Opt Quant ElectronDOI 10.1007/s11082-013-9871-1

The preparation of Bell state using ground state of �-typeRb atoms in two optical cavities

Wenlin Li · Chong Li · Heshan Song

Received: 31 October 2013 / Accepted: 25 December 2013© Springer Science+Business Media New York 2014

Abstract The preparation process of quantum Bell states |ψ±〉 is investigated using two�-type Rydberg atoms for the carriers. Two ground states | f 〉 and |g〉 of the �-type atoms aredefined as logical states |0〉 and |1〉 to establish the physical model for preparing quantum statebased on the quantum Zeno dynamics theory. Further, the theoretical analysis and derivationfor the physical model are completed, and preparation of Bell state is obtained. Finally,feasibility analysis about the preparation scheme of Bell states is made. The transitionsbetween atomic energy levels can be avoided after completing preparation of Bell statebecause the logic states |0〉 and |1〉 are all the ground states in this work.

Keywords Quantum optics and optical information processing · Quantum entanglement ·Bell state · conversion operation · �-type Rd atom

1 Introduction

In recent years, the investigation of the quantum communications has attracted comprehensiveattention and become the main content of quantum information filed because it shows manyunique advantages, such as high security, high capacity, and so on (Alcaraz and Rajabpour2013; Rips and Hartmann 2013; Rohde et al. 2013; Wang et al. 2009; Rosset et al. 2012;Ren et al. 2013). Bell states

∣∣φ±⟩

= 1√2(|00〉 ± |11〉) and

∣∣ψ±⟩

= 1√2(|01〉 ± |10〉) are most

commonly entanglement resources in quantum computation and quantum information (Jinet al. 2013; Najarbashi and Maleki 2011; Räsänen et al. 2012), which is playing an importantrole in the reported works on schemes of quantum communication, for examples, quantumsecure direct communication (QSDC) scheme (Beige et al. 2002; Deng et al. 2003), quantumteleportation scheme (Bennett et al. 1993). Therefore, it becomes key problem to realizequantum communication whether the Bell states can be successfully prepared, measured,

W. Li · C. Li (B) · H. SongSchool of Physics and Optoelectronic Engineering,Dalian University of Technology, Dalian 116024, Chinae-mail: liwenlinll@126.com

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saved and convertibly operated. Some typical techniques of preparing Bell states have beenproposed, including single-photon polarization state as well as fiber remote cavity method(Li et al. 2008; Ren et al. 2013). However, it is difficult to achieve the preparation of states|φ±〉 by spatially separated cavities linked with optical fibers owing to the cause in whichthe double excitation processes need to be considered. To this end, Boström et al. proposeda quasi-secure quantum direct communication scheme based only on states |ψ±〉 (Boströmand Felbinger 2002). Compared with conventional quantum secure direct communicationscheme, the scheme reported by Boström skillfully avoided the double excited processesbecause states |φ±〉 are not considered, and it solved the problem to achieve quantum securedirect communication using cavities linked with optical fibers. On this basis, Zheng and Guo(2000), Yin and Li (2007) have already completed the researches of preparing quantum stateswith two-atoms, two-level systems and multi-atoms, two-level systems in spatially separatedcavities linked with optical fibers, respectively. Recently, Ye et al. (2008) extended suchschemes to an entanglement state with higher dimension.

Communication schemes reported at present are basically confined on using atomic excitedstate and the ground state to prepare the quantum state, which maybe lead to the occurrentprobability of transition or the spontaneous emission of atoms after preparation and destructthe desired prepared quantum states (Yin and Li 2007; Zou et al. 2003; Wu et al. 2013).Therefore, it is of theoretical significance and practical value work to propose a more effectivepreparation scheme and to avoid the transition and the spontaneous emission of atoms.

In this paper, the preparation process of quantum states |ψ±〉 is investigated in whichtwo �-type Rydberg atoms are used for the carrier. Compared with the general schemes,the technique in the work is to use the relevant theory of quantum Zeno dynamics and theevolution of quantum state is limited to a special Zeno subspace. Correspondingly, two groundstates | f 〉 and |g〉 of the�-type atom are defined as logical states |0〉 and |1〉. The advantageof the definition is there is no atomic transition between energy levels after the operation iscompleted, which can avoid the confusion of quantum states. On the other hand, the highexcited Rydberg-state of Rb atoms are used to substitute the low excited state atoms reportedin general schemes. Rydberg atom can better satisfy the requirements of the quantum Zenodynamics on the system because is takes on excellent properties, including longer excitedstate lifetime, smaller spacing between energy levels and the strong interaction between thebodies (Saffman et al. 2010). Therefore, Rydberg atoms often are used to achieve the quantumlogic gates of quantum bits (Saffman and Walker 2005), which also make our scheme easierto be verified by the experiment.

This paper is organized as follows: In Sect. 2, the physical model of preparation quantumstate is established. In Sect. 3, the theoretical analysis and derivation for physical model arecompleted, and preparation of Bell state is obtained. Discussion of the results and feasibilityanalysis will be made in Sect. 4 Finally, the summary and the prospects are given.

2 The brief introduction of model

Two�-type atoms A and B are confined in two remote cavities C1 and C2, respectively, andthe cavities are linked with optical fiber B1 in order to constitute the carrier of preparing Bellstate, as shown in Fig. 1. The transitions between the energy levels of�-type atom are shownin Fig. 2. It can be known from Fig. 2 that in each atom there are two ground states | f 〉, |g〉and one excited state |e〉, in which | f 〉 → |e〉 is excited by using quantum field and |g〉 → |e〉is excited using classic field. As a special type of Rydberg atom, Rb atom integrates manyexcellent properties of Rydberg atom. For example, the lifetime of Rydberg states of Rb atom

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The preparation of Bell state using ground state of �-type Rb atoms

Fig. 1 Sketch of theexperimental setup

C1 C2

B1

A B

Fig. 2 Transition of�-type atom

Ωg

e

fg

Fig. 3 Level configurationinvolved of Rb atom

0Fm =

21/292 S

21/292 P

1Fm =1Fm = −

in low energy level l can reach ms magnitude order. So, Rydberg states of Rb atom can beselected as particlesAandB.

Although the �-type energy level need two ground states, they are obtained using addi-tional magnetic field to split ground state energy level. Therefore, the two split energy level.(m j = 1,m j = −1) at 922S1/2 of Rb atom in magnetic field are select as ground states| f 〉, |g〉 and 922 P1/2 as excited state |e 〉, as shown in Fig. 3.

3 Theoretical analysis of model and preparation of Bell states

3.1 Theoretical analysis of �-type energy level structure of Rb atom

The Hamiltonian of the atom-field system of the above-mentioned model under interactionrepresentation can be written as

Htotal = Hlaser + HI (1)

Of them, Hlaser , HI are the Hamiltonian of laser filed and interaction between laser andatom, respectively, and Htotal is total Hamiltonian. Hlaser and HI can be written explicitlyas

Hlaser = �1|e〉A〈g| +�2|e〉B〈g| + H.c.

HI = g1a1|e〉A〈 f | + g2a2|e〉B〈 f | + λ[b+(a1 + a2)] + H.c. (2)

where a1 and a2 are the annihilation operator for the cavities C1 and C2; b+is the creationoperator for the fiber B1; λ is the coupling coefficient between the fiber and cavities. Forsimplicity, supposing that λ,�1,�2 all are real number and g1 = g2 = g.

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Once the initial state of the system is selected as | f g〉AB |00〉C1C2|0〉B1 , the subspace S1

of the system can be represented by a complete orthogonal basis

|φ1〉 = | f g〉AB |00〉C1C2 |0〉B1 ,

|φ2〉 = | f e〉AB |00〉C1C2 |0〉B1 ,

|φ3〉 = | f f 〉AB |01〉C1C2 |0〉B1 ,

|φ4〉 = | f f 〉AB |10〉C1C2 |0〉B1 ,

|φ5〉 = | f f 〉AB |00〉C1C2 |1〉B1 ,

|φ6〉 = |g f 〉AB |00〉C1C2 |0〉B1 ,

|φ7〉 = |e f 〉AB |00〉C1C2 |0〉B1 . (3)

Under this representation, HI can be rewritten as

H2,3I = H3,2

I = H4,7I = H7,4

I = g

H3,5I = H5,3

I = H4,5I = H5,4

I = λ (4)

and Hlaser can be rewritten as

H1,2laser = H2,1

laser = �2

H6,7laser = H7,6

laser = �1 (5)

where H j,kI (laser) = 〈φ j |HI (laser)|φk〉.

If �1,2 � g, λ, the system can be further simplified by using quantum Zeno dynamics(Facchi and Pascazio 2002; Facchi et al. 2009). At the same time, the space S1 can be dividedinto the following five Zeno subspaces:

Z1 = {|ψ1〉 , |ψ2〉 , |ψ3〉}Z2 = {|ψ4〉}Z3 = {|ψ5〉}Z4 = {|ψ6〉}Z5 = {|ψ7〉} (6)

and the eigenvectors corresponds to the each space in Eq. (6) are as

|ψ1〉 = |φ1〉,|ψ2〉 = λ

√

g2 + 2λ2(|φ2〉 − g

λ|φ5〉 + |φ7〉),

|ψ3〉 = |φ6〉,|ψ4〉 = 1

2(|φ2〉 + |φ3〉 − |φ4〉 − |φ7〉),

|ψ5〉 = 1

2(|φ2〉 − |φ3〉 + |φ4〉 − |φ7〉),

|ψ6〉 = λ√

g2 + 2λ2

(

g

2λ|φ2〉 +

√

g2 + 2λ2

2λ|φ3〉 +

√

g2 + 2λ2

2λ|φ4〉 + |φ5〉 + g

2λ|φ7〉

)

,

|ψ7〉 = λ√

g2 + 2λ2

(

g

2λ|φ2〉 −

√

g2 + 2λ2

2λ|φ3〉 −

√

g2 + 2λ2

2λ|φ4〉 + |φ5〉 + g

2λ|φ7〉

)

.

(7)

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The preparation of Bell state using ground state of �-type Rb atoms

All eigenvalues corresponds to the eigenvectors areζ1 = ζ2 = ζ3 = 0, ζ4 = gζ5 = −g, ζ6 = √

g2 + 2λ2, ζ7 = −√

g2 + 2λ2.According to related derivation of Zeno dynamics, the Hamiltonian in Eq. (1) can be

reduced to

Htotal ∼∑

k

∑

η,θ

(ζk Pηk + Pηk Hlaser Pθk )

= (�2δ |ψ1〉 〈ψ2| +�1δ |ψ2〉 〈ψ3| + H.c.)+7

∑

k=4

ζk |ψk〉 〈ψk | (8)

where δ = λ√g2+2λ2

. Pk is the projection operator projected to the kth Zeno space and its

expression is:Pk =

∑

k

Pηk =∑

η

|η〉 〈η| (|η〉 ∈ Zk) (9)

If initial state is prepared as

|ψ1〉 = | f g〉AB |00〉C1C2|0〉B1 (10)

then, entire system will be controlled to evolve in a subspace Z1, and the reduced Hamiltonianis:

Hef f = �2δ |ψ1〉 〈ψ2| +�1δ |ψ2〉 〈ψ3| + H.c. (11)

So under the action of Hamiltonian Eq. (11), the initial state will evolve with time asfollows

|ψ(t)〉= 1

�2

[(

�21 +�2

2 cos�δt) |ψ1〉 − i��2 sin�δt |ψ2〉 +�1�2(−1 + cos�δt) |ψ3〉

]

(12)where �2 = �2

1 + �22.

3.2 Preparation of Bell state

The evolution state |ψ(t)〉 discussed in the previous section can be divided into two kinds ofsituation.

(1) Let ts = π�δ�1 = (

√2 − 1)�2. The evolution of the initial state becomes:

|ψ(tS)〉 = 1√2(|ψ1〉 + |ψ3〉) = 1√

2(| f g〉AB + |g f 〉AB)⊗ |00〉C1C2

|0〉B1 (13)

And taking | f 〉 as |0〉 and |g〉 as |1〉, then

|ψ〉AB = 1√2(|01〉AB + |10〉AB) = |ψ+〉 (14)

(2) Let ts = π�δ�1 = (

√2 + 1)�2. The evolution of the initial state is:

|ψ(tS)〉 = 1√2(|ψ1〉 − |ψ3〉) = 1√

2(| f g〉AB − |g f 〉AB)⊗ |00〉C1C2

|0〉B1 (15)

And taking | f 〉 as |0〉 and |g〉 as |1〉, then

|ψ〉AB = 1√2(|01〉AB − |10〉AB) = ∣

∣ψ−⟩

(16)

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It is found that the Bell states can be prepared using the initial state when the time ts isselected appropriately, and the transformation of states |ψ〉± in system can be realized byadjusting the relationship between �1 and �2.

4 Discussion of the results and feasibility analysis

Fidelity is an important concept in quantum communication (Jozsa 1994), which can be usedto describe the degree of similarity between two quantum states. In this section, the fidelitybetween |ψ(t)〉 and the target state |ψ〉+ or |ψ〉− is discussed firstly in order to intuitiveunderstanding the evolution results of quantum state in the system changing with time. Thesimulation analysis is achieved according to the definition of fidelity and the evolutions offidelity between |ψ(t)〉 and the target states |ψ〉+, |ψ〉− with time are shown in Figs. 4 and5, respectively. It can be seen from Figs. 4 and 5 that the initial state will evolve to the targetstate when ts = π

�δ. If the fidelity is only required to exceed 90 %, then ts can be allowed to

exist a large error range.In the above discussion, the condition �1,2 � g, λ is required in order to simplify calcu-

lations. Afterwards, the evolution of the initial state with time is analyzed under the actionof simplified Hamiltonian Hef f . However, the magnitude order of �1,2, g and λ needs to bediscussed concretely in actual experiments. Suppose that state |φ(tS)〉 is the evolution stateof initial state witch is governed by the total Hamiltonian Htotal at ts = π/�δ. Then thefidelity between state |φ(tS)〉 and state |ψ(tS)〉 can be calculated, shown in Fig. 6 (In order tofacilitate the calculation, let g = λ). It can be known from Fig. 6 that, under the conditions�1 = (

√2 − 1)�2, the value of fidelity F declines gradually with the value x = �1/g

increases. Within the range of x < 0.13, the fidelity can keep above 90 %, which also meansthe derivation and simplification in above discussion is acceptable if x < 0.13. It is unnec-essary to obtain very remarkable splitting of the ground state energy level, in other words,

Fig. 4 Evolution of fidelity between |ψ(t)〉 and the target states |ψ〉+(�1 = (√

2 − 1)�2)

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The preparation of Bell state using ground state of �-type Rb atoms

Fig. 5 Evolution of fidelity between |ψ(t)〉 and the target states |ψ〉−(�1 = (√

2 + 1)�2)

Fig. 6 Relationship between F and �1/g(�1 = (√

2 − 1)�2, g = λ)

the condition �1,2 < 0.13g will be achieved easy. This is one of the reasons why Rydbergatoms are chosen for the preparation of quantum States. Compared to low excited States ofatoms, Rydberg atoms have smaller spacing between energy levels.

In actual operation, it is difficult to ensure the value of adjustable parameter g is adjustedaccurate. The state |ψ(tS)〉 will change and the research results will be affected while thereis an error� g, i.e., g′ = �g + g. Therefore, the discussion of fidelity under considering error�g is necessary. The function relationship between fidelity and error x =�g/g is shown in

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Fig. 7 Relationship between F and �g/g(�1 = (√

2 − 1)�2, g = λ,�1/g = 0.13)

Fig.7 when the parameters are taken as g = λ, �1 = (√

2 − 1)�2. It can be seen that thefidelity is bigger than 90 % once �g/gε[−0.15, 0.15].

5 Conclusion

A method of preparation Bell states using Rb atom is presented in this paper. Firstly, twoground states | f 〉 and |g〉of �-type atom are defined as logical states |0〉 and |1〉, and thephysical model of preparation quantum state is established based on the quantum Zenodynamics theory. Further, the theoretical analysis and derivation for the physical model arecompleted and preparation of Bell states is obtained. Secondly, the evolution relationshipwith time between fidelity and error is computed numerically. Finally, feasibility analysis ismade about the preparation scheme of Bell states.

Comparison is analyzed between the preparation scheme of Bell states proposed in thiswork with that by the common two-level system, and our work are of the following advan-tages: (1) the expected target state can be prepared only using one step operation; (2) the logicstates |0〉 and |1〉 all are the ground states, which may avoid the transition between atomicenergy levels after completed preparation; (3) the experimental data of relevant energy levelsare given which makes the preparation scheme easier to be realized experimentally.

Acknowledgments This research was supported by the National Natural Science Foundation of China(Grant No 11175033).

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