. The tangle for the multipartite of the system can be achieved. InSection 4, another identical atom passes through the cavity; thus, the entangled atomscan be attained. With the wave functions of the system in hand, we calculate thetangles for each of the bipartite partitions of the multipartite of the system. Meantime,we can get the EPR state. In the last section, we summarize our results.

2. The model

Let us consider a three-level atomic configuration that is showed in Fig. 1. Theatom passes through a two-mode Fock state cavity field. The Hamiltonian underrotating wave approximation can be written as

H ¼ _½oegjeihej þ ofgjf ihf j þ o1aþa þ o2b

þb

þ ðg1jeihgja þ g2jeihf jb þ g1aþjgihej þ g2b

þjf ihejÞ� , ð1Þ

where, a, b are the annihilation operators of the two modes interact with jei2jgi andjei2jf i transitions, respectively. They are corresponding to the different polarization

g

f

∆

a

b

e

g

Fig. 1. Three-level atomic configuration with levels jgi; jf i and jei interacting with two orthogonal modes

of the cavity, described by annihilation operators a and b. Here, jgi; jf i correspond to the ground hyperfinelevels and jei represents the excited level of the atom. g stands for the atom-cavity coupling of the modes

with the corresponding transitions, D is the detuning of the modes from the corresponding atomic

transition.

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directions. olgðl 2 e; f Þ is the atomic transition frequency, oiði 2 1; 2Þ is the frequencyof the cavity modes, and gi is the atom-cavity coupling constant, which is assumedreal. The interaction Hamiltonian in the interaction picture can be written as

H 0 ¼ _ðg1jeihgjaeiD1t þ g2jeihf jbe

iL2t þ g1aþjgiheje�iD1t þ g2b

þjf iheje�iD2tÞ ,

(2)

where, Di ¼ oeg;f � oiði 2 1; 2Þ is the detuning of the cavity modes from thecorresponding atomic transition.Now, we take g1 ¼ g2 ¼ g, D1 ¼ D2 ¼ D and _ ¼ 1. Under the adiabatical

approximation and the large detuning approximation, the effective Hamiltonian canbe given as

Heff ¼g2

Dðjgihgjaþa þ jgihf jaþb þ jf ihgjbþa þ jf ihf jbþbÞ . (3)

Here, the first term and the last term represent the Stark shifts and the rest twoterms demonstrate the interaction leading to a transition from the initial state to thefinal state. From the effective Hamiltonian, it is apparent that the Stark shift termsare of the same order of the magnitude as the coupling term. So they cannot beignored from the effective Hamiltonian.

3. The interaction of one atom with the cavity

Now, we let a Lambda-type atom which is showed in Fig. 1 passes though the two-mode cavity. It is assumed that the two-mode cavity be in the state j10i initially andthe atom arrive at t ¼ 0 in the state jgi and leave it at t ¼ t1. For the chosen initialconditions, the relevant subspace of the states we have to consider is fjgij10i,jf ij01ig. Under the interaction of the effective Hamiltonian, one can acquire the nextequations which display the evolution of the state.

jgij10i ! a1ðt1Þjgij10i þ b1ðt1Þjf ij01i, (4)

where,

a1ðt1Þ ¼1

21þ Cos

2g2

Dt1 � iSin

2g2

Dt1

� �, (5)

b1ðt1Þ ¼1

2�1þ Cos

2g2

Dt1 � iSin

2g2

Dt1

� �. (6)

The probability that the three-level atom is found in the state jgi at t ¼ t1 can beobtained from Eq. (5).

wð1Þg ðt1Þ ¼ ja1ðt1Þj

2 ¼1

21þ Cos

2g2

Dt1

� �, (7)

where, the superscript ‘‘(1)’’ and the subscript ‘‘1’’ refer to the first atom.

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It is known from the above equation that the probability of finding the atom in thestate jgi depends on the coupled constant g, interaction time t1 and the detuning ofthe cavity-mode from the atom transition D. If we set Cosð2g2=DÞt1 ¼ �1, that is tosay, when t1 ¼ ð2k þ 1ÞpD=2g2 (k is an arbitrary integral), the incident atom istransferred from the state jgi to the state jf i under the effect of the quantified cavitymodes. The procedure is just like the Raman procedure.Now, let us introduce a new measurement of entanglement which is called as

‘‘tangle’’ to describe the degree of correlation between the multipartite of the system.Coffman et al. [11] defined the tangle t2 for a system of two qubits as

t2ðrÞ � maxf0; l1 � l2 � l3 � l4g2 , (8)

where the l0isði ¼ 1; 2; 3; 4Þ are square roots of the eigenvalues of the non-Hermitianoperator r ~r, which are ordered in decreasing order. Here, r ¼

Pi pijciihcij and the

tilde displays the spin flip of the quantum state, i.e., j ~cii � sy � syjc�i i and the

asterisk symbolizes complex conjugation in the standard basis.Indeed, the definition can be extended directly to the result of Rungta et al. [15] in

order to get an analytic form for the tangle t of a bipartite system in a pure state witharbitrary subsystem dimensions,

tðcABÞ � C2ðcABÞ ¼ 2nAnB½1� trðr2AÞ� . (9)

At this point we note that the scale factors nA and nB, which may in general dependon the dimensions of the subsystems DA and DB, respectively, are usually set to oneso that agreement with the two-qubit case is maintained, and so that the addition ofextra unused Hilbert-space dimensions has no effect on the value of the tangle.Now, under the assumption that the system is in an overall pure state, using Eq.

(9), the tangle for the multipartite of the system can be acquired as

tAðf 1f 2Þðt1Þ ¼ 2½1� trðr2AÞ�

¼ 2½1� trðr2f 1f 2Þ� ¼ �2 Sin4

g2

Dt1 þ Cos4

g2

Dt1 � 1

� �, ð10Þ

tf 2ðAf 1Þðt1Þ ¼ 2½1� trðr2f 2Þ�

¼ 2½1� trðr2Af 1Þ� ¼ �2 Sin4

g2

Dt1 þ Cos4

g2

Dt1 � 1

� �, ð11Þ

where we have use the fact the (non-zero) eigenvalue spectra of the two marginaldensity operators for a bipartite division of a pure state are identical [16,17] inattaining the rightmost equalities. The above expressions show us that if

Sing2

Dt1 ¼ �

ffiffiffi2

p

2,

that is to say, when t1 ¼ ðð8k � 1Þp=4g2ÞD (k is an arbitrary integer), the tangle forthe atom and the cavity-mode and that for one-mode and the rest of the system areboth arrive the max value. It can be proved to be unity easily. These tangles can beapplied to the quantum control of atomic ensembles. Because the overall system is

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pure, any correlation between the cavity and the ensemble is necessarily in one-to-one corresponding to the amount of entanglement between these two subsystems.The quantum backaction on cavity due to measurement of the atom is quantified byEq. (10). On the other hand, a measurement of one of the cavity-modes, whichinduces backaction on the remaining subsystem, is described by Eq. (11).Next, let us come to talk about the situation that the atom arrives at t ¼ 0 in the

state jf i and the two-mode cavity is in the Fock state j01i initially. The wavefunction after the evolution can be given as

jf ij01i ! a01ðt

01Þjgij10i þ b0

1ðt01Þjf ij01i , (12)

where

a01ðt

01Þ ¼

1

2Cos

2g2

Dt01 � iSin

2g2

Dt01 � 1

� �, (13)

b01ðt

01Þ ¼

1

2Cos

2g2

Dt01 � iSin

2g2

Dt01 þ 1

� �. (14)

From the above equations, it is easy to calculate the probability that the atom isfound in the state jgi at t ¼ t01:

w0ð1Þg ðt01Þ ¼ ja0

1ðt01Þj

2 ¼1

21� Cos

2g2

Dt01

� �. (15)

In the same way, if the interaction time t01 ¼ ð2k0þ 1ÞpD=2g2 (k0 is an arbitrary

integer), the probability of finding the three-level atom in the state jgi when it leavethe cavity is unity. Under the above condition, the atom is transferred from the statejf i to the state jgi with the interaction of the two-mode cavity. It is similar to theabove situation.Next, let us pay attention to the tangle between the atom and the cavity-mode and

that between one cavity-mode and the remainder of the system under the initialcondition. Following the above expressions, it can be easily attained that:

t0Aðf 1f 2Þðt01Þ ¼ �2 Sin4

g2

Dt01 þ Cos4

g2

Dt01 � 1

� �, (16)

t0f 2ðAf 1Þðt01Þ ¼ �2 Sin4

g2

Dt01 þ Cos4

g2

Dt01 � 1

� �. (17)

Similarly, it can be proved that if t01 ¼ ðð8k0� 1Þp=4g2ÞD (k0 is an arbitrary

integer); the tangles for the multipartite of the system achieve the max value which isunity. Corresponding to the symmetrical initial conditions, the time evolutions foreach of the different tangles have the same form.The probability of finding the first atom in the state jgi when it leaves the cavity,

corresponding to the initial condition jgij10i, is shown in Fig. 2(a). Fig. 2(b) showsthe time evolution of the tangle for the atom and the two-mode cavity field and thetangle for one mode and the remainder of the system under the same initialcondition. If the system is in the state jf ij01i initially, the probability of finding the

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0.2 0.4 0.6 0.8 1

t1

0.2

0.4

0.6

0.8

1

wg

0.1 0.2 0.3 0.4 0.5t1

0.2

0.4

0.6

0.8

1

Tan

gle

(a) (b)

Fig. 2. (a) The probability evolution for atom in the state jgi1 and the field in the state j10i initially with

D ¼ 10g; ðg=2pÞ � 25MHz [18]. (b) The time evolution of the tangle for the atom and the two-mode cavity

field and the tangle for one mode and the remainder of the system for the same condition.

0.2 0.4 0.6 0.8 1t1

0.2

0.4

0.6

0.8

1

wg

0.1 0.2 0.3 0.4 0.5t1

0.2

0.4

0.6

0.8

1

Tan

gle

(a) (b)

Fig. 3. (a) The probability evolution for atom in the state jf i1 and the field in the state j01i initially with

D ¼ 10g; ðg=2pÞ � 25MHz [18]. (b) The time evolution of the tangle for the atom and the two-mode cavity

field and the tangle for one mode and the remainder of the system for the same condition.


first atom in the state jgi when it leaves the cavity is shown in Fig. 3(a). Fig. 3(b)shows the time evolution of the tangle for the atom and the two-mode cavity fieldand the tangle for one mode and the remainder of the system under the same initialcondition.The figures of the above functions of the system are all the sinusoidal oscillation,

which shows the observable periodic behavior.

4. The interaction of the second atom with the cavity

When the first atom leaves the cavity, we let the second identical atom passthrough the two-mode cavity. Via the quantum field inside the cavity, the two atomsbecome entangled. Eq. (3) can also be applied to the second atom when it enters thecavity and interacts with the field. If jgi1jgi2j10i is chosen as the initial state, thewave function in the interaction picture of the two atoms and the two field modes

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now has the form:

jcðt1; t2Þi ¼ a1ðt1Þa2ðt2Þjgi1jgi2j10i

þ a1ðt1Þb2ðt2Þjgi1jf i2j01i þ b1ðt1Þjf i1jgi2j01i , ð18Þ

where

a1ðt1Þ ¼ e�iðg2=DÞt1Cosg2

Dt1 , (19a)

a2ðt2Þ ¼ e�iðg2=DÞt2Cosg2

Dt2 , (19b)

b1ðt1Þ ¼ �ie�iðg2=DÞt1Sing2

Dt1 , (19c)

b2ðt2Þ ¼ �ie�iðg2=DÞt2Sing2

Dt2 . (19d)

The superscript ‘‘(j)’’ and the subscript ‘‘(j)’’ (j ¼ 1; 2) refer to the jth atom,respectively.Following the wave function of the new system consisting of two identical three-

level atoms and the two field modes, it can be attained that the probability that thesecond atom leaves the cavity in the state jf i

wð2Þf ðt1; t2Þ ¼ ja1ðt1Þb2ðt2Þj

2

¼1

16Cos

2g2

Dt2 � 1

� �Cos

2g2

Dt1 þ 1

� �� 2(

þ Sin22g2

Dt1 1� Cos

2g2

Dt2

� �2

þSin22g2

Dt2 1þ Cos

2g2

Dt1

� �2

þ Sin22g2

Dt1Sin

2 2g2

Dt2

)

¼1

41þ Cos

2g2

Dt1

� �1� Cos

2g2

Dt2

� �. ð20Þ

The probability that the second atom is in the state jf i when it leaves the cavity isdelivered as the function of the interaction time t1 that the first atom interacts withthe cavity field and the interaction time t2 that the second atom interacts with thecavity field. The above probability does not depend on whether the first atomis detected before, during, or after the interaction of the second atom with thecavity field.

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Next, using Eq. (9) and let nA ¼ nB ¼ 1, the field-ensemble tangle, the one-atom-remainder tangle and the one-mode-remainder tangle can be calculated to be

tABðf 1f 2Þ ¼ 2ð1� trr2ABÞ

¼ 2ð1� trr2f 1f 2Þ ¼ �2 Cos4

g2

Dt1 Cos

4 g2

Dt2 þ Cos4

g2

Dt1 Sin

4 g2

Dt2

�

þ2Cos2g2

Dt1 Sin

2 g2

Dt1 Sin

2 g2

Dt2 þ Sin4

g2

Dt1 � 1

�, ð21Þ

tAðBf 1f 2Þ ¼ 2ð1� trr2AÞ ¼ 2ð1� trr2Bf 1f 2Þ

¼ �2 Cos4g2

Dt1 Cos

4 g2

Dt2 þ Cos4

g2

Dt1 Sin

4 g2

Dt2

�

þ2Cos4g2

Dt1 Cos

2 g2

Dt2 Sin

2 g2

Dt2 þ Sin4

g2

Dt1 � 1

�, ð22Þ

tf 2ðABf 1Þ ¼ 2ð1� trr2f 2Þ ¼ 2ð1� trr2ABf 1Þ

¼ �2 Cos4g2

Dt1Cos

4 g2

Dt2 þ Cos4

g2

Dt1 Sin

4 g2

Dt2

�

þ2Cos2g2

Dt1 Sin

2 g2

Dt1 Sin

2 g2

Dt2 þ Sin4

g2

Dt1 � 1

�. ð23Þ

The subscript ‘‘A;B’’ in the above equations represent the two identical three-levelatoms respectively. The subscripts f 1; f 2 denote the two modes of the quantifiedcavity field. It is apparent that the tangle between the atom ensemble and the cavityfield depend on the interaction time t1; t2, the coupled constant g and the detuning ofthe frequency of the cavity-mode from the transfer frequency of the atom. Similarly,Eq. (21) represents the quantum backaction on the ensemble due to the measurementof the cavity field. On the same time, the backaction which is leaded by themeasurement of one atom on the remaining subsystem is described by Eq. (22).While Eq. (23) denotes the quantum backaction resulted by the measurement of onemode of the cavity field on the remainder of the subsystem. The two modes exchangethe energy via the interaction with the atoms. This produces the entanglementsbetween the cavity field and the ensemble and that in the one-mode-remainder andone-atom-remainder partitions.Now let us talk about the tangle for the two atoms. Tracing over the field

subsystem, the two-atom mixed state rABðt1; t2Þ can be acquired. If j00i; j01i; j10i; j11iis chosen as the basis, following the expression in the Section 3, the ‘‘spin flip’’ of thetwo-atom mixed state rABðt1; t2Þ can be achieved. Thus, the Eq. (8) can be applied tocalculating the time evolution of the tangle for the two atoms. It gives as

t2ðrABðt1; t2ÞÞ � maxf0; l1 � l2 � l3 � l4g2 ¼ 4Cos2g2

Dt1 Sin

2 g2

Dt1 Sin

2 g2

Dt2 ,

(24)

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where l1; l2; l3; l4 is the square roots of the eigenvalues of the non-Hermitian operatorrABðt1; t2Þ ~rABðt1; t2Þ. It is obvious that when

t1 ¼ð8k þ 1ÞpD

4g2and t2 ¼

ð4m þ 1ÞpD2g2

.

(k;m are arbitrary integers), the tangle between the two atoms achieve the max value.It can be prove to be unity. At this time, Eq. (18) can be transformed as

jc0ðt1; t2Þi ¼

ffiffiffi2

p

2ðjgi1jf i2 þ jf i1jgi2Þj01i . (25)

Via the procedure, the maximally entangled atoms can be created. The pheno-menon can be explained as follows: When the first three-level atom passes through thecavity, it leaves the information in the cavity field. Similarly, when the secondatom passes through the cavity, it stores the information in the cavity field, too. Sothe two atoms become entangled with each other via the cavity field. Thisatom–atom correlation displays non-classical features although the two atoms followtwo spatially separated paths. Specifically, the atom–atom tangle quantifies thedegree to which the ensemble behaves as a collective entity, rather than as twoindividual particles.The probability of finding the second atom in the state jf i when it leaves the

cavity, corresponding to the initial condition jgi1jgi2j10i, is shown in Fig. 4(a).Fig. 4(b) shows the time evolution of the tangle between the two-mode cavity fieldand the atom ensemble and that between one mode and the remainder of the systemconsisting of ensemble and the field under the same initial condition. Fig. 4(c)displays the time evolution of the tangle between one atom and the remainder of thesystem of atom ensemble and the two mode cavity field for the same condition. Thetime evolution of the tangle for the two atoms is shown in Fig. 4(d).It is supposed that the initial state be jf i1j01i and the second atom be in the state

jf i2 when it enters the cavity. Similarly, the wave function of the system comprisingtwo atoms and two-mode field has the form as

jc0ðt01; t

02Þi ¼ � iSin

g2

Dt01 exp �i

g2

Dt01

� �jgi1jf i2j10i

� iCosg2

Dt01 Sin

g2

Dt02 exp �i

g2

Dðt01 þ t02Þ

� �jf i1jgi2j10i

þ Cosg2

Dt01 Cos

g2

Dt02 exp �i

g2

Dðt01 þ t02Þ

� �jf i1jf i2j01i . ð26Þ

The probability of finding the second atom in the state jgi when it leaves the cavitycan be acquired:

w0ð2Þg ðt01; t

02Þ ¼ Cos2

g2

Dt01 Sin

2 g2

Dt02 . (27)

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00.25

0.50.75

1 0

0.25

0.5

0.75

1

t 2

00.250.5

0.751

wf

t1

00.25

0.50.75

1

t10

0.25

0.5

0.75

1

t 2

00.250.5

0.751

Tan

gle

00.25

0.50.75

1

t10

0.25

0.5

0.75

1

t 2

00.250.5

0.751

Tan

gle

00.25

0.50.75

1

t10

0.25

0.5

0.75

1

t 2

00.250.5

0.751

Tan

gle

(a)(b)

(c) (d)

Fig. 4. (a) The probability evolution for atom ensemble in the state jgi1jgi2 and the field in the state j10i

initially with D ¼ 10g; ðg=2pÞ � 25MHz [18]. (b) The time evolution of the tangle between the atom

ensemble and the two-mode cavity field and the tangle between one mode and the remainder of the system

for the same condition. (c) The time evolution of the tangle between one atom and the remainder of the

system of atom ensemble and the two mode cavity field for the same initial condition. (d) The tangle

between the two atoms.


The time evolutions for the tangle between the cavity field and the ensemble andthe one-atom-remainder tangle and the one-mode-remainder tangle can be given as

t0ABðf 1f 2Þ¼ 2ð1� trr02ABÞ

¼ 2ð1� trr02f 1f 2Þ ¼ �2 Cos4

g2

Dt01 Cos

4 g2

Dt02 þ Cos4

g2

Dt01 Sin

4 g2

Dt02

�

þ2Cos2g2

Dt01 Sin

2 g2

Dt01 Sin

2 g2

Dt02 þ Sin4

g2

Dt01 � 1

�, ð28Þ

t0AðBf 1f 2Þ¼ 2ð1� trr02AÞ ¼ 2ð1� trr02Bf 1f 2

Þ

¼ �2 Cos4g2

Dt01 Cos

4 g2

Dt02 þ Cos4

g2

Dt01 Sin

4 g2

Dt02

�

þ2Cos4g2

Dt01 Cos

2 g2

Dt02 Sin

2 g2

Dt02 þ Sin4

g2

Dt01 � 1

�, ð29Þ

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t0f 2ðABf 1Þ¼ 2ð1� trr02f 2 Þ

¼ 2ð1� trr02ABf 1Þ ¼ �2 Cos4

g2

Dt01 Cos

4 g2

Dt02 þ Cos4

g2

Dt01 Sin

4 g2

Dt02

�

�2Cos2g2

Dt01 Sin

2 g2

Dt01 Sin

2 g2

Dt02 þ Sin4

g2

Dt01 � 1

�. ð30Þ

Apparently, the time evolutions of the tangle for the multipartite of the systemunder the initial state jf i1jf i2j01i and the state jgi1jgi2j10i have the same form.In the same way, the tangle for the two atoms has the form as

t02ðr0ABÞ ¼ 4Cos2

2g2

Dt01 Sin

2 2g2

Dt01 Sin

2 2g2

Dt02 . (31)

The probability of finding the second atom in the state jgi when it leaves thecavity, corresponding to the initial condition jf i1jf i2j01i, is shown in Fig. 5.Now, if the second atom is in the state jgi2 when entering the cavity, the above

wave function changes as

jc00ðt1; t2Þi ¼ � iSin

g2

Dt001 Cos

g2

Dt002 exp �i

g2

Dðt001 þ t002Þ

� �jgi1jgi2j10i

� Sing2

Dt001 Sin

g2

Dt002 exp �i

g2

Dðt001 þ t002Þ

� �jgi1jf i2j01i

þ Cosg2

Dt001 exp �i

g2

Dt001

� �jf i1jgi2j01i . ð32Þ

Under this initial condition, the probability of finding the second atom in the statejf i2 when it leaves the cavity can be attained as

w00ð2Þf ðt001 ; t

002Þ ¼ Sin2

g2

Dt001 Sin

2 g2

Dt002 . (33)

00.25

0.50.75

1

t10

0.25

0.5

0.75

1

t 2

00.250.5

0.751

wg

Fig. 5. The probability evolution for atom ensemble in the state jf i1jf i2 and the field in the state j01i

initially with D ¼ 10g; ðg=2pÞ � 25MHz [18].

ARTICLE IN PRESS


Correspondingly, the ensemble-field tangle and one-atom-remainder tangle andone-mode-remainder tangle change as

t00ABðf 1f 2Þ¼ 2ð1� trr002ABÞ ¼ 2ð1� trr002f 1f 2

Þ

¼ � 2 Sin4g2

Dt001 Sin

4 g2

Dt002 þ Sin4

g2

Dt001 Cos

4 g2

Dt002

�

þ2Cos2g2

Dt001 Sin

2 g2

Dt001 Sin

2 g2

Dt002 þ Cos4

g2

Dt001 � 1

�, ð34Þ

t00AðBf 1f 2Þ¼ 2ð1� trr002A Þ ¼ 2ð1� trr002Bf 1f 2

Þ

¼ �2 Sin4g2

Dt001 Sin

4 g2

Dt002 þ Sin4

g2

Dt001 Cos

4 g2

Dt002

�

þ2Sin4g2

Dt001 Cos

2 g2

Dt002 Sin

2 g2

Dt002 þ Cos4

g2

Dt001 � 1

�, ð35Þ

00.25

0.50.75

1t1

0

0.25

0.5

0.75

1

t 2

00.250.5

0.751

wg

00.25

0.50.75

1

t10

0.25

0.5

0.75

1

t 2

00.250.5

0.751

Tan

gle

00.25

0.50.75

1t1 0

0.25

0.5

0.75

1

t 2

00.250.5

0.751

Tan

gle

(a)(b)

(c)

Fig. 6. (a) The probability evolution for atom ensemble in the state jf i1jgi2 and the field in the state j01i

initially with D ¼ 10g; ðg=2pÞ � 25MHz. (b) The time evolution of the tangle between the atom ensemble

and the two-mode cavity field and that between one mode and the remainder of the system two mode

cavity field for the same condition. (c) The time evolution of the tangle between one atom and the

remainder of the system.

ARTICLE IN PRESS


t00f 2ðABf 1Þ¼ 2ð1� trr002f 2

Þ ¼ 2ð1� trr002ABf 1Þ

¼ �2 Sin4g2

Dt001 Sin

4 g2

Dt002 þ Sin4

g2

Dt001 Cos

4 g2

Dt002

�

þ2Cos2g2

Dt001 Sin

2 g2

Dt001 Sin

2 g2

Dt002 þ Cos4

g2

Dt001 � 1

�. ð36Þ

It is different from the situation when the system is in the state jgi1jgi2j10i orjf i1jf i2j01i initially.While the atom–atom tangle still has the same form with the above results.

t2ðr00ABðt1; t2ÞÞ � maxf0; l001 � l002 � l003 � l004g2 ¼ 4Cos2

g2

Dt001 Sin

2 g2

Dt002 Sin

2 g2

Dt001 .

(37)

This result denotes the fact that whether the system in the states jgi1jgi2j10i,jf i1jf i2j01i or in the state jf i1jgi2j01i, the tangles for the two atoms arrive the maxvalue at the same time.The probability that the second atom is in the state jf i when it leaves the cavity,

related to the initial condition jf i1jgi2j01i, is shown in Fig. 6(a). The time evolutionsof the tangle for the cavity field and the atom ensemble and that for one mode andthe remainder of the system under the same initial condition are showed in Fig. 6(b).Fig. 6(c) displays the time evolution of the tangle between one atom and theremainder of the system.

5. Conclusion

In this paper, we consider a system consisting of l-type three level atoms and two-mode Fock state cavity field. Under the adiabatical approximation and the largedetuning condition, the effective Hamiltonian of the system can be acquired.Following the effective Hamiltonian, the wave functions of the system under thedifferent initial conditions can be obtained. When we let another atom pass throughthe cavity, the entangled atoms can be generated. Furthermore, the tangles for theatom ensemble and the two-mode cavity field are presented under the three differentinitial conditions. Meantime, we can give out the tangle for one atom and theremainder of the system and that between one mode and the remainder of the system.The atom–atom tangles under the different initial conditions which are given out inthis paper all have the same form. When the interaction time is chosen appropriately,the maximally entangled atoms can be generated, which is an essential resource forquantum computation. Highly entangled state plays a key role in an efficientrealization of quantum information processing including quantum teleportation,cryptography, dense coding and computation. In these protocols, the maximallyentangled states are required. The phenomenon that entanglement cannot be freelydistributed among subsystems in a multipartite system is also displayed in this paper.The research of the phenomenon can be applied to the quantum control.

ARTICLE IN PRESS


Acknowledgements

This work was funded by the National Fundamental Research Program (GrantNo. 2001CB309300), the National Natural Science Foundation (Grant No.10204020) and the Innovation Funds from the Chinese Academy of Sciences.

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the tangle of the lambda-type three levels atoms and the two-mode cavity field

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