Nonidentical Coupling－based N－atom W Entanglement

ZHANG Ying－qiao

（Department of Physics，College of Science，Yanbian University，Yanji 133002，China）

Nonidentical Coupling－based N－atom W Entanglement

ZHANG Ying－qiao

（Department of Physics，College of Science，Yanbian University，Yanji 133002，China）

Based on the nonidentical coupling of N atoms with a single－mode cavity，a scheme for generation of an N－atom W state is proposed in cavity QED.Under the condition of the large－detuned interactionΔ?gj，the influence of cavity decay is negligible.Such multiple－qubit entanglement will have wide applications to largescale quantum communication and quantum computation.

N－atom W state；three－level atom；cavity QED

1 Introduction

Quantum entanglement is an important physical resource in quantum communication［1－3］and quantum computation.So far，a lot of schemes［4－11］have been proposed to generate entanglements based on various of physical systems both in theory and experiment.Most of these schemes are focused on generation of two－qubit or three－qubit entanglements.For example，Turchette et al［6］have produced the deterministic entangled states of two trapped ions.In this scheme，the two qubits are constituted by the internal states of two trapped ions in both the Bell－like singlet and triplet states.Reina et al［7］have prepared the maximally entangled Bell and Greenberger－Horne－Zeilinger states by optically driven the coupled quantum dots.An alternative scheme［8］to generate three－qubit W state via nonlinear optical state truncation using quantum scissors is proposed.Using atom interferometric techniques，the generation of Bell，NOON and W states of an electromagnetic field in high－Q cavities are also proposed［9］.

However，multiple－qubit entanglements are often required to a large－scale quantum information processing.Therefore，the generation of multiple－qubit enanglements have attracted increasingly attentions.For example，an experimental demonstration of five－photon entanglement［10］was provided by generating a four－photon entangled state with two entangled photon pairs and then combining itwith a single－photon state.Based on rf SQUID，a simple scheme was presented to generate the N－qubit W states by adiabatic passage［11］.

Based on the nonidentical coupling of the three three－level atoms with a resonant single－mode vacuum cavity field［12］and the large－detuning interaction of two identical two－level atoms with a singlemode cavity field［13］in cavity QED，as well as the combination of the coherent dynamics of the above two schemes［14］，we propose a scheme for generation of an N－atom W state.Though it is a generalization of the previous three－atom W state scheme［15］，it is universal to multiple－qubit case.

This paper is organized as follows：In Sec.Ⅱ we describe the coupling dynamics of N atoms interaction with a single－mode cavity.In Sec.Ⅲ we describe the generation of an N－atom W state.Conclusion is given in Sec.Ⅳ.

2 Coupling Dynamics of N Atoms Interaction with a Single－mode Cavity

We consider that N three－level atoms simultaneously interact with a single－mode cavity field.Each atom has one excited state｜e〉and two ground states｜g〉and｜r〉.In our scheme，the state｜r〉is not involved in the interaction with the cavity mode.In the rotating－wave approximation，the interaction Hamiltonian in the interaction picture can be written as（assuming?＝1）

where S＋j＝｜e〉j〈g｜，S－j＝｜g〉j〈e｜（j＝1，2，…，N）.a?and a are the creation and annihilation operators of the single－mode cavity，respectively.gjis the coupling constant of the j th atom to the cavity mode.Δ＝ω0－ωwithω0andωbeing the atomic transition frequency and cavity frequency，respectively.In the case ofΔ?gj，there is no energy exchange between the atomic system and the cavity，the effective Hamiltonian is

whereλj＝g2j／Δandλjk＝gjgk／Δ.The first two terms denote the photon－number dependent Stark shifts，and the third term denotes the cavity mode－induced dipole coupling among the N atoms.Assuming that the coupling constant for the first atom with the cavity is g1and the coupling constants for the other atoms with the cavity are g ＝gj（j＝2，3，…，N）.If the cavity field is in the vacuum state initially，the effective Hamiltonian becomes

whereλ1＝g21／Δand g2／Δ.

3 Generation of an N－atom W State

We assume that the N atoms are in the state〉，initially.The state evolution is

where

If the interaction time satisfy t＝ （2k＋1）π／［（N－1）l＋1］λ1（k is an integer）and the parameters that related to nonidentical atom－cavity coupling satisfywe obtain

Therefore we have generated an N－atom W state

4 Conclusion

In conclusion，we have proposed an universal scheme for generation of an N－atom W state based on the nonidentical coupling of N atoms with a single－mode cavity.Under the large－detuned interaction of the N atoms with the single－mode cavity（Δ?gj），the influence of cavity decay is negligible.Such multiple－qubit entanglements are often required to a large－scale quantum communication and quantum computation.

［1］ Bennett C H，Brassard G，Crépeau C，et al.Teleporting an Unknown Quantum State via Dual Classical and Einstein－Podolsky－Rosen Channels［J］.Phys Rev Lett，1993，70：1895－1899.

［2］ Boschi D，Branca S，Martini F D，et al.Experimental Realization of Teleporting an Unknown Pure Quantum State via Dual Classical and Einstein－Podolsky－Rosen Channels［J］.Phys Rev Lett，1998，80：1121－1125.

［3］ Ekert A K.Quantum Cryptography based on Bell’s Theorem［J］.Phys Rev Lett，1991，67：661－663.

［4］ Osnaghi S，Bertet P，Auffeves A，et al.Coherent Control of an Atomic Collision in a Cavity［J］.Phys Rev Lett，2001，87：037902（4）.

［5］ Eibl M，Kiesel N，Bourennane M，et al.Experimental Realization of a Three－qubit Entangled W State［J］.Phys Rev Lett，2004，92：077901－077904.

［6］ Turchette Q A，Wood C S，King B E，et al.Deterministic Entanglement of Two Trapped Ions［J］.Phys Rev Lett，1998，81：3631－3634.

［7］ Reina J H，Quiroga L，Johnson N F.Quantum Entanglement and Information Processing via Excitons in Optically Driven Quantum Dots［J］.Phys Rev A，2000，62：012305（8）.

［8］ Said R S，Wahiddin M R B，Umarov B A.Generation of Three－qubit Entangled W－state by Nonlinear Optical State Truncation［J］.J Phys B：At Mol Opt Phys，2006，39：1269－1274.

［9］ Islam R，Khosa A H，Saif F.Generation of Bell，NOON and W states via Atom Interferome－try［J］.J Phys B：At Mol Opt Phys，2008，41：035505－035514.

［10］ Zhao Z，Chen Y A，Zhang A N，et al.Experimental Demonstration of Five－photon Entanglement and Open－Destination Teleportation［J］.Nature，2004，430：54－58.

［11］ Deng Z J，Gao K L，F(xiàn)eng M.Generation of N－qubit W States with rf SQUID Qubits by Adiabatic Passage［J］.Phys Rev A，2006，74：064303（4）.

［12］ Shao X Q，Zhu A D，Zhang S，et al.Efficient Scheme for Implementing an N－qubit Toffoli Gate by a Single Resonant Interaction with Cavity Quantum Electrodynamics［J］.Phys Rev A，2007，75：034307（4）.

［13］ Zheng S B，Guo G C.Efficient Scheme for Two－atom Entanglement and Quantum Information Processing in Cavity QED［J］.Phys Rev Lett，2000，85：2392－2395.

［14］ Zhang Y Q，Zhang S.Generation of an N－qubit Phase Gate via Atom－Cavity Nonidentical Coupling［J］.Chin Phys B，2009，18：4683－4689.

［15］ Zhang Y Q.Generation of a Three－qubit W State via Atoms－Cavity Nonidentical Coupling［J］.Journal of Yanbian University（Natural Science），2010，36（2）：146－148.

1004－4353（2011）03－0242－03

基于非全同耦合的N－ 原子W糾纏態(tài)

張英俏
（延邊大學(xué)理學(xué)院 物理系，吉林 延吉133002）

在腔QED中利用N個(gè)原子與一個(gè)單模腔的非全同耦合提出一個(gè)制備N(xiāo)原子W糾纏 態(tài)的方案.在大失諧相互作用Δ?gj的條件下，腔衰減的影響可以忽略.此類(lèi)多比特糾纏態(tài)可廣泛應(yīng)用于級(jí)聯(lián)量子通訊和量子計(jì)算中.

N原子W態(tài)；三能級(jí)原子；腔QED

2011 -05 -14

張英俏（1978—），女，理學(xué)博士，講師，研究方向?yàn)榱孔有畔W(xué).

0365；4250

A