# Generation of field cluster states through collective operation of cavity QED disentanglement eraser

**ABSTRACT** We investigate effects of a collective disentanglement eraser performed over

states of two pairs of pre-entangled cavities tagged independently with two

identical three-level atoms. It is shown that the collective disentanglement

operation ensures not only the recovery of initial coherence but also its

extension from the initial two to four qubits, generating four-qubit field

cluster states. We also propose a cavity QED scheme to generate an arbitrary

field graph state by means of a collective operation of disentanglement erasers.

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**ABSTRACT:**We present a quantum CNOT logic gate based on interaction of a three-level cesium atom with a two-mode electromagnetic field in a high-Q superconducting cavity. The three-level atom acts as a control qubit and the two-mode electromagnetic field serves as a target qubit. Presently available QED experiments make it feasible to realize the theoretical suggestion in the laboratory. We determine the feasibility of our proposal by calculating the fidelity.Journal of Russian Laser Research 01/2008; 29(6):538-543. · 0.71 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We present a controlled quantum secure direct communication protocol by using cluster states via swapping quantum entanglement and local unitary operation. In the present scheme, the sender transmit the secret message to the receiver directly and the secret message can only be recovered by the receiver under the permission of the controller.International Journal of Theoretical Physics 48(10):2971-2976. · 1.09 Impact Factor - SourceAvailable from: Farhan Saif[Show abstract] [Hide abstract]

**ABSTRACT:**We present an experimentally feasible method, based on currently available cavity QED technology, to generate n-partite linear cluster and graph states in external degree of freedom of atoms. The scheme is based on first tagging n two-level atoms with the respective cavity fields in momentum space. Later on an effective Ising interaction between such tagged atoms, realized through consecutive resonant and dispersive interactions of auxiliary atoms with the remanent cavity fields, can generate the desired atomic momenta states. The procedure is completed when the auxiliary atoms after passing through Ramsey zones are detected in either of their internal states. We also briefly explain the generation of weighted graph states in the atomic external degree of freedom.Quantum Information Processing 01/2013; 12(1). · 1.75 Impact Factor

Page 1

Eur. Phys. J. D (2008)

DOI: 10.1140/epjd/e2008-00095-1

THE EUROPEAN

PHYSICAL JOURNAL D

Generation of field cluster states through collective operation

of cavity QED disentanglement eraser

R. Ul-Islam1,2, A.H. Khosa3,a, H.-W. Lee3,4, and F. Saif1

1Department of Electronics, Quaid-i-Azam University, Islamabad, Pakistan

2Photonics Division, National Institute of Lasers and Optronics, PINSTECH, Nilore, Islamabad, Pakistan

3Centre for Quantum Physics, COMSATS Institute of Information Technology, Islamabad, Pakistan

4Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea

Received 2nd January 2008

Published online 7 May 2008 – c ? EDP Sciences, Societ` a Italiana di Fisica, Springer-Verlag 2008

Abstract. We investigate effects of a collective disentanglement eraser performed over states of two pairs

of pre-entangled cavities tagged independently with two identical three-level atoms. It is shown that the

collective disentanglement operation ensures not only the recovery of initial coherence but also its extension

from the initial two to four qubits, generating four-qubit field cluster states. We also propose a cavity QED

scheme to generate an arbitrary field graph state by means of a collective operation of disentanglement

erasers.

PACS. 03.67.-a Quantum information – 03.67.Mn Entanglement production, characterization, and ma-

nipulation – 42.50.Pq Cavity quantum electrodynamics; micromasers

1 Introduction

Quantum entanglement was justly referred to as the char-

acteristic trait of quantum mechanics [1], because it ex-

hibits all the dilemmas related to the conflict between

the theory and local realism in a most profound man-

ner [2]. The concept of quantum eraser [3] has also played

its due part in this quest for an in-depth understanding

of what happens at the microscopic level, especially con-

cerning the relationship of time and quantum dynamics as

well as interconnections among information eraser, com-

plementarity and entanglement [4–6]. In this respect, var-

ious types of quantum erasers have been proposed and

experimentally demonstrated both with photons as well

as with atom-field systems [4,7,8]. All of these are basi-

cally concerned with the restoration or revival of initial

coherence that gets destroyed due to coupling of some tag

to a coherent superposition state. Therefore, an optimal

quantum eraser may be taken as comprised of coupling of

a tag qubit via its external, internal or intrinsic degrees of

freedom to any given superposition state, which binds the

tag and the superposition state under non-local, entangled

correlations through interactions of the labelling system

i.e. the tag with either one or both components of the orig-

inal superposition. This consequently enlarges the Hilbert

space of the new non-factorizable system. The mutual or-

thogonality of the tag qubit and the initial superposition

effectively destroys the interference pattern characteristic

ae-mail: ashfaq khosa@yahoo.com

of the untagged state. This interference loss is attributed

to the possible distinguishability of the final tag states.

The measurement of the tag qubit in its original basis,

say |α? and |β?, collapses the state, washing out interfer-

ence fringes completely. The fringe revival, however, can

be achieved if the tag qubit is measured in a rotated basis,

i.e.,

tion the class of quantum erasers introduced by Garisto

and Hardy [9] that not just deals with the restoration of

interference but also suggests a method for the recovery

of coherence of an initially two-partite entangled state by

removing the unwanted tag qubit. Such a type of eraser

is usually termed as disentanglement eraser and has been

experimentally verified in NMR based systems [10].

Suppose we have an initially entangled Bell state

???Φ(±)

An extra tag qubit can get entangled with the system

either through its interaction with the component A or B

or both. The state will therefore comes to be

1

√2

1

√2(|α? + |β?) [7]. Here we specifically want to men-

A,B

?

=

1

√2(|0A,0B? ± |1A,1B?).

(1)

|ΨA,B,T? =

????0A,0B,p(T)?

±

???1A,1B,q(T)??

,

(2)

with p,q = 0, 1 and p ?= q. This tagged information

poses a potential threat to coherence of the initial state

???Φ(±)

A,B

?

unless it is removed or erased in a satisfactory

manner. As stated earlier, the usual method to erase such

Page 2

2 The European Physical Journal D

tagged information is to measure the ancilla qubit in an

obliquely rotated basis, i.e.,

tion of such disentanglement eraser for cavity QED based

atom-field systems has already been proposed with exper-

imentally feasible schematics for coupling and removal of a

tag atomic qubit imposed over an initially entangled field

Bell state [11].

In present article we extend the procedure proposed

in [11] and show that operation of a collective eraser over

two tagged states of the type expressed in equation (2)

not only safeguards coherence of the states but also lead

to a four-partite field cluster state. Collective eraser is

performed through consecutive resonant and dispersive in-

teractions of the atoms with the cavity field, followed by

their passage through Ramsey zone, prior to detection.

Subsequent detection of atoms in excited or ground state

culminates into the generation of cluster state. These re-

cently introduced so called cluster states [12] exhibit some

novel characteristics. The states are found to be more re-

sistant to decoherence as compared to their counterpart

GHZ states [13]. Furthermore, the cluster states exhibit a

rich nonlocality structure different from that possessed by

the GHZ states. For example, one can construct new types

of Bell inequalities that are maximally violated by the

four-qubit cluster states but not violated at all by the four-

qubit GHZ states [14,15]. The emerging interest in cluster

states is linked with the newly proposed one-way quan-

tum computing model whose operatibility depends upon

such states [16]. The model performs universal quantum

computing through local single-qubit measurements of the

cluster states and its experimental feasibility has also been

verified through the operational exploration of the four-

photon cluster state [17]. Thus, owing to their evident im-

portance, many different schemes of generating the cluster

states in photonic, atomic and solid-state systems have

been proposed employing various techniques [12,18–20].

Owing to their relative easiness to implement, the schemes

utilizing linear optics techniques have already been ex-

perimentally demonstrated [17,21]. Such techniques [18],

however, suffer from the fact that they are inherently prob-

abilistic. Whereas schemes based on cavity QED tech-

nologies [19] have an ideal success probability of unity,

although in reality such schemes are subjected to exper-

imental imperfections such as cavity photon loss, atomic

spontaneous emission, and violation of the Lamb-Dicke

condition.

The scheme we consider here is also a cavity QED

based scheme and generates the four-qubit field cluster

state in four spatially separated cavities. In Section 2, we

briefly describe the tagging procedure. The collective op-

eration of a disentanglement eraser on the tagged system,

leading to the generation of the four-qubit cluster state,

is then described in Section 3. A schematic view of our

scheme for tagging and erasing information is presented

in Figure 1, which will be described in detail in Sections 2

and 3. In Section 4 we show that an arbitrary field graph

state can be generated by an appropriate preparation of

initially tagged field states and collective operation of dis-

entanglement erasers upon these fields. Finally in Sec-

??p(T)?±??q(T)?. Implementa-

Fig. 1. Schematic representation of the proposed setup for

tagging and operating disentanglement eraser.

tion 5 we conclude with a discussion covering experimental

feasibility of the proposed method.

2 Tagging procedure

Suppose we are provided with two independent but iden-

tical field Bell states

j,k

where j,k = A,B(C,D), and |0? and |1? refer to the vac-

uum and one-photon state, respectively. Such a state can

be prepared by sending successively an excited two-level

atom through two initially vacuum state cavities. Atom

interacts resonantly with first cavity for π/2 Rabi pulse.

After leaving the cavity, atom is passed through a Ramsey

classical field for a time corresponding to π pulse. The

atom finally interacts resonantly with the second cavity,

again for a π pulse. The detection of atom in ground state,

after emerging from second cavity ensures the generation

of desired state. This can be done, for example, using a

set up similar to reference [22].

In order to place a tag over each pair of the field Bell

states

j,k

, we take two identical three-level cascade

atoms, initially prepared into a symmetric superposition

of the two lower levels |l1? and |l2?, and pass them sepa-

rately through cavities A and C, respectively (see Fig. 1).

The atoms are selected such that the lower atomic tran-

sition |l1? → |l2? is far detuned and hence effectively de-

coupled from the cavity field contained in cavities A and

C, whereas the upper transition |l2? → |l3? only yields

dispersive interactions through an atom-field detuning of

δ. The effective Hamiltonian for such a set of system will

be

?

???Φ(+)

?

=

1

√2(|0j,0k? + |1j,1k?),

???Φ(+)

?

H(Tq)

eff

=?µ2

δ

aa†???l(Tq)

3

??

l(Tq)

3

??? − a†a

???l(Tq)

2

??

l(Tq)

2

???

?

.

(3)

Page 3

R. Ul-Islam et al.: Generation of field cluster states3

Here superscript Tq(q = 1 or 2) denotes the tag atom (1

or 2), a†a stands for the Fock field number operator, with

a†a|n? = n|n?, and µ is the atom-field coupling constant.

The corresponding state vector for an interaction time τq

may be written as

|Ψj,k,Tq(τq)? =Cl(Tq)

1

0j,0k(τq)

+ Cl(Tq)

???0j,0k,l(Tq)

???1j,0k,l(Tq)

???0j,1k,l(Tq)

???2j,1k,l(Tq)

1

?

2

0j,0k(τq)

+ Cl(Tq)

???0j,0k,l(Tq)

???1j,1k,l(Tq)

???1j,1k,l(Tq)

2

?

?

?

?

?

?

1

1j,0k(τq)

+ Cl(Tq)

1

1

1j,1k(τq)

+ Cl(Tq)

1

2

0j,1k(τq)

+ Cl(Tq)

2

2

1j,1k(τq)

+ Cl(Tq)

2

1

2j,1k(τq)

1

.

(4)

We note that atom-1 interacts with cavity A and atom-

2 interacts with cavity C. Schr¨ odinger’s equation under

initial conditions Cl(Tq)

Cl(Tq)

gives the following state vector for an interaction time τq,

1

0j,0k(0) = Cl(Tq)

1j,0k(0) = Cl(Tq)

2

0j,0k(0) = Cl(Tq)

0j,1k(0) = Cl(Tq)

1

1j,1k(0) =

2

1j,1k(0) = 1/2 and Cl(Tq)

121

2j,1k(0) = 0

|Ψj,k,Tq(τq)? =

1

√2

????l(Tq)

2

1

?

⊗

⊗

√2(|0j,0k? + eiµ2τq

1

√2(|0j,0k? + |1j,1k?)

+

???l(Tq)

?

1

δ

|1j,1k?)

?

,

(5)

where j,k,q = A,B,1(C,D,2). We select the interaction

times of the atoms in cavities A and C such that µ2τq/δ

= π for both q = 1 and 2. The atoms after completing

dispersive interaction with their respective cavities then

pass through Ramsey zone R1and R2as shown in Fig. 1,

where their internal states transform to

???l(Tq)

2

1

?

?

→

1

√2

1

√2

????l(Tq)

1

1

?

?

+

???l(Tq)

2

2

??

??

,

???l(Tq)

→

????l(Tq)

−

???l(Tq)

,

(6)

and the state listed in (5) consequently becomes

??Ψj,k,Tq

?=

1

√2

????0j,0k,l(Tq)

1

?

+

???1j,1k,l(Tq)

2

??

.

(7)

This completes the tagging procedure as atom-1 and

atom-2 are now tagged with the initial entangled states

???Φ(+)

A,B

?

and

???Φ(+)

C,D

?

, respectively, through their internal

degrees of freedom forming three partite GHZ states. The

product of these two GHZ states may be expressed as

???ΨC,D,T2

2

???l(T1)

Here we have taken |X1? = |0A,0B,0C,0D?, |X2? =

|0A,0B,1C,1D?, |X3? = |1A,1B,0C,0D? and |X4? =

|1A,1B,1C,1D?.

A,B,T1

?

= |ΨA,B,T1? ⊗ |ΨC,D,T2?

?

+ |X3? ⊗

=1

|X1? ⊗

???l(T1)

,l(T2)

1

,l(T2)

1

?

?

+ |X2? ⊗

???l(T1)

,l(T2)

1

,l(T2)

2

??

?

.

21

+ |X4? ⊗

???l(T1)

22

(8)

3 Generation of cluster states through

operation of disentanglement eraser

In this section, we discuss the evolution of the tagged

systems under application of a collective disentanglement

eraser. This is done by passing atom-1 through an initially

empty cavity E where it goes through a resonant π Rabi

cycle and hence exits out in the ground state with its state

effectively transferred to the cavity (see Fig. 1). The state

of the system thus comes to be

???ΨD,E,T2

1

A,B,C,T1

?

=1

2[|X1? ⊗

???1E,l(T2)

???0E,l(T2)

1

?

+ |X2? ⊗

???1E,l(T2)

???0E,l(T2)

2

?

− i|X3? ⊗

?

− i|X4? ⊗

2

?

] ⊗

???l(T1)

1

?

.

(9)

Atom-1, detected in the ground state, is consequently

traced out of the system. Atom-2, then traverses cav-

ity E, such that the transition |l1?

couples dispersively with the cavity field. This can

be done, for example, by applying an external Stark

field that induces a detuning ∆ between field and

atomic transition frequencies, and/or by moving the

cavity mirrors if necessary. Such dispersive interac-

tions are again described by the Hamiltonian H(T2)

?λ2

∆

22

notes the atom-field coupling constant in this case. The

corresponding state vector of the system after an arbi-

trary interaction time t, may be expressed as

→|l2? now

eff

=

?

aa†???l(T2)

??

l(T2)

??? − a†a

???l(T2)

1

??

l(T2)

1

???

?

, where λ de-

|Ψ (t)? =Cl(T2)

1

X1,0E(t)

+ Cl(T2)

???X1,0E,l(T2)

???X2,1E,l(T2)

???X3,0E,l(T2)

???X4,2E,l(T2)

1

?

2

X2,0E(t)

+ Cl(T2)

???X2,0E,l(T2)

???X2,1E,l(T2)

???X4,1E,l(T2)

2

?

?

?

?

?

?

1

X2,1E(t)

+ Cl(T2)

1

1

X3,1E(t)

+ Cl(T2)

1

2

X3,0E(t)

+ Cl(T2)

2

2

X4,1E(t)

+ Cl(T2)

2

1

X4,2E(t)

1

.

(10)

Page 4

4The European Physical Journal D

Schr¨ odinger’s

Cl(T2)

−i/2 and Cl(T2)

the following expression for the state vector

equation

X2,0E(0) = 1/2, Cl(T2)

X2,1E(0) = Cl(T2)

underinitial conditions

1

X1,0E(0) = Cl(T2)

21

X3,1E(0) = Cl(T2)

X3,0E(0) = Cl(T2)

2

X4,1E(0) =

121

X4,2E(0) = 0 yields

|Ψ (t)? =1

iλ2

2

????X1,0E,l(T2)

1

?

?

+ e−iλ2

∆t???X2,0E,l(T2)

2

?

−ie

∆t???X3,1E,l(T2)

Atom-2 is then passed through a Ramsey zone R3which

produces internal state transformations similar to (6). If

the interaction time of the second atom with cavity E is

selected such that λ2t/∆ = π/2, then the state of the

system after atom’s passage through R3becomes

1

− ie−2iλ2

∆t???X4,1E,l(T2)

2

??

.

(11)

|Ψ? =

1

√2

?1

2[|X1,0E? − i|X2,0E? + |X3,1E?

???l(T2)

+i|X4,1E?] ⊗

1

?

+1

2[|X1,0E? + i|X2,0E?

+|X3,1E? − i|X4,1E?] ⊗

???l(T2)

1

2

??

or

.

(12)

Thus, when atom-2 is detected in either

we get the corresponding equally probable entangled field

state. However, since our goal is to perform a collective

eraser, therefore to complete the procedure duly, we pass

another atom, say atom-3, initially in its ground state,

through cavity E, where it goes through a resonant inter-

action with the cavity mode for a time corresponding to

a π Rabi cycle. This operation effectively sweeps the field

information to the atom while leaving cavity E in vac-

uum state. This atom, after emerging out of the cavity E,

passes through R3and is finally detected at its respective

state-sensitive detector D3 as depicted in Figure 1. The

final expression of the state vector therefore comes to be

?1

+1

2[|X1? − i|X2? + i|X3? − |X4?] ⊗

+1

2[|X1? + i|X2? − i|X3? − |X4?] ⊗

+1

2[|X1? + i|X2? + i|X3?

+|X4?] ⊗

???l(T2)

?

???l(T2)

2

?

,

|ΨF? =1

22[|X1? − i|X2? − i|X3? + |X4?] ⊗

???l(T2)

1

???l(T2)

1

,l(3)

1

?

???l(T2)

2

,l(3)

2

?

?

,l(3)

1

???l(T2)

???l(T2)

2

,l(3)

2

??

⊗ |0E?.

(13)

Thus, upon detection of atoms through state sensitive de-

tectors D2and D3in combination of the internal states of

either

1

,l(3)

1

,

1

,l(3)

2

we get any one of the corresponding equally probable

linear field cluster states that can subsequently be con-

verted into the standard form through local operations, if

needed [12,23]. One sees that the two independent pairs

of two-field Bell states have now been merged to form a

four-field cluster state. Thus, the collective operation of

???l(T2)

??

,

???l(T2)

2

,l(3)

1

?

,or

???l(T2)

2

,l(3)

2

?

,

a disentanglement eraser can generate four-qubit coher-

ence out of the initial coherence of a pair of two entangled

qubits.

The proposal presented here may be generalized in a

straightforward way to generate (2n)-qubit field cluster

states of the form,

????X(2n)

where

???X(2n)

???X(2n)

|ΨF? =1

2

1

?

+

???X(2n)

2

?

+

???X(2n)

3

?

−

???X(2n)

4

??

,

(14)

1

?

= |0A1,0A2,...,0An,0C1,0C2,...,0Cn?,

= |0A1,0A2,...,0An,1C1,1C2,...,1Cn?,

= |1A1,1A2,...,1An,0C1,0C2,...,0Cn?,

= |1A1,1A2,...,1An,1C1,1C2,...,1Cn?,

provided we are in possession of a pair of tagged states of

the type

???X(2n)

???X(2n)

2

?

3

?

4

?

(15)

??Ψj1,j2,...,jn,Tq

?=

1

√2

????0j1,0j2,...,0jn,l(Tq)

1

?

+

???1j1,1j2,...,1jn,l(Tq)

2

??

,

(16)

where

(C1,C2,...,Cn) represent n cavities initially entan-

gled with one another and tagged with atom-1 (atom-2).

Through collective operation of a disentanglement eraser,

(2n) entangled qubits can be generated out of a pair of

(n) entangled qubits.

j,q

=

A,1(C,2),and

A1,A2,...,An

4 Generation of graph states

In the scheme described above, collective operation of a

disentanglement eraser on two independent cavity fields

and the resulting generation of coherence between them

are accomplished by tagging the two fields with an atom

each and letting the two tag atoms interact one after the

other with a common radiation field. Alternatively, it can

be achieved by tagging one cavity field with an atom and

utilizing interaction between this atom and the field of the

other cavity. Such a tagged field-state that serves as the

basic building block for the generation of field graph states

can be engineered by passing an excited two-level atom

through an initially vacuum state cavity for a time corre-

sponding to π/2 Rabi pulse. The state is engineered when,

atom after emergence from the cavity, is passed through

a Ramsey zone for a π Rabi oscillation. In particular, the

controlled-Z operation between two cavity fields can be

accomplished by letting the atom tagged with one cavity

field go through a dispersive interaction with the field of

the other cavity and pass through a Ramsey zone. Con-

sider, for example, cavity-1 tagged with atom-1 in state

1

√2

12

metric superposition

????01,l(1)

?

+

???11,l(1)

??

and cavity-2 prepared in a sym-

1

√2(|02? + |12?). After a dispersive

Page 5

R. Ul-Islam et al.: Generation of field cluster states5

interaction of atom-1 with the field of cavity-2, the state

of the system becomes

???Ψ(2)(τ(d))

?

=1

2

????01,02,l(1)

2

1

?

+ e

iλ2

∆τ(d)???01,12,l(1)

1

?

+e

−iλ2

∆

τ(d)???11,02,l(1)

?

+ e

−2iλ2

∆

τ(d)???11,12,l(1)

2

??

.

(17)

Here τ(d)stands for the time dispersive interaction of the

atom with the second cavity. We then let atom-1 pass

through a Ramsey zone. The state of the system then

becomes

?1

+e

+1

2

−2iλ2

∆

???Ψ(2)?

=

1

√2

−iλ2

∆

2

?

|01,02? + e

iλ2

∆τ(d)|01,12?

−2iλ2

∆

τ(d)|11,02? + e

|01,02? + e

τ(d)|11,12?

?

τ(d)|11,02?

???l(1)

⊗

???l(1)

??

1

?

?

iλ2

∆τ(d)|01,12? − e

τ(d)|11,12?

−iλ2

∆

−e

?

⊗

2

.

(18)

Thus the detection of the atom in either of the state i.e.

???l(1)

interaction time aided with subsequent local operations,

we can convert the state, if needed, into standard form

1

2(|01,02?+|01,12?+|11,02?−|11,12?). In general, the ini-

tial state for the generation of n-partite linear field cluster

state may be expressed as follows:

1

?

or

???l(1)

2

?

, we get the corresponding linear two par-

tite field cluster state. With appropriate selection of the

???Ψ(n)(0)

The state engineering procedure is quite easy to follow.

The kth atom, tagged with the kth cavity field, inter-

acts dispersively with the (k +1)th cavity field for a time

τ(d)

kth atom passes through Ramsey zone prior to its de-

tection. The procedure continues until (n − 1)th atom

completes its dispersive interactions with the nth cav-

ity field and then detected after its traversal through the

Ramsey zone. Culmination of the process subsequently

yields any one of the 2n−1equally probable n-partite lin-

ear cluster states corresponding to the specific pattern of

the recorded atoms out of the 2n−1possible permutations.

In order to elaborate it a little bit further, we present the

case of four partite linear field cluster state. Here, firstly

atom-1, tagged with cavity-1, interacts dispersively with

the field in cavity-2 for a time τ(d)

the Ramsey zone. Next, atom-2, tagged with cavity-2,

performs dispersive interactions with cavity-3 for a time

τ(d)

2

and then goes through the Ramsey zone after which

it is duly recorded in a state selective detector. Finally,

atom-3 which was initially tagged with the cavity-3 in-

teracts dispersively with the field in cavity-4 and then it

also passes through the Ramsey zone. It should be noted

that being dispersive in nature, the temporal ordering of

?

=

1

2

n

2

n−1

?

k=1

????0k,l(k)

1

?

+

???1k,l(k)

2

??

⊗ (|0n?+|1n?).

(19)

k. After the completion of dispersive interactions, this

1

and passes through

the interactions is neither specific nor important. Thus

the atoms may interact with the fields simultaneously or

in any arbitrary sequence deemed experimentally feasi-

ble. Since three atoms are involved in the generation of

four-partite field cluster state, so 24−1= 8 equally prob-

able states are possible corresponding to the recording of

any one out of the eight atomic internal state detection

possibilities

k

, for i, j, k = 1,2. The four-

partite linear field cluster state generated when atoms are

detected in, for example

???l(1)

i,l(2)

j,l(3)

?

???l(1)

1,l(2)

2,l(3)

1

?

pattern is given by

???Ψ(4)?

=1

4(|01,02,03,04? + e

iλ2

∆

2

iλ2

∆τ(d)

3

|01,02,03,14?

+ e

?

?

?

?

?

?

τ(d)

−τ(d)

3

?

|01,02,13,04?

?

?

+τ(d)

3

|01,12,03,14?

−2τ(d)

|01,12,13,04?

−2τ(d)

+ e

iλ2

∆

τ(d)

2

−2τ(d)

3

|01,02,13,14?

|01,12,03,04?

?

−τ(d)

?

|11,02,03,04?

τ(d)

13

|11,02,03,14?

τ(d)

123

− e

− e

− e

− e

+ e

iλ2

∆

τ(d)

1

−τ(d)

2

iλ2

∆

τ(d)

1

−τ(d)

2

iλ2

∆

τ(d)

123

?

iλ2

∆

τ(d)

12

−2τ(d)

3

|01,12,13,14?

−iλ2

∆

τ(d)

1

?

?

?

?

?

?

+ e

−iλ2

∆

−τ(d)

?

+τ(d)

+ e

−iλ2

∆

−τ(d)

?

|11,02,13,04?

?

|11,12,03,04?

−τ(d)

|11,12,03,14?

+τ(d)

2

+

2

|11,12,13,04?

+τ(d)

23

|11,12,13,14?).

+ e

−iλ2

∆

τ(d)

1

−τ(d)

2

+2τ(d)

3

|11,02,13,14?

− e

− e

−iλ2

∆

2τ(d)

1

+τ(d)

2

?

−iλ2

∆

2τ(d)

1

+τ(d)

23

?

− e

− e

−2iλ2

∆

τ(d)

1

τ(d)

3

?

−2iλ2

∆

?

τ(d)

1

+τ(d)

?

(20)

Since in dispersive interactions we are free to select arbi-

trary values for the interaction times τ(d)

so we may engineer weighted entangled states along with

standard cluster states. The above n = 4 case corresponds

to the four-qubit cluster state we considered in the previ-

ous section.

Now, with the availability of Ising interactions based

on controlled-Z gate as described above, we are in a posi-

tion to engineer any arbitrary graph state [23,24]. Here, for

the sake of demonstration, we consider the case of n-qubit

star graph states (e.g., No. 3 (n = 4), No. 5 (n = 5), No. 9

(n = 6) of Fig. 4 of Ref. [23]). These states (Fig. 2) are

local unitarily equivalent to their counterpart GHZ states.

For the generation of such states, we prepare cavity-1 into

superposition of zero and one photon whereas the remain-

ing (n−1) cavity fields are taken to be tagged with respec-

tive atoms. Therefore the initial state for such a system

1 ,τ(d)

2

and τ(d)

3 ,

Page 6

6 The European Physical Journal D

Fig. 2. A n-partite cavity field graph state. Dots denote ver-

tices 1,2,...,n and A which are high-Q cavities containing

Fock field superpositions. Vertex A forms the sole edge of this

graph.

will be

???Ψ(n)(0)

The procedure for the generation of desired star graph

state is now quite simple. All the tagged atoms inter-

act dispersively in a successive manner with the fields

in cavity-1 and then pass through Ramsey zones, prior

to their detection. The procedure finally culminates into

the generation of required graph states whose weights

can be chosen appropriately by proper selection of dis-

persive atom-field interaction times. Similarly for the

generation of n-qubit ring-type graph state (e.g., No. 8

(n = 5), No. 18 (n = 6) of Fig. 4 of Ref. [23]), we

propose to prepare all n cavities in the tagged state

1

√2

12

tag atom interact dispersively with the (k − 1)th-cavity

field and then traverses through a Ramsey zone. Finally

atom-1 interacts dispersively with the nth cavity field and

pass through a Ramsey zone. This generates the tomo-

graphically desired state by effectively closing the ring. It

is now clear that an appropriate application of the col-

lective disentanglement eraser allows one to generate any

arbitrary field graph state one desires. Furthermore, our

scheme can be used to generate weighted graph states [24]

that result when particles and fields interact for different

interaction times, because in our scheme atom-field inter-

action times can be controlled precisely and fixed to any

arbitrary value we desire.

?

=

1

2

n

2

n

?

k=2

????0k,l(k)

1

?

+

???1k,l(k)

2

??

⊗ (|01?+|11?).

(21)

????0k,l(k)

?

+

???1k,l(k)

??

where k = 1,2,...,n. Now kth

5 Discussion

We have demonstrated that operation of a disentangle-

ment eraser not only restores coherence of the original

untagged state but can also bind coherent subsystems

bearing independent tags into an extended single coherent

entangled system occupying effectively doubled Hilbert

space, if performed collectively over a set of mutually fac-

torizable states. A new feature in our scheme originates

from an addition of one more pair of entangled cavity fields

to the scheme of Zubairy et al. [11]. By tagging each pair

of the cavity fields with an atom and exploiting interac-

tion of the two tagged atoms with a common radiation

field, two independent pairs of entangled fields have been

transformed to four fields all entangled with one another

in the form of the four-field cluster state. Experimental

execution of the proposed scheme seems promising un-

der the prevailing research status of cavity QED. Cavity

quality factors in the range of 1012have been reported

leading to lifetimes up to seconds in the microwave re-

gion [25], whereas dispersive atom-field interaction times

are of the order of a few microseconds and resonant inter-

actions usually take times in nanoseconds. All the tech-

niques invoked in the present proposal have already been

perfected experimentally [22,26] with dispersive interac-

tions fully explored for a variety of experiments includ-

ing the successful demonstration of the quantum phase

gate [27]. Furthermore, Zubairy et al. [11] have already

proved the feasibility of tag addition and removal over in-

dividual Bell states. These atomic tags, however, can also

be implemented through resonant interactions on atomic

transition

2

→

2π Rabi rotation. Since a resonant 2π transition requires

a comparatively smaller interaction time than a dispersive

interaction, such an alteration in the scheme will make it

more resistant to decoherence and experimentally more

feasible [11]. This 2π-phase flip has already been experi-

mentally demonstrated [28]. Although a sufficiently pre-

cise control over synchronized atomic dynamics has been

achieved [29], such stringent criteria are not needed here.

What we need is that the three atoms should pass through

cavity E consecutively one after another. This allows us to

utilize a time window between the atoms. The operational

complexity as well as atom-cavity resource utilization in-

voked in the present work is quite matchable with that

of [11]. Zubairy et al. in [11] have used disentanglement

eraser for teleportation whereas our proposed collective

eraser culminates into the generation of four-field linear

cluster state. Furthermore, although in Section 3, we have

mentioned five cavities but only three of them take part in

the atom-field interactions. Thus, if we take two-mode en-

tangled state in a single cavity [30], then the same task of

four-field cluster state generation can be accomplished by

using only three cavities. Indeed cavity QED based pro-

posals bearing equivalent or even higher complexity have

already been published much earlier [31]. Experimental

cavity QED has progressed substantially during last 10–

12 years. Hence the proposed scheme can be safely ex-

ecuted, at least, under an experimental scenario where

high-Q cavities are placed close to one another.

???l(Tq)

?

???l(Tq)

3

?

for a time corresponding to a

RUI is thankful to Higher Education Commission, Government

of Pakistan for financial support through Merit Scholarship.

AHK thanks COMSTECH for their support. HWL thanks col-

leagues at COMSATS Institute of Information Technology for

numerous helpful discussions on various aspects of this work

during his visit.

Page 7

R. Ul-Islam et al.: Generation of field cluster states7

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