A simple scheme for realizing six-photon entangled state based on cavity quantum electrodynamics

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1 J. At. Mol. Sci. doi: /jams a Vol. 3, No. 1, pp February 2012 A simple scheme for realizin six-photon entanled state based on cavity quantum electrodynamics Den-Yu Zhan, Shi-Qin Tan, Xin-Wen Wan, and Li-Jun Xie Department of Physics and Electronic Information Science, Henyan Normal University, Henyan, Hunan , China Received 17 April 2011; Accepted (in revised version) 10 May 2011 Published Online 28 September 2011 Abstract. A simple scheme is presented for eneratin six-photon entanled state with resonant interaction between a cascade type four-level atom and two three-mode cavities. In the proposed protocol, the quantum information is encoded on Fock states of the cavity fields. We solve Schrödiner equation and obtain quantum states of interaction system.the detection of the atom can collapse the cavity to the desired six-photon entanled state. PACS: a, Lx, p Key words: quantum information, cascade four-level atom, three-mode cavity, entanled state 1 Introduction Cavity quantum electrodynamics (QED) is an ideal candidate for implementin quantum information processin[1]. The reason is based on the followin two points. (i) Photons are ideal carriers for fast and reliable communication over lon distances, and the atoms are ood memorizers for storin and processin quantum information. Thus the combination of atoms and photons can be useful in quantum computation. (ii) The atoms trapped in a hih-q cavity have lon decoherence time[2]. Entanlement of two or more particles is the most intriuin characteristic of quantum mechanics. In recent years, several physical system have been suested to enerate quantum entanlement[3 5], amon which cavity quantum electrodynamics (QED) system is viewed as a promisin tool in that recent development in cavity QED techniques have made us to produce quantum entanlement between cavity fields[6] and between atoms[7]. Entanled states not only are reconized as an essential inredient for testin the foundation of quantum mechanics, but also have many sinificant applications in quantum-information processin (QIP)[8]. Generally, the more particles that can be entanled, the more clearly nonclassical effects are exhibited, and the more useful the states Correspondin author. address: ÝÞÒ¾ ÓÙºÓÑ (D. Y. Zhan) 73 c 2012 Global-Science Press

2 74 D. Y. Zhan, S. Q. Tan, X. W. Wan, and L. J. Xie/J. At. Mol. Sci. 3 (2012) are for quantum applications[9]. Thus eneration and manipulation of multipartite entanled states are very important tasks in QIP and have been attractin much attention. Dür et al. have shown that there are two inequivalent classes of tripartite entanlement states, i.e., the GHZ class and the W class, under stochastic local operations and classical communications[10]. GHZ type of entanled state has many interestin properties. For example, the three-particle GHZ state is maximally stable aainst noise, maximally violates Bell inequalities, and can be used to implement perfect teleportation[11]. Men et al have proposed a scheme for preparin an N atoms GHZ entanled state throuh the interaction between N atoms and a cavity[12]. Althouh many schemes for tripartite entanled states have been studied[13,14], the report about preparin multi-photon (N>3) entanled state is very few. 2 Realization of six-photon entanled state We consider the resonant interaction of a cascade type four-level atom with a three-mode cavity field is shown in Fi. 1. The interaction Hamitonian for such a system can be described as ( h=1)[15] H 1 = 1 a + i +a i + 2 b + i j + b j i + 3 c + j e +c e j, (1) where a + (b +,c + ) and a(b,c) are the creation and annihilation operators for the cavity fields, respectively, = 1 = 2 = 3 is the couplin constant of the interaction of atom with the cavity mode. e i j ÙÖ ½ ØÓÑ ÐÚÐ ØÖÙØÙÖ Ó ØÝÔ ÓÙÖ¹ÐÚÐ ØÓÑ ÓÙÔÐÒ ØÓ ØÖ¹ÑÓ ÚØݺ For simplicity, assume that the cavities are initially prepared in 0,0,0 1, 0,0,0 2, the atom is initially in e, we obtain the system initially prepared in the state φ(t=0) = e 0,0,0 1 0,0,0 2. (2)

3 D. Y. Zhan, S. Q. Tan, X. W. Wan, and L. J. Xie/J. At. Mol. Sci. 3 (2012) We consider a eneral state of φ(t) =C1 (t) e,0,0,0 +C 2 (t) j,0,0,1 +C 3 (t) i,0,1,1 +C 4 (t),1,1,1. (3) Usin the Schrodiner equation, we can et a roup of first-order time differential equations of C 1 (t),c 2 (t),c 3 (t),c 4 (t) i C 1 t = C 2, i C 2 t = (C 1+C 3 ), i C 3 t = (C 2+C 4 ), i C 4 t = C 3. (4a) (4b) (4c) (4d) On eliminatin C 1 (t),c 2 (t),c 3 (t), we obtain the followin fourth-order time differential equation for C 4 4 C 4 t C 4 t C 4 =0. (5) The initial condition for C 4 at t=0 are C 4 (0)=0, We consider a solution of the form C 4 (0) t 2 =0, 2 C 4 (0) t 2 =0, 3 C 4 (0) t 3 =. (6) C 4 (t)=ae ω 1 t +Be ω 2 t +Ce ω 3 t +De ω 4 t, (7) whereω 1,ω 2,ω 3 andω 4 are the roots the fourth-order polynomial equation z z =0. (8) The A,B,C,D are as follows A= (ω 1 ω 2 )(ω 1 ω 3 )(ω 1 ω 4 ), (9a) B= (ω 1 ω 2 )(ω 2 ω 3 )(ω 3 ω 4 ), (9b) C= (ω 1 ω 3 )(ω 2 ω 3 )(ω 3 ω 4 ), (9c) D= (ω 1 ω 4 )(ω 2 ω 4 )(ω 3 ω 4 ). (9d)

4 76 D. Y. Zhan, S. Q. Tan, X. W. Wan, and L. J. Xie/J. At. Mol. Sci. 3 (2012) From C 4, we can et C 1,C 2,C 3. C 1 as follows C 1 =1+ i A(ω ) ω 1 (e ω 1 t 1)+ B(ω ) ω 2 (e ω 2t 1) + C(ω ) ω 3 (e ω 3t 1)+ D(ω ) ω 4 (e ω 4 t 1). (10) Accordin to the effective Hamiltonian (1), we solve correspondin Schrödiner equation and obtain expression of the state. We send the atom throuh the first cavity, after an interaction time t 1, the state evolves into φ(t) = C 1 (t 1 ) e 0,0,0 1 +C 2 (t 1 ) j 0,1,1 1 +C 3 (t 1 ) i 0,1,1 +1+C 4 (t 1 ) 1,1,1 1 0,0,0 2. (11) Then we let the atom pass throuh the second cavity, and after an interaction time t 2,the atom-cavity system evolves into the state φ(t 1 +t 2 ) = 1 2 C 1 (t 1 ) 0,0,0 1 C1 (t 2 ) e 0,0,0 2 +C 2 (t 2 ) j 0,0,1 2 +C 3 (t 2 ) i 0,1,1 2 +C 4 (t 2 ) 1,1,1 2 +C2 (t 1 ) 0,0,1 1 cos( 2t2 )+1 j 0,0,0 2 2isin( 2t 2 ) i 0,1,0 2 + cos( 2t 2 ) 1 1,1,0 2 +C 3 (t 1 ) 0,1,1 cos(t 2 ) i 0,0,0 2 isin(t 2 ) 1,0,0 2 +C 4 (t 1 ) 1,1,1 1 0,0,0 2. (12) By choosin C 3 (t 1 )=0, 2t 2 =2nπ (n is an inteer), we can obtain state φ(t1 +t 2 ) =C 1 (t 1 ) 0,0,0 1 C1 (t 2 ) e 0,0,0 2 +C 2 (t 2 ) j 0,0,1 2 +C 3 (t 2 ) i 0,1,1 2 +C 4 (t 2 ) 1,1,1 2 +C2 (t 1 ) 0,0,1 cos( 2t2 )+1 j 0,0,0 2 2isin( 2t 2 ) i 0,1,0 2 +C 4 (t 1 ) 1,1,1 1 0,0,0 2. (13) We can perform a measurement on the atom, if the atom is detected in the state the cavity field collapses into the followin state φ(t1 +t 2 ) C 1 (t 1 )C 4 (t 2 ) 0,0,0 1 1,1,1 2 +C 4 (t 1 ) 1,1,1 1 0,0,0 2. (14) The states φ(t 1 +t 2 ) is six-photon entanled state, when C 1 (t 1 )C 4 (t 2 )=C 4 (t 1 ), φ(t 1 + t 2 ) is six-photon GHZ state.

5 D. Y. Zhan, S. Q. Tan, X. W. Wan, and L. J. Xie/J. At. Mol. Sci. 3 (2012) Conclusions The implementation of the quantum entanled state requires the passae of a cascade type four-level atom throuh two three-mode cavity. As with any proposal for quantum computin implementation, its success ultimately depends on bein able to complete many coherent dynamics durin the decoherence time, the atomic and cavity lifetimes should bein the larer than the interaction time of the atoms with the cavity fields. The photon number in the two cavity modes should also remain unaltered durin interaction with the atom. In summary, we have proposed a simple method for eneration of six-photon entanled state via cavity QED. Comparin with previous schemes, our proposal is more simple and feasible since the detection of atom can collapse the cavity to the desired six-photon entanled state. Acknowledments. The project is supported by the National Natural Science Foundation of China under Grant No , the Key Scientific Research Fund of Hunan Provincial Education Department under Grant No. 09A013, and the Science and Technoloy Research Foundation of Hunan Province under Grant No. 2010FJ4120. References [1] J. M. Raimond, M. Brune, and S. Haroche, Rev. Mod. Phys. 73 (2001) 565. [2] S. Q. Tan, D. Y. Zhan, L. J. Xie, et al., Chinese Phys. Lett. 26 (2009) [3] C. Z. Wan and M. F. Fan, Chinese Phys. 12 (2003) 287. [4] K. H. Son and G. C. Guo, Acta Opt. Sin. 48 (1999) 661 (in Chinese). [5] S. H. Xian, J. Huaihua Univ. 23 (2004) 22. [6] A. Rauschenbeutel, P. Bertet, S. Osnahi, et al., Phys. Rev. A 64 (2001) (R). [7] A. Rauschenbeutel, G. Noues, S. Osnahi, et al., Science 288 (2000) [8] Y. D. Zhan, Principle of Quantum Information Physics (Science Press, Beijin, 2005) (in Chinese). [9] X. W. Wan, Y. G. Shan, L. X. Xia, and M. W. Lu, Phys. Lett. A 364 (2007) 7. [10] W. Dür, G. Vidal, and J. I. Cirac, Phys. Rev. A 62 (2000) [11] A. Khrennikov, Contextual Approach to Quantum Fomalism (Spriner, Berlin, 2009). [12] F. Y. Men, A. D. Zhu, K. H. Yeon, and S. C. Yu, Phys. Scr. 81 (2010) [13] S. B. Zhen, Phys. Rev. Lett. 87 (2001) [14] C. F. Wu, J. L. Chen, L. C. Kwek, and C. H. Oh, Phys. Rev. A 73 (2006) [15] J. T. Chan and M. S. Zubairy, Phys. Rev. A 77 (2008)