Tripartite entanglement generation using cavity-QED and its dynamics in dissipative environments

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Tripartite entanglement generation using cavity-QEDand its dynamics in dissipative environmentsBilal Farooq a , Rabia Tahira a & Manzoor Ikram ba Centre for Quantum Physics, Department of Physics, COMSATS Institute of InformationTechnology, Islamabad, Pakistanb National Istitute of Lasers and Optronics, Nilore, Islamabad, PakistanPublished online: 25 Oct 2012.

To cite this article: Bilal Farooq , Rabia Tahira & Manzoor Ikram (2012): Tripartite entanglement generation using cavity-QEDand its dynamics in dissipative environments, Journal of Modern Optics, 59:19, 1664-1673

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Journal of Modern OpticsVol. 59, No. 19, 10 November 2012, 1664–1673

Tripartite entanglement generation using cavity-QED and its dynamics in

dissipative environments

Bilal Farooqa, Rabia Tahiraa* and Manzoor Ikramb

aCentre for Quantum Physics, Department of Physics, COMSATS Institute of Information Technology, Islamabad,Pakistan; bNational Istitute of Lasers and Optronics, Nilore, Islamabad, Pakistan

(Received 20 June 2012; final version received 20 September 2012)

We investigate the generation of tripartite field states inside the high-Q cavities using the cavity QED. The maingoal is to successfully generate the entanglement in tripartite systems by passing two-level atoms through threeidentical high-Q cavities. Our scheme gives the successful generation of entangled tripartite W and GHZ statesfor pre-determined interaction times of atoms with the cavity fields. The dynamics of initial entangled states isstudied as the system evolves in the dissipative environments.

Keywords: tripartite entanglement; entanglement generation; entanglement dynamics; sudden death ofentanglement; quantum information processing; cavity QED

1. Introduction

Quantum entanglement is an important resource ofquantum information processing (QIP) and is veryuseful in many quantum protocols. Two or moresubsystems of a quantum system in an entangled statecan be used as a quantum information channel toperform computational and cryptographic tasks whichare not possible for a classical system. The well-knownexample of an entangled state is shown by Bell. Nodoubt quantum entanglement lies at the heart ofquantum information but it is a fact that it is toofragile to play a role in the real world. In an idealsituation entanglement required for quantum informa-tion processing must be steady in time. But in reality aquantum system unwantedly interacts with its envi-ronment, which is the cause of entanglement degrada-tion. The real QIP tasks require successful generationof entanglement and study of the decoherencemechanism.

The multipartite entangled states are importantto study because of their vital role in QIP tasks.Multipartite entanglements are investigated in manyfundamental aspects of quantum theory such asviolation of nonlocality [1–3]. A specific class ofmultipartite states such as GHZ and W states areimplied for many important applications such asteleportation [4,5], dense coding [6,7] and quantumkey distribution [8,9]. Many studies have been done onthe generation of multipartite entangled states [10–12].The cavity QED based techniques play a vital role in

generating atom–atom, atom–field and field–field

entanglement [13–15]. Schemes of generating multi-

partite GHZ and W states have been proposed by

using second-order off-resonant Bragg’s diffraction

[16]. Bragg’s diffraction is an interference induced

process that coherently transforms a quantum mechan-

ical system from one state to another equal energy state

by a periodic interaction. Tripartite GHZ-state of two

atoms and a cavity mode was constructed step by step

which implies a programmable sequence of operations

to generate and demonstrate the entanglement of

complex systems [17]. The tripartite entanglement is

exploited for foundational issues [18] and is experi-

mentally investigated [19–21]. The study of decoher-

ence or entanglement dynamics of these states is

important as quantum systems always interact with

the surrounding environment.Entanglement dynamics may be asymptotic or may

vanish at some finite time; the so-called sudden death

of entanglement (SDE) [22]. It is studied that entan-

glement dynamics depends on the initial preparation of

states and the surrounding environments [23]. Earlier,

we have studied that SDE depends upon the probabil-

ity of the components with one photon in each cavity

in bipartite systems [24]. It is investigated that SDE

always occurs in a thermal environment. In this paper,

we extended the study to multipartite systems. There

are different quantitative measures of entanglement for

bipartite and multipartite systems. The well-known

multipartite measures are tangle [25], geometric

*Corresponding author. Email: [email protected]

ISSN 0950–0340 print/ISSN 1362–3044 online

� 2012 Taylor & Francis

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measure of entanglement [26] and hyperdeterminant[27]. Other reliable methods to quantify entanglementinclude negativity [28], entropy [29,30] and Woottersconcurrence [31] but concurrence is constrained onlyfor a two-qubit system.

The paper is organized as follows. In Section 2, wepresent the theoretical model for the generation anddynamics of entanglement for tripartite systems. InSections 3 and 4, we first present the model to generateW-state and GHZ state and then study the dynamics ofentangled W-state and GHZ state, respectively. Section5 includes the concluding remarks. Equations ofmotion in terms of density matrix elements and theirsolutions are shown in Appendix 1.

2. Model

The generation of an entangled field state for a fixednumber of photons between two cavities has alreadybeen investigated [32]. In the same context, we nowgenerate tripartite field states such as W and GHZstates in the three high-Q cavities. The atom passesthrough three cavities and interacts with the resonantmodes of the electromagnetic fields inside the threecavities via the Jaynes–Cummings Hamiltonian [33].In order to generate the entangled states, we have tocontrol the interaction times of the atoms with thefields inside the cavities. The general tripartite fieldstate can be represented in the Fock basis as

Cj i ¼Xi, j,k

Ci, j,k i, j, k��

,ð1Þ

where the Ci, j,k’s are the probability amplitudes withPi, j,k Ci, j,k

�� 2¼ 1. The unitary operator for a singletwo-level atom interacting with a quantized cavity field[34] is given by

U �ið Þ ¼ cos gi�i ayi ai þ 1� �1=2� �

aj i ah j

þ cos gi�i ayi ai

� �1=2� �bj i bh j

� isinð gi�iða

y

i ai þ 1Þ1=2Þaijaihbj

ðayi ai þ 1Þ1=2

� iayisinð gi�iða

y

i ai þ 1Þ1=2Þjbihaj

ðayi ai þ 1Þ1=2, ð2Þ

where gi is the coupling constant, �i is the interactiontime of the atom with the field, ai and ayi are theannihilation and creation operators for the ith cavitywhere i¼A, B and C. The tripartite entangled state(Equation (1)) or any subset can be generated forpredetermined interaction times of the each atom in thethree cavities. Once the entanglement is generated, the

study of its dynamics as a result of interaction with theenvironment becomes equally important.

We assume that the interaction between the fieldsand the reservoirs is weak and there is no back reactioneffect which shows that reservoirs remain in thethermal equilibrium mixture of photon number statesall the time. Also we consider that the correlation timebetween the fields and the reservoirs is much shorterthan the characteristic time of the evolution dynamicsof the fields such as spontaneous emission time andentanglement sudden death time so that the Markovapproximation is valid. By using Born–Markov, rotat-ing wave and dipole–dipole approximations the masterequation can be derived from the general reservoirtheory. The reduced density matrix for the field insidethe cavities surrounding the same thermal environment[34] can be obtained as:

_� ¼ �X

i¼A,B,C

�i2

�n aiay

i �� 2ayi �ai þ �aiay

i

� �

�X

i¼A,B,C

�i2ð �nþ 1Þ ayi ai�� 2ai�a

y

i þ �ay

i ai

� �, ð3Þ

where �n is the mean number of photons in the thermalreservoir and the �i’s are cavity decay rates.

We use the basis: 0A, 0B, 0Cj i ! 1j i, 0A, 0B, 1Cj i !

2j i, 0A, 1B, 0Cj i ! 3j i, 1A, 0B, 0Cj i ! 4j i, 0A, 1B,j

1Ci ! 5j i, 1A, 0B, 1Cj i ! 6j i, 1A, 1B, 0Cj i ! 7j i, and1A, 1B, 1Cj i ! 8j i to define the density matrix of thethree-qubit system. The solution of Equation (3) isrepresented by the following matrix

�ðtÞ ¼

�11 �12 �13 �14 �15 �16 �17 �18

�21 �22 �23 �24 �25 �26 �27 �28

�31 �32 �33 �34 �35 �36 �37 �38

�41 �42 �43 �44 �45 �46 �47 �48

�51 �52 �53 �54 �55 �56 �57 �58

�61 �62 �63 �64 �65 �66 �67 �68

�71 �72 �73 �74 �75 �76 �77 �78

�81 �82 �83 �84 �85 �86 �87 �88

266666666666664

377777777777775:

ð4Þ

The matrix elements �ij are given in Appendix 1. In thepresent scheme we are considering the identical cavi-ties, so the cavity decay rates are assumed to be thesame, i.e. �A ¼ �B ¼ �C ¼ �. After successful genera-tion of the required entangled state, the main task is toquantify the amount of entanglement during thesystem evolution with time. There are some techniquesfor the quantitative measure of entanglement formultipartite systems [35,36]. Negativity is a usefultechnique to quantify the entanglement among bipar-tite [28] and tripartite [36,37] quantum states. Thenegativity is based upon the positive partial

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transposition (PPT) criterion which is a necessary

condition to check the separability of bipartite and

multipartite states [38,39]. The negativity ranges from

N¼ 0 for a separable state to N¼ 1 for a maximally

entangled state. The explicit expression for negativity is

NðtÞ ¼ max 0, �2Xi

�iðtÞ

" #, ð5Þ

where the �iðtÞ’s are the eigenvalues of the partial

transposed matrix and t is the interaction time of the

cavity fields with the environment.

3. Generation and dynamics of W-state

The striking feature of the W-state [40] is that it

remains entangled over the loss of one qubit. A

multiqubit W-state exhibits stronger nonclassical

behavior than a GHZ-state [41]. This unique charac-

teristic of the W-state makes it an ideal resource for

communication based on multinodal networks [42],

teleportation, dense coding [43] and optimal universal

quantum cloning [44]. It has a remarkable application

in quantum memories [45]. The shape of the maximally

tripartite entangled W-state is

CWj i ¼1

31=2ð 0, 0, 1j i þ 0, 1, 0j i þ 1, 0, 0j iÞ: ð6Þ

In Figure 1, the scheme to generate the entangled

W-state is shown. This state can be generated by

passing an excited two-level atom through three

identical high-Q cavities A, B and C for pre-calculated

interaction times. The initial state of the atom–field

system is given by

C�¼0ðA,B,CÞ��

¼ a, 0A, 0B, 0Cj i: ð7Þ

After interaction the atomic–field state can be calcu-

lated by using Equation (2), as:

C�ðA,B,CÞ��

¼ U �Cð ÞU �Bð ÞU �Að Þ C�¼0ðA,B,CÞ��

: ð8Þ

After passing the excited atom through the threecavities, the final state of the whole system may bewritten as:

C �A, �B, �Cð Þ�� ¼ E000 a, 0A, 0B, 0Cj i þ G001 b, 0A, 0B, 1Cj i

þ G010 b, 0A, 1B, 0Cj i þ G100 b, 1A, 0B, 0Cj i, ð9Þ

where

E000 ¼ cos g�Að Þ cos g�Bð Þ cos g�Cð Þ ð10Þ

is the probability amplitude for the excited state atomand

G001 ¼ �i cos g�Að Þ cos g�Bð Þ sin g�Cð Þ, ð11Þ

G010 ¼ �i cos g�Að Þ sin g�Bð Þ, ð12Þ

G100 ¼ �i sin g�Að Þ, ð13Þ

are the probability amplitudes for the ground state.The probability of detecting the atom in the groundstate which is the necessary condition of our scheme,can be calculated as

PG ¼ 1� cos2 g�Að Þ cos2 g�Bð Þ cos2 g�Cð Þ: ð14Þ

Thus, the probability of detecting the atom in theground state is maximum if either of g�A, g�B or g�C is2nþ 1ð Þðp=2Þ, where n is an integer. If the atom isdetected in the ground state, then the normalized stateof the field inside the three cavities becomes

CW �A, �B, �Cð Þ�� ¼ K1½G001 b, 0A, 0B, 1Cj i þ G010 b, 0A, 1B, 0Cj i

þ G100 b, 1A, 0B, 0Cj i�, ð15Þ

where K1 is the normalization constant

K1 ¼1

G001j j2þ G010j j2þ G100j j2� 1=2 : ð16Þ

In order to calculate the interaction times of the atomwith the three cavities for successful generation of theW-state, we assume all the probability amplitudes ofthe ground state the same, i.e.

G001 ¼ G010 ¼ G100: ð17Þ

For another equation, we assume the probability of thedetection of an atom in the excited state is zero, i.e.

E000 ¼ 0:

This condition also makes the success probabilitymaximum equal to unity. By using the above condi-tions, the calculated interaction times are: g�1A ¼0:61548, g�1B ¼ 0:785398 and g�1C ¼ 1:5708. Hence,our scheme proposed the successful generation of theW-state for the selected interaction times.

Figure 1. A model to generate the W-state shows that asingle excited atom passes through three independent andidentical cavities. (The color version of this figure is includedin the online version of the journal.)

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Now, let us consider the dynamics of an initialentangled field state in terms of arbitrary values of theinitial probability amplitudes, given by

C1j i ¼ C001 0, 0, 1j i þ C010 0, 1, 0j i þ C100 1, 0, 0j i,

ð18Þ

where C001j j2þ C010j j2þ C100j j2¼ 1. The tripartite neg-ativity [36,37] for this W-like state in terms of arbitraryvalues of the initial probability can be calculated asN ¼ ðNANBNCÞ

1=3, where Ni is the negativity aftertaking the partial transposition over the ith qubit,given as

NAðtÞ ¼ max

0, ðexpð��tÞ � 1Þ

�1þ ð1þ 4 expð�2�tÞ

��expð��tÞ � 1

�2C001j j2ð1� C001j j2Þ

�1=2� ,

ð19Þ

NBðtÞ ¼ max

0, ðe��t � 1Þ

�1þ ð1þ 4e�2�tðe��t � 1Þ�2

� C010j j2ð1� C010j j2Þ

�1=2� , ð20Þ

and

NCðtÞ ¼ max

0,�expð��tÞ � 1

�1þ ð1þ 4 expð�2�tÞ

��expð��tÞ � 1

�2C100j j2ð1� C100j j2Þ

�1=2� :

ð21Þ

From Equations (19)–(21), it is obvious that partialtransposition on each qubit gives the weight ofprobability of that qubit. For C001j j ¼ C010j j ¼

C100j j ¼ 3�1=2, we have a maximally entangledW-state. In Figure 2(a), there is a plot of negativityNCðtÞ against initial probability C100j j2 and time. Wecan observe the asymptotic dynamics of an initialentangled state for all values of the initial probabilityC100j j2. The same dynamics can be observed fornegativities NAðtÞ and NBðtÞ against correspondingprobabilities C001j j2 and C010j j2, respectively.Figure 2(b) shows the plot of entanglement for amaximally entangled W-state. It is obvious that SDEalways occurs in the thermal environment and suddendeath time (SDT) decreases as mean photon number inthe environment increases.

4. Generation and dynamics of GHZ-state

The GHZ-state which is a non-classical state was firststudied by Greenberger, Horne and Zeiligner in 1989.It is a maximally entangled state of at least

three qubits. However, entanglement will be destroyedwhen any one of the three qubit is traced out. There areso many applications of GHZ-state as a state withmultipartite entanglement because entanglementbetween the several parties is the most essential featurein quantum communication and computation proto-cols. A maximally entangled tripartite GHZ-state[46,47] is

CGHZj i ¼1

21=2ð 0, 0, 0j i þ 1, 1, 1j iÞ: ð22Þ

In Figure 3, the scheme for the generation of aGHZ-state is presented. Interaction of atom-1 with thecavities A and B and atom-2 with the cavities A and Cfor predetermined interaction times can successfully

(a)

(b)

Figure 2. The figure shows the entanglement dynamics for(a) initial W-like state for �n ¼ 0: The asymptotic behavior isobvious at all values of the initial probability C100j j2; (b)maximally entangled W-state for �n ¼ 0 (solid black), �n ¼ 0:1(red dashed) and �n ¼ 0:2 (blue dotted-dash). (The colorversion of this figure is included in the online version of thejournal.)


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generate a GHZ-state as in Equation (22). In the case

of the GHZ-state, we consider two two-level atoms in

the ground states and the cavity A is in the coherent

superposition of the Fock states 0j i and 3j i. Then, the

initial state of the combined atoms–fields system can be

written as

Cð1Þ�¼0�� E

¼ b1, b2j i �1

21=20Aj i þ 3Aj ið Þ � 0B, 0Cj i: ð23Þ

After passing the first atom through cavities A and B,

the final state obtained becomes the initial state for the

second atom, which is

Cð2Þ�¼0�� E

¼1

21=2b2j i � ð b1, 0A, 0Bj i

þ cosð g�1A31=2Þ b1, 3A, 0Bj i

� sinð g�1A31=2Þ sinð g�1BÞ b1, 2A, 1Bj i

� i sinð g�1A31=2Þ cosð g�1BÞ a1, 2A, 0Bj iÞ � 0Cj i:

ð24Þ

When the second atom passes through the cavities A

and C the final atom–field state in terms of probability

amplitudes is

C �1A,�1B,�2A,�2Cð Þ�� ¼

1

21=2ðE100 a1,a2,1A, 0B, 0Cj iþA101 a1,b2,1A, 0B, 1Cj i

þB110 b1,a2,1A, 1B, 0Cj iþA200 a1,b2,2A, 0B, 0Cj i

þB200 b1,a2,2A, 0B, 0Cj iþG000 b1,b2,0A, 0B, 0Cj i

þG210 b1,b2,2A, 1B, 0Cj iþG201 b1,b2,2A, 0B, 1Cj i

þG300 b1,b2,3A, 0B, 0Cj iþG111 b1,b2,1A, 1B, 1Cj iÞ,

ð25Þ

where the values of probability amplitudes are:

E100 ¼�sinðg�1A31=2Þcosðg�1BÞ sinðg�2A2

1=2Þcosðg�2CÞ,

ð26Þ

A101 ¼ i sinð g�1A31=2Þ cosð g�1BÞ sinð g�2A2

1=2Þ sinð g�2CÞ,

ð27Þ

B110 ¼ i sinð g�1A31=2Þ sinð g�1BÞ sinð g�2A2

1=2Þ cosð g�2CÞ,

ð28Þ

A200 ¼ �i sinð g�1A31=2Þ cosð g�1BÞ cosð g�2A2

1=2Þ, ð29Þ

B200 ¼ �i cosð g�1A31=2Þ sinð g�2A3

1=2Þ cosð g�2CÞ, ð30Þ

G000 ¼ 1, ð31Þ

G210 ¼ � sinð g�1A31=2Þ sinð g�1BÞ cosð g�2A2

1=2Þ, ð32Þ

G201 ¼ � cosð g�1A31=2Þ sinð g�2A3

1=2Þ sinð g�2CÞ, ð33Þ

G300 ¼ cosð g�1A31=2Þ cosð g�2A3

1=2Þ, ð34Þ

G111 ¼ sinð g�1A31=2Þ sinð g�1BÞ sinð g�2A2

1=2Þ sinð g�2CÞ:

ð35Þ

The interaction times for each atom in three cavities

can be calculated using the necessary condition that

both atoms should be detected in the ground state.

It follows that

PG ¼ 1� sin2ð g�1A31=2Þ½1� sin2ð g�1BÞ

� ð1� sin2ð g�2A21=2Þ cos2ð g�2CÞÞ�

� sin2ð g�2A31=2Þ cos2ð g�2CÞ cos

2ð g�1A31=2Þ:

ð36Þ

In order to calculate the interaction times of the atoms

with the cavities for the successful generation of the

GHZ-state, we assume

G000 ¼ G111, ð37Þ

and

G210 ¼ G201 ¼ G300 ¼ 0: ð38Þ

By using conditions in Equations (37) and (38), the

calculated interaction times are: g�1A ¼ 0:9069,g�1B ¼ 1:5708, g�2A ¼ 1:11072 and g�2C ¼ 1:5708.Hence by selecting these interaction times we can

generate the GHZ-state in the cavities with unit

probability.

Figure 3. A model to generate the GHZ-state in which aground state atom-1 passes through cavity A and its identicalcavity B. Another ground state atom-2 passes throughcavities A and C. (The color version of this figure is includedin the online version of the journal.)


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Now, let us study the dynamics of the GHZ-like

state. The initial entangled GHZ-like state is

C2j i ¼ C000 000j i þ C111 111j i, ð39Þ

where C000 and C111 are arbitrary initial probability

amplitudes with C000j j2¼ 1� C111j j2. The negativity

obtained for the GHZ-like state is given by

NðtÞ ¼ max�0,�expð��tÞ C111j j2ðexpð��tÞ � 1Þ

þ expð�3�tÞ C111j j�4 expð3�tÞ

þ C111j j2ð4� 12 expð�tÞ þ 13 expð2�tÞ

� 10 expð3�tÞ þ expð4�tÞÞ1=2��

: ð40Þ

For C111j j ¼ C000j j ¼ 2�1=2, we have maximallyentangled GHZ state. The plot in Figure 4(a) showsthe entanglement dynamics with time against the initialprobability C111j j2. As the probability C111j j2¼ 1�C000j j2 is small, i.e. C111j j5 C000j j, entanglementdynamics follows the asymptotic behavior. WhenC111j j4 C000j j, we observe SDE. We can also observethat sudden death time decreases as the probabilityC111j j2 for the initial entangled state increases. InFigure 4(b), there is a plot of entanglement for themaximally entangled GHZ state. It is obvious thatSDE always occur in a thermal environment and SDTdecreases as the mean photon number in the environ-ment increases.

5. Conclusion

The paper addressed the two major areas of recentstudy in the field of quantum information processing.One is the successful generation of entanglement inmultipartite systems and the other is the dynamics ofinitial entangled multipartite states. In this regard, wepresented the schemes to successfully generate theentanglement in tripartite states using cavity QED.These schemes are experimentally viable as alreadyinvestigated with atom–atom, atom–field and field–field entanglement in high-Q cavities [13–15]. Westudied the generation of tripartite W and GHZstates and observed their dynamics. Using cavityQED, we investigated that by controlling the interac-tion times perfectly in each cavity we can easily reachour goal to generate the tripartite states. The dynamicsof tripartite W and GHZ states are investigated byusing the negativity which is based on the positivepartial transposition (PPT) criterion. Negativity is usedas a measure of entanglement for tripartite systems[37]. The study showed that locally equivalent statesfollow different dynamics which depends upon theinitial preparation of the entangled states.

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(a)

(b)

Figure 4. The figure shows the entanglement dynamics for(a) GHZ-like state for �n ¼ 0: It depends upon the initialpreparation of the state, SDE is obvious at larger values ofC111j j2; (b) maximally entangled GHZ-state for �n ¼ 0 (solidblack), �n ¼ 0:1 (red dashed) and �n ¼ 0:2 (blue dotted-dash).(The color version of this figure is included in the onlineversion of the journal.)


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Appendix 1. Equations of motion of the density

matrix elements and their solutions

The equations of motion in terms of density matrix elements

for the general state Equation (3) are as follows:

_�11 ¼ � �nð�A þ �B þ �CÞ�11

þ ð �nþ 1Þ �C�22 þ �B�33 þ �A�44ð Þ, ð41Þ

_�12 ¼ �1

2½ �n 2�A þ 2�B þ 7�Cð Þ þ 4�C��12

þ ð �nþ 1Þ 2�B�35 � �A�46ð Þ, ð42Þ

_�13 ¼ �1

2½2 �n �A þ 2�B þ �Cð Þ þ �B��13

þ ð �nþ 1Þ �C�25 þ �A�47ð Þ, ð43Þ

_�14 ¼ �1

2½2 �n 2�A þ �B þ �Cð Þ þ �A��14

þ ð �nþ 1Þ �C�26 þ �B�37ð Þ, ð44Þ

_�15 ¼ �1

2½2 �nð�A þ 2�B þ 2�CÞ

þ �B þ �Cð Þ��15 þ ð �nþ 1Þ�A�48, ð45Þ


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_�16 ¼ �1

2½2 �nð2�A þ �B þ 2�CÞ

þ ð�A þ �CÞ��16 þ ð �nþ 1Þ�B�38, ð46Þ

_�17 ¼ �1

2½2 �nð2�A þ 2�B þ �CÞ

þ ð�A þ �BÞ��17 þ ð �nþ 1Þ�C�28, ð47Þ

_�18 ¼ �1

24 �nþ 1ð Þð�A þ �B þ �CÞ�18, ð48Þ

_�22 ¼ �n�C�11 � ½ �nð�A þ �B þ 3�CÞ þ �C��22

þ ð �nþ 1Þ �B�55 þ �A�66ð Þ, ð49Þ

_�23 ¼ �1

2½2 �nð�A þ 2�B þ 2�CÞ

þ ð�B þ �CÞ��23 þ ð �nþ 1Þ�A�67, ð50Þ

_�24 ¼ �1

2½2 �n 2�A þ �B þ 2�Cð Þ

þ ð�A þ �CÞ��24 þ ð �nþ 1Þ�B�57, ð51Þ

_�25 ¼1

2�n�C�13 �

1

2½2 �nð�A þ 2�B þ 3�CÞ

þ ð�B þ 2�CÞ��25 þ ð �nþ 1Þ�A�68, ð52Þ

_�26 ¼ �n�C�14 �1

2½2 �nð2�A þ �B þ 3�CÞ�26

þ ð�A þ 2�CÞ��26 þ ð �nþ 1Þ�B�58, ð53Þ

_�27 ¼ �1

24 �nþ 1ð Þð�A þ �B þ �CÞ�27, ð54Þ

_�28 ¼ �n�C�17 �1

2½2 �n 2�A þ 2�B þ 3�Cð Þ

þ ð�A þ �B þ 2�CÞ��28, ð55Þ

_�33 ¼ �n�B�11 � ½ �n �A þ 3�B þ �Cð Þ þ �B��33

þ ð �nþ 1Þð�C�55 þ �A�77Þ, ð56Þ

_�34 ¼ �1

2½2 �nð2�A þ 2�B þ �CÞ

þ ð�A þ �BÞ��34 þ ð �nþ 1Þ�C�56, ð57Þ

_�35 ¼ �n�B�12 �1

2½2 �n �A þ 3�B þ 2�Cð Þ

þ ð2�B þ �CÞ��35 þ ð �nþ 1Þ�A�78, ð58Þ

_�36 ¼ �1

24 �nþ 1ð Þð�A þ �B þ �CÞ�36, ð59Þ

_�37 ¼ �n�B�14 �1

2½2 �n 2�A þ 3�B þ �Cð Þ

þ ð�A þ 2�BÞ��37 þ ð �nþ 1Þ�C�58, ð60Þ

_�38 ¼ �n�B�16 �1

2½2 �n 2�A þ 3�B þ 2�Cð Þ

þ �A þ 2�B þ �Cð Þ��38, ð61Þ

_�44 ¼ �n�A�11 � �n½ 3�A þ �B þ �Cð Þ þ �A��44

þ ð �nþ 1Þð�C�66 þ �B�77Þ, ð62Þ

_�45 ¼ �1

24 �nþ 1ð Þð�A þ �B þ �CÞ�45, ð63Þ

_�46 ¼ �n�A�12 �1

2½2 �n 3�A þ �B þ 2�Cð Þ

þ ð2�A þ �CÞ��46 þ ð �nþ 1Þ�B�78, ð64Þ

_�47 ¼ �n�A�13 �1

2½2 �n 3�A þ 2�B þ �Cð Þ

þ ð2�A þ �BÞ��47 þ ð �nþ 1Þ�C�68, ð65Þ

_�48 ¼ �n�A�15 �1

2½2 �n 3�A þ 2�B þ 2�Cð Þ

þ ð2�A þ �B þ �CÞ��48, ð66Þ

_�55 ¼ �n�B�22 þ �n�C�33 � ½ �nð�A þ 3�B þ 3�CÞ

þ ð�B þ �CÞ��55 þ ð �nþ 1Þ�A�88, ð67Þ

_�56 ¼ �n�C�34 �1

2½2 �nð2�A þ 2�B þ 3�CÞ

þ ð�A þ �B þ 2�CÞ��56, ð68Þ

_�57 ¼ �n�B�24 �1

2½2 �nð2�A þ 3�B þ 2�CÞ

þ ð�A þ 2�B þ �CÞ��57, ð69Þ

_�58 ¼ �n�B�26 þ �n�C�37 �1

2½2 �nð2�A þ 3�B þ 3�CÞ

þ ð�A þ 2�B þ 2�CÞ��58, ð70Þ

_�66 ¼ �n�B�22 þ �n�C�44 � ½ �nð3�A þ �B þ 3�CÞ

þ ð�A þ �CÞ��66 þ ð �nþ 1Þ�B�88, ð71Þ

_�67 ¼ �n�A�23 �1

2½2 �n 3�A þ 2�B þ 2�Cð Þ

þ ð2�A þ �B þ �CÞ��67, ð72Þ

_�68 ¼ �n�A�25 þ �n�C�47 �1

2½2 �nð3�A þ 2�B þ 3�CÞ

þ ð2�A þ �B þ 2�CÞ��68, ð73Þ

_�77 ¼ �n�A�33 þ �n�B�44 � ½ �nð3�A þ 3�B þ �CÞ

þ ð�A þ �BÞ��77 þ ð �nþ 1Þ�C�88, ð74Þ

_�78 ¼ �n�A�35 þ �n�B�46 �1

2½2 �nð3�A þ 3�B þ 2�CÞ

þ ð2�A þ 2�B þ �CÞ��78, ð75Þ


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_�88 ¼ �n �A�55 þ �B�66 þ �C�77ð Þ

� ð3 �nþ 1Þð�A þ �B þ �CÞ�88: ð76Þ

In order to obtain the solutions of the above density

matrix elements we consider that the three reservoirs are

vacuum, i.e. �n ¼ 0, and they have the same decay rates, i.e.

�A ¼ �B ¼ �C ¼ �.

�11 tð Þ ¼ 1� C111j j2expð�3t�Þ þ expð�2t�Þð C011j j2þ C101j j2

þ C110j j2þ3 C111j j2Þ � expð�t�Þ½ð1� C000j j2Þ

þ C011j j2þ C101j j2þ C110j j2þ2 C111j j2�, ð77Þ

�12 tð Þ ¼ exp �1

2t�

� �½C000C001þ

�1� expð�t�Þ

ðC010C011

þC100C101ÞþC110C111

�1� 2expð�t�Þþ expð�2t�Þ

�,

ð78Þ

�13 tð Þ ¼ exp �1

2t�

� �½C000C010 þ


þ ðC001C011 � C100C110Þ�, ð79Þ

�14 tð Þ ¼ exp �1

2t�

� ��C000C100þ


ðC001C101

þC010C110ÞþC000C011

�1� 2expð�t�Þþ expð�2t�Þ

�,

ð80Þ

�15 tð Þ ¼C000C011 expð�t�ÞþC100C111 expð�t�Þ�1� expð�t�Þ

,

ð81Þ


,

ð82Þ

�17 tð Þ ¼ C111 expð�t�Þ C000 þ C001


� , ð83Þ

�18 tð Þ ¼ C000C111 exp �3

2t�

� �, ð84Þ

�22 tð Þ ¼ expð�t�Þ ½ C001j j2þ�1� expð�t�Þ

ð C011j j2þ C101j j2Þ

þ C111j j2�1� 2 expð�t�Þ þ expð�2t�Þ

�, ð85Þ


,

ð86Þ


,

ð87Þ

�25 tð Þ ¼ exp �3

2t�

� �C001C011 þ C101C111 exp �

3

2t�

� �

� ð1� expð�t�Þ , ð88Þ

�26 tð Þ ¼ exp �3

2t�

� �½C001C101 þ 2C010C110

þ C000C011


�, ð89Þ

�27 tð Þ ¼ C001C110 exp �3

2t�

� �, ð90Þ

�28 tð Þ ¼ C001C111 expð�2t�Þ, ð91Þ




�, ð92Þ


,

ð93Þ

�35 tð Þ ¼ exp �3

2t�

� �½C010C011 þ 3C110C111

þ C100C101


�, ð94Þ

�36 tð Þ ¼ C010C101 exp �3

2t�

� �, ð95Þ

�37 tð Þ ¼ C010C110 exp �3

2t�

� �þ C000C011 exp �

3

2t�

� �� ð1� expð�t�Þ

, ð96Þ

�38 tð Þ ¼ C010C111 expð�2t�Þ, ð97Þ




�, ð98Þ

�45 tð Þ ¼ C100C011 exp �3

2t�

� �, ð99Þ

�46 tð Þ ¼ C100C101 exp �3

2t�

� �þ C110C111 exp �

3

2t�

� �� ð1� expð�t�Þ

, ð100Þ

�47 tð Þ ¼ C100C110 exp �3

2t�

� �þ C101C111 exp �

3

2t�

� ��1� expð�t�Þ

, ð101Þ

�48 tð Þ ¼ C100C111 expð�2t�Þ, ð102Þ

�55 tð Þ ¼ C011j j2expð�2t�Þþ C111j j2expð�2t�Þ�1� expð�t�Þ

,

ð103Þ

�56 tð Þ ¼ C011C101 expð�2t�Þ, ð104Þ

�57 tð Þ ¼ C011C110 expð�2t�Þ, ð105Þ


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�58 tð Þ ¼ C000C011 exp �5

2t�

� �, ð106Þ

�66 tð Þ ¼ C101j j2expð�2t�Þ þ C111j j2expð�2t�Þ�1� expð�t�Þ

,

ð107Þ

�67 tð Þ ¼ C101C110 expð�2t�Þ, ð108Þ

�68 tð Þ ¼ C101C111 exp �5

2t�

� �, ð109Þ

�77 tð Þ ¼ C110j j2expð�2t�Þ þ C111j j2expð�2t�Þ�1� expð�t�Þ

,

ð110Þ

�78 tð Þ ¼ C110C111 exp �5

2t�

� �, ð111Þ

�88 tð Þ ¼ C111j j2expð�3t�Þ: ð112Þ


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Tripartite entanglement generation using cavity-QED and its dynamics in dissipative environments

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Transcript of Tripartite entanglement generation using cavity-QED and its dynamics in dissipative environments