- Email: [email protected]
Generation correlated four-mode states in cavity QED
Optics Communications 277 (2007) 103–108 www.elsevier.com/locate/optcom
Generation correlated four-mode states in cavity QED Zhi-Rong Zhong Department of Electronic Science and Applied Physics, Fuzhou University, Fuzhou 350002, PR China Received 13 November 2006; received in revised form 4 April 2007; accepted 27 April 2007
Abstract A scheme for preparing correlated four-mode states with controllable weighting factors is presented. In the scheme, a sequence of suitably prepared four-level atoms are orderly sent through two bimodal cavities, the detection of all atoms in ground state collapses cavity fields to the desire state. The distinct advantage of our scheme is that the interaction time can be greatly shortened, which is important in view of decoherence. 2007 Elsevier B.V. All rights reserved. PACS: 42.50.DV Keywords: Correlated four-mode states; Four-level atom; Resonant interaction
One of the central topics in quantum optics is quantum state engineering. So far, a number of schemes have been presented for generating various quantum states. In cavity QED, schemes have been proposed for the generation of Schrodinger cat states of a cavity field [1–3]. On the other hand, many schemes have been proposed to prepare any Fock state superposition of an electromagnetic field. Vogel et al. [4] have firstly proposed a method for producing an arbitrary superposition of n + 1 photon number states from the vacuum state by injecting n approximately prepared atoms into a cavity and detecting all of them in the ground state. Parkins et al. [5] have proposed a scheme for generating such superposition states via adiabatic passage. Zheng [6] has shown that such states can be generated via the interaction of a multilevel atom with a single-mode cavity field. Recently, the correlated quantum states of a multi-mode field have aroused much interest and many schemes have been presented for preparing such states [7–11]. Sanders et al. [8] have presented a scheme to generate entangled coherent states with the help of a Mach–Zehnder interferometer. Davidovich et al. [9] have proposed a method for producing quantum superpositions of coherent microwave field states located simultaneously in two cavities by using two quantum switches. More recently, many people are interested in the multi-dimensional (more than two) quantum systems. The proof of Bell’s theorem without the inequalities presented by Greenberger, Horne, and Zeilinger was extended to multiparticle multi-dimensional systems [12,13]. It has been shown that quantum key distributions based on N-dimensional systems are more secure than those based on two-dimensional systems [14,15]. It also has been demonstrated that violations of local realism caused by two entangled N-dimensional (N P 3) systems are stronger than that by two-dimensional systems [16]. In the context of cavity QED, various schemes have been proposed for preparing maximally entangled states. Zheng [17] has proposed an alternative scheme for generating multi-dimensional entanglement between two or more multilevel atoms in a thermal cavity. Shu et al. [18] have presented a scheme for generating four-mode multiphoton entangled states. In this paper, we propose a scheme to generate correlated four-mode states with controllable weighting factors via fourlevel atom resonantly interacting with two bimodal cavities, which are initially prepared in the two-mode vacuum state. In E-mail address: [email protected] 0030-4018/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.04.054
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Z.-R. Zhong / Optics Communications 277 (2007) 103–108
the scheme, atoms which are produced in zone B by the excitation of a velocity selected atomic beam effusing from oven O, pass through the first cavity. Before the atoms enter the second cavity, they are sent through two classical fields F1 and F2 in turn. As is described in Fig. 1. The distinct advantage of our scheme is that the interaction time can be greatly shortened, which is important in view of decoherence. As the application of this scheme, we show how to generate four-mode multiphoton maximally entangled states. Suppose the atom has four states denoted by jei, jgi, jii, and jg 0 i, with the energy-level xi, xg, xe and xg0 , respectively. Let us consider the resonant interaction of a four-level atom with a two-mode field, as sketched in Fig. 2. The transition between jei and jgi is coupled to cavity mode 1, while the transition between jgi and jii is coupled to cavity mode 2. The transition between jei (jgi, jii) and jg 0 i is highly detuned from both the cavity modes, respectively, thus the state jg 0 i is a auxiliary level which performs the transformation jgi to jg 0 i so that the interaction of the atom with the second cavity is frozen if the atom is in the state jgi after exit the first cavity. The Hamiltonian for such a system is written as ( h = 1) [19] þ þ þ H ¼ xe jeihej þ xg jgihgj þ xi jiihij þ x1 aþ 1 a1 þ x2 a2 a2 þ g1 ða1 jgihej þ a1 jeihgjÞ þ g2 ða2 jiihgj þ a2 jgihijÞ
ð1Þ
aþ i
where and ai (i = 1, 2) are the creation and annihilation operators for the cavity fields, g1 (g2) is the coupling constant of the transition between jei (jii) and jgi with the cavity field, x1, x2 are the frequencies of two-mode cavity field. In the interaction picture, the interaction Hamiltonian is given by þ H I ¼ kðaþ 1 jgihej þ a2 jiihgjÞ þ H :c;
ð2Þ
we here assume that g1 = g2 = k. The basis states for the system are of the form jn1 ; n2 ; n3 ; n4 ; si ¼ jn1 ijn2 ijn3 ijn4 ijsi;
ð3Þ
where ni (i = 1, 2, 3, 4) refers to the number of excitations in modes of two bimodal cavities and s refers to the state of fourlevel atom.
C2
C1
B
O
F1
F2
D
Fig. 1. The displays of the set-up for engineering correlated four-mode states. The atom is produced in zone B by the excitation of a velocity selected atomic beam effusing from oven O, F1 and F2 are two classical fields tuned to the transitions jei M jii, and jgi M jg 0 i, respectively, C1, C2 are the two-mode cavities, D is the detector for the state jii.
Fig. 2. Schematic diagram of energy level of four-level atom with corresponding transition (jei, jgi, jii and jg 0 i denote the states of the atom, g1, g2 are the coupling coefficients).
Z.-R. Zhong / Optics Communications 277 (2007) 103–108
105
We assume that the first cavity field is initially in two-mode vacuum state j0, 0i. Now we send the first atom initially in the state jW1a i ¼ jei;
ð4Þ
through the cavity. After an interaction time t1 the atom-cavity system evolves into the state pffiffiffi pffiffiffi pffiffiffi pffiffiffi 1 i 2 cos 2kt1 þ 1 je; 0; 0i þ cos 2kt1 1 ji; 1; 1i sin 2kt1 jg; 1; 0i: jW1 i ¼ 2 2 2
ð5Þ
After the atom exits from the cavity, it traverses two classical fields in turn, which leads to the transition, respectively. jei ! jii;
jii ! jei;
0
jgi ! jg i:
ð6Þ ð7Þ
These lead to pffiffiffi pffiffiffi pffiffiffi pffiffiffi 1 i 2 cos 2kt1 þ 1 ji; 0; 0i cos 2kt1 1 je; 1; 1i sin 2kt1 jg0 ; 1; 0i: ð8Þ jW1 i ¼ 2 2 2 Then we let the atom pass through the second cavity which is also initially in two-mode vacuum state, after the same interaction time t1, the atom-cavity system evolves into " pffiffiffi 1 pffiffiffi pffiffiffi pffiffiffi 1 2 cos 2kt1 þ 1 ji; 0; 0; 0; 0i cos 2kt1 þ 1 je; 0; 0; 1; 1i cos 2kt1 1 jW1 i ¼ 2 2 2 # pffiffiffi pffiffiffi pffiffiffi pffiffiffi i i 2 cos 2kt1 1 ji; 1; 1; 1; 1i sin 2kt1 jg; 1; 0; 1; 1i sin 2kt1 jg0 ; 1; 0; 0; 0i: ð9Þ þ 2 2 2 Now we perform a measurement on the atom. If this atom is detected in the state jii, the cavity field collapses onto the state 2 pffiffiffi pffiffiffi 1 1 1 cos 2kt1 þ 1 j0; 0; 0; 0i cos 2kt1 1 j1; 1; 1; 1i; jWf i ¼ N 1 ð10Þ 2 2 where N1 is a normalization factor given by ( ) 2 1 2 2 1=2 pffiffiffi pffiffiffi 1 cos 2kt1 þ 1 cos 2kt1 1 þ : N1 ¼ 2 2
ð11Þ
Now, we send the second atom in the state jW2a i ¼ jei;
ð12Þ
through the first cavity, after interaction time t2, the atom-cavity field system evolves into the state (
" pffiffiffi 1 pffiffiffi pffiffiffi pffiffiffi 2 1 jW2 ðtÞi ¼ N 1 cos 2kt2 1 ji; 1; 1i cos 2kt1 þ 1 cos 2kt2 þ 1 je; 0; 0i þ 2 2 2 # " 2 1 pffiffiffi pffiffiffi pffiffiffi i 1 2 cos 4kt2 þ 2 je; 1; 1i sin 2kt2 jg; 1; 0i j0; 0i cos 2kt1 1 2 2 4 # ) p ffiffi ffi pffiffiffi pffiffiffi pffiffiffi 6 2 cos 4kt2 1 ji; 2; 2i sin 4kt2 jg; 2; 1i j1; 1i : þ 4 4i
ð13Þ
Then we let the atom pass through two classical fields in turn, where they undergo the transition, respectively. jei ! jii; jii ! jei; jgi ! jg0 i;
ð14Þ ð15Þ
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Z.-R. Zhong / Optics Communications 277 (2007) 103–108
the cavity field evolves into the state ( " pffiffiffi 1 pffiffiffi pffiffiffi pffiffiffi 2 1 pffiffiffi i cos 2kt2 1 je; 1;1i sin 2kt2 jg0 ;1;0i j0;0i cos 2kt1 þ 1 cos 2kt1 þ 1 ji;0;0i jW2 ðtÞi ¼ N 1 2 2 2 2 # ) p ffiffiffi p ffiffi ffi 2 1 pffiffiffi pffiffiffi 1 pffiffiffi 6 pffiffiffi 2 2 cos 4kt2 þ 2 ji;1; 1i cos 4kt2 1 je; 2;2i sin 4kt2 jg0 ;2;1i j1;1i : þ cos 2kt1 1 2 4 4 4i ð16Þ Then we let the atom pass through the second cavity, after the same interaction time t2 the atom-cavity system evolves into the state ( ( " pffiffiffi 1 1 pffiffiffi pffiffiffi pffiffiffi pffiffiffi 1 2 cos 2kt1 þ 1 cos 2kt2 þ 1 ji; 0; 0; 0; 0i cos 2kt2 þ 1 je; 0; 0i cos 2kt2 1 jW2 ðtÞi ¼ N 1 2 2 2 2 # ) pffiffiffi pffiffiffi pffiffiffi pffiffiffi i i 2 0 cos 2kt2 1 ji; 1; 1i pffiffiffi sin 2kt2 jg; 1; 0i pffiffiffi sin 2kt2 jg ; 1; 0i þ 2 2 2 ( 2 1 pffiffiffi pffiffiffi pffiffiffi pffiffiffi 1 1 1 þ cos 2kt1 1 2 cos 4kt2 þ 2 j1; 1i: cos 2kt2 þ 1 ji; 1; 1i þ cos 2kt2 1 je; 0; 0i 2 4 2 2 " pffiffiffi pffiffiffi pffiffiffi pffiffiffi 6 i 1 pffiffiffi sin 2kt2 jg; 1; 0i cos 4kt2 1 j2; 2i 2 cos 4kt2 þ 2 je; 1; 1i 4 4 2 # pffiffiffi )) pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi i 2 6 2 sin 4kt2 jg; 2; 1i cos 4kt2 1 ji; 2; 2i i sin 4kt2 jg0 ; 2; 1; 1; 1i : ð17Þ þ 4 4 4 If the atom is detected in the state jii, the cavity field collapses onto the state 2 pffiffiffi pffiffiffi pffiffiffi pffiffiffi 1 1 cos 2kt1 þ 1 cos 2kt2 þ 1 j0; 0; 0; 0i þ cos 2kt2 1 jW2f i ¼ N 2 cos 2kt1 þ 1 4 4 2 pffiffiffi pffiffiffi pffiffiffi 1 þ cos 4kt2 þ 1 cos 2kt2 þ 1 cos 2kt1 1 j1; 1; 1; 1i 8 2 2 pffiffiffi pffiffiffi 3 cos 4kt2 1 cos 2kt1 1 j2; 2; 2; 2i ; þ 16 where N2 is given by ( 2 1 2 pffiffiffi pffiffiffi pffiffiffi pffiffiffi 1 cos 2kt1 þ 1 cos 2kt2 þ 1 þ cos 2kt2 1 N2 ¼ cos 2kt1 þ 1 4 4 ) 2 2 3 2 2 2 pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 1 þ cos 4kt2 þ 1 cos 2kt2 þ 1 cos 2kt1 1 cos 4kt2 1 þ cos 2kt1 1 : 8 16
ð18Þ
ð19Þ
Suppose we repeat the procedure N times, and each time we detect the atom in the same state jii. In this way, we can convert an initial two-mode vacuum state of the cavity field into correlated four-mode states. After the detection of the (N 1)th atom in the state jii, the cavity field is in the state jWfN 1 i ¼ N N 1
N 1 X
C KN 1 jK; K; K; Ki:
ð20Þ
K¼0
Suppose the Nth atom is initially prepared in the state jWNa i ¼ jei;
ð21Þ
After it interacts with the field for an interaction time tn and is detected in the state jii, the field state reads jWNf i ¼ N N
N X
C NK jK; K; K; Ki;
K¼0
where NN is a normalization factor given by
ð22Þ
Z.-R. Zhong / Optics Communications 277 (2007) 103–108
" NN ¼
N X N X
107
#1=2 C NJ ðC NK Þ
;
ð23Þ
J ¼0 K¼0
where pffiffiffi 1 cos 2ktN þ 1 C 0N 1 ; ðC 00 ¼ 1Þ; 2 2 pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi K þ 1 1 N N 1 cos 2K ktN 1 C K1 CK ¼ cos 2K ktN þ 1 cos 2kt0N þ 1 C KN 1 ; 4K 4 pffiffiffiffiffiffi 0 pffiffiffiffiffiffiffi N þ 1 N 1 cos 2N ktN 1 cos 2K ktN 1 C NN 1 CN ¼ : N
C N0 ¼
ð24Þ ð25Þ ð26Þ
In order to generate the state jWNf i with the desired coefficients C NK , we have to obtain the coefficients C NK 1 for the state jWfN 1 i firstly. According to Eqs. (24)–(26), we can express the unknown coefficients C KN 1 in terms of C NK by pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi
2
4KðK þ 1Þ cos 2K ktN 1 C NK1 4K 2 cos 2K ktN þ 1 cos 2ktN þ 1 C NK C KN 1 ¼ : ð27Þ pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi
2 h
2 i 2 K cos 2K ktN þ 1 cos 2ktN þ 1 þ ðK þ 1Þ cos 2K ktN 1 We take jWfN 1 i as a new desired state which can be obtained by sending N 1 atoms through the cavity. For the state jWfN 1 i we do the same calculations and obtain N 1 coefficients C KN 2 for the state jWfN 2 i. We repeat the process until the cavity field is in two-mode vacuum state, we get a series of parameters t1, t2, . . ., tN which determine the interaction time as to obtain the desired state jWNf i. The probability of finding all N atoms in the jii state is given by P¼
N Y
Pn ¼
n¼1
1 ðN n Þ2
;
ð28Þ
where Pn is the probability to find the nth atom in the jii state. As an example, we show how to four-mode multiphoton maximally entangled states. According to Eq. (18), if pffiffigenerate ffi p ffiffiffi the interaction time is chosen as 2kt1 ¼ p5 ; 2kt2 ¼ p2, the state reads 1 jWi ¼ pffiffiffi ½j0; 0; 0; 0i þ j1; 1; 1; 1i þ j2; 2; 2; 2i; 3
ð29Þ
here the state jWi is the four-mode multiphoton maximally entangled states. The success probability to prepare this state is 0.33. It is necessary to give a brief discussion on the experimental realization of the proposed scheme. To generate four-mode multiphoton maximally entangled states, we may consider a four-level atom with states jei, jgi, jii, and jg0 i, of which the radiative lifetimes are of the order of Tat = 30 ms. In the experiment reported in Ref. [20], the photon lifetime in the cavity Q is T c ¼ 2pv 130 ms, thus the lifetime for the two-photon state is T 2p ¼ T2c ¼ 75 ms. By using a velocity selector and applying a Stark field adjustment, we can make the atom resonant with the cavity field for the right amount of time. Setting k = 2p · 25 kHz, we have t1 ¼ 5ppffiffi2k ¼ 3 ls and t2 ¼ 2ppffiffi2k ¼ 7 ls in our scheme. Considering the travelling time of the atom [21], we obtain that the total time required to complete the process is about 100 ls, which is much shorter than Tat and T2P. Therefore, there is sufficient time to complete the process to generate four-mode multiphoton maximally entangled states. In summary, we have proposed a scheme for generating four-mode correlated states. It is based on the injection of a sequence of four-level atoms into two bimodal cavities one by one and the detection of them are in the ground state. The distinct advantage of our scheme is that the interaction time can be greatly shortened, which is important in view of decoherence. Furthermore, we show how to generate four-mode multiphoton maximally entangled states. Acknowledgements This work supported by National Natural Science Foundation of China Under Grant No. 10674025, and Funds from Key Lab of Quantum Information, University of Science and Technology of China, and FuJian Department of Education under Grant No. JB06043. References [1] M. Brune, S. Haroche, J.M. Raimond, L. Davidovich, N. Zagury, Phys. Rev. A 45 (1992) 5193. [2] B.M. Garraway, B. Sherman, H. Moya-Cessa, P.L. Knight, G. Kurizki, Phys. Rev. A 49 (1994) 535. [3] C.C. Gerry, Phys. Rev. A 53 (1996) 3818.
108 [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
Z.-R. Zhong / Optics Communications 277 (2007) 103–108
K. Vogel, V.M. Akulin, W.P. Schleich, Phys. Rev. Lett. 71 (1993) 1816. A.S. Parkins, A. Marte, P. Zoller, O. Carnal, H.J. Kimble, Phys. Rev. A 51 (1995) 1578. S.B. Zheng, Opt. Commum. 154 (1998) 290. B. Deb, G. Gamgopadhyay, D.S. Ray, Phys. Rev. A 51 (1995) 2651. B.C. Sanders, G.J. Milburn, Phys. Rev. A 39 (1989) 694. L. Davidovich, A. Maali, M. Brune, J.M. Raimond, S. Haroche, Phys. Rev. Lett. 71 (1993) 2360. E. Solano, G.S. Agarwal, H. Walther, Phys. Rev. L 90 (2003) 027903. C.C. Gerry, Phys. Rev. A 54 (1996) R2529. A. Cabello, Phys. Rev. A 63 (2001) 022104. N.J. Cerf, S. Massar, S. Pironio, Phys. Rev. Lett. 89 (2002) 080402. A. Karlsson, G. Bjork, Phys. Rev. A 64 (2001) 012306. D. Kaszlikowski1, P. Gnaciski1, M. Zukowski, W. Miklaszewski, A. Zeilinger, Phys. Rev. Lett. 85 (2000) 4418. N.J. Cerf, M. Bourennane, A. Karlsson, N. Gisin, Phys. Rev. Lett. 88 (2002) 127902. S.B. Zheng, Phys. Rev. A 68 (2003) 035801. J. Shu, X.B. Zou, Y.F. Xiao, G.C. Guo, Phys. Rev. A 74 (2006) 044301. Y. Wu, X.X. Yang, Phys. Rev. A 56 (1997) 2443; Y. Wu, Phys. Rev. A 54 (1996) 1586. [20] S. Kuhr1,. S. Gleyzes, C. Guerlin, J. Bernu, et al, arxiv:quant-ph/0612138 v1. [21] A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J.M. Raimond, S. Haroche, Science 288 (2000) 2024.
Generation correlated four-mode states in cavity QED Zhi-Rong Zhong Department of Electronic Science and Applied Physics, Fuzhou University, Fuzhou 350002, PR China Received 13 November 2006; received in revised form 4 April 2007; accepted 27 April 2007
Abstract A scheme for preparing correlated four-mode states with controllable weighting factors is presented. In the scheme, a sequence of suitably prepared four-level atoms are orderly sent through two bimodal cavities, the detection of all atoms in ground state collapses cavity fields to the desire state. The distinct advantage of our scheme is that the interaction time can be greatly shortened, which is important in view of decoherence. 2007 Elsevier B.V. All rights reserved. PACS: 42.50.DV Keywords: Correlated four-mode states; Four-level atom; Resonant interaction
One of the central topics in quantum optics is quantum state engineering. So far, a number of schemes have been presented for generating various quantum states. In cavity QED, schemes have been proposed for the generation of Schrodinger cat states of a cavity field [1–3]. On the other hand, many schemes have been proposed to prepare any Fock state superposition of an electromagnetic field. Vogel et al. [4] have firstly proposed a method for producing an arbitrary superposition of n + 1 photon number states from the vacuum state by injecting n approximately prepared atoms into a cavity and detecting all of them in the ground state. Parkins et al. [5] have proposed a scheme for generating such superposition states via adiabatic passage. Zheng [6] has shown that such states can be generated via the interaction of a multilevel atom with a single-mode cavity field. Recently, the correlated quantum states of a multi-mode field have aroused much interest and many schemes have been presented for preparing such states [7–11]. Sanders et al. [8] have presented a scheme to generate entangled coherent states with the help of a Mach–Zehnder interferometer. Davidovich et al. [9] have proposed a method for producing quantum superpositions of coherent microwave field states located simultaneously in two cavities by using two quantum switches. More recently, many people are interested in the multi-dimensional (more than two) quantum systems. The proof of Bell’s theorem without the inequalities presented by Greenberger, Horne, and Zeilinger was extended to multiparticle multi-dimensional systems [12,13]. It has been shown that quantum key distributions based on N-dimensional systems are more secure than those based on two-dimensional systems [14,15]. It also has been demonstrated that violations of local realism caused by two entangled N-dimensional (N P 3) systems are stronger than that by two-dimensional systems [16]. In the context of cavity QED, various schemes have been proposed for preparing maximally entangled states. Zheng [17] has proposed an alternative scheme for generating multi-dimensional entanglement between two or more multilevel atoms in a thermal cavity. Shu et al. [18] have presented a scheme for generating four-mode multiphoton entangled states. In this paper, we propose a scheme to generate correlated four-mode states with controllable weighting factors via fourlevel atom resonantly interacting with two bimodal cavities, which are initially prepared in the two-mode vacuum state. In E-mail address: [email protected] 0030-4018/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.04.054
104
Z.-R. Zhong / Optics Communications 277 (2007) 103–108
the scheme, atoms which are produced in zone B by the excitation of a velocity selected atomic beam effusing from oven O, pass through the first cavity. Before the atoms enter the second cavity, they are sent through two classical fields F1 and F2 in turn. As is described in Fig. 1. The distinct advantage of our scheme is that the interaction time can be greatly shortened, which is important in view of decoherence. As the application of this scheme, we show how to generate four-mode multiphoton maximally entangled states. Suppose the atom has four states denoted by jei, jgi, jii, and jg 0 i, with the energy-level xi, xg, xe and xg0 , respectively. Let us consider the resonant interaction of a four-level atom with a two-mode field, as sketched in Fig. 2. The transition between jei and jgi is coupled to cavity mode 1, while the transition between jgi and jii is coupled to cavity mode 2. The transition between jei (jgi, jii) and jg 0 i is highly detuned from both the cavity modes, respectively, thus the state jg 0 i is a auxiliary level which performs the transformation jgi to jg 0 i so that the interaction of the atom with the second cavity is frozen if the atom is in the state jgi after exit the first cavity. The Hamiltonian for such a system is written as ( h = 1) [19] þ þ þ H ¼ xe jeihej þ xg jgihgj þ xi jiihij þ x1 aþ 1 a1 þ x2 a2 a2 þ g1 ða1 jgihej þ a1 jeihgjÞ þ g2 ða2 jiihgj þ a2 jgihijÞ
ð1Þ
aþ i
where and ai (i = 1, 2) are the creation and annihilation operators for the cavity fields, g1 (g2) is the coupling constant of the transition between jei (jii) and jgi with the cavity field, x1, x2 are the frequencies of two-mode cavity field. In the interaction picture, the interaction Hamiltonian is given by þ H I ¼ kðaþ 1 jgihej þ a2 jiihgjÞ þ H :c;
ð2Þ
we here assume that g1 = g2 = k. The basis states for the system are of the form jn1 ; n2 ; n3 ; n4 ; si ¼ jn1 ijn2 ijn3 ijn4 ijsi;
ð3Þ
where ni (i = 1, 2, 3, 4) refers to the number of excitations in modes of two bimodal cavities and s refers to the state of fourlevel atom.
C2
C1
B
O
F1
F2
D
Fig. 1. The displays of the set-up for engineering correlated four-mode states. The atom is produced in zone B by the excitation of a velocity selected atomic beam effusing from oven O, F1 and F2 are two classical fields tuned to the transitions jei M jii, and jgi M jg 0 i, respectively, C1, C2 are the two-mode cavities, D is the detector for the state jii.
Fig. 2. Schematic diagram of energy level of four-level atom with corresponding transition (jei, jgi, jii and jg 0 i denote the states of the atom, g1, g2 are the coupling coefficients).
Z.-R. Zhong / Optics Communications 277 (2007) 103–108
105
We assume that the first cavity field is initially in two-mode vacuum state j0, 0i. Now we send the first atom initially in the state jW1a i ¼ jei;
ð4Þ
through the cavity. After an interaction time t1 the atom-cavity system evolves into the state pffiffiffi pffiffiffi pffiffiffi pffiffiffi 1 i 2 cos 2kt1 þ 1 je; 0; 0i þ cos 2kt1 1 ji; 1; 1i sin 2kt1 jg; 1; 0i: jW1 i ¼ 2 2 2
ð5Þ
After the atom exits from the cavity, it traverses two classical fields in turn, which leads to the transition, respectively. jei ! jii;
jii ! jei;
0
jgi ! jg i:
ð6Þ ð7Þ
These lead to pffiffiffi pffiffiffi pffiffiffi pffiffiffi 1 i 2 cos 2kt1 þ 1 ji; 0; 0i cos 2kt1 1 je; 1; 1i sin 2kt1 jg0 ; 1; 0i: ð8Þ jW1 i ¼ 2 2 2 Then we let the atom pass through the second cavity which is also initially in two-mode vacuum state, after the same interaction time t1, the atom-cavity system evolves into " pffiffiffi 1 pffiffiffi pffiffiffi pffiffiffi 1 2 cos 2kt1 þ 1 ji; 0; 0; 0; 0i cos 2kt1 þ 1 je; 0; 0; 1; 1i cos 2kt1 1 jW1 i ¼ 2 2 2 # pffiffiffi pffiffiffi pffiffiffi pffiffiffi i i 2 cos 2kt1 1 ji; 1; 1; 1; 1i sin 2kt1 jg; 1; 0; 1; 1i sin 2kt1 jg0 ; 1; 0; 0; 0i: ð9Þ þ 2 2 2 Now we perform a measurement on the atom. If this atom is detected in the state jii, the cavity field collapses onto the state 2 pffiffiffi pffiffiffi 1 1 1 cos 2kt1 þ 1 j0; 0; 0; 0i cos 2kt1 1 j1; 1; 1; 1i; jWf i ¼ N 1 ð10Þ 2 2 where N1 is a normalization factor given by ( ) 2 1 2 2 1=2 pffiffiffi pffiffiffi 1 cos 2kt1 þ 1 cos 2kt1 1 þ : N1 ¼ 2 2
ð11Þ
Now, we send the second atom in the state jW2a i ¼ jei;
ð12Þ
through the first cavity, after interaction time t2, the atom-cavity field system evolves into the state (
" pffiffiffi 1 pffiffiffi pffiffiffi pffiffiffi 2 1 jW2 ðtÞi ¼ N 1 cos 2kt2 1 ji; 1; 1i cos 2kt1 þ 1 cos 2kt2 þ 1 je; 0; 0i þ 2 2 2 # " 2 1 pffiffiffi pffiffiffi pffiffiffi i 1 2 cos 4kt2 þ 2 je; 1; 1i sin 2kt2 jg; 1; 0i j0; 0i cos 2kt1 1 2 2 4 # ) p ffiffi ffi pffiffiffi pffiffiffi pffiffiffi 6 2 cos 4kt2 1 ji; 2; 2i sin 4kt2 jg; 2; 1i j1; 1i : þ 4 4i
ð13Þ
Then we let the atom pass through two classical fields in turn, where they undergo the transition, respectively. jei ! jii; jii ! jei; jgi ! jg0 i;
ð14Þ ð15Þ
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Z.-R. Zhong / Optics Communications 277 (2007) 103–108
the cavity field evolves into the state ( " pffiffiffi 1 pffiffiffi pffiffiffi pffiffiffi 2 1 pffiffiffi i cos 2kt2 1 je; 1;1i sin 2kt2 jg0 ;1;0i j0;0i cos 2kt1 þ 1 cos 2kt1 þ 1 ji;0;0i jW2 ðtÞi ¼ N 1 2 2 2 2 # ) p ffiffiffi p ffiffi ffi 2 1 pffiffiffi pffiffiffi 1 pffiffiffi 6 pffiffiffi 2 2 cos 4kt2 þ 2 ji;1; 1i cos 4kt2 1 je; 2;2i sin 4kt2 jg0 ;2;1i j1;1i : þ cos 2kt1 1 2 4 4 4i ð16Þ Then we let the atom pass through the second cavity, after the same interaction time t2 the atom-cavity system evolves into the state ( ( " pffiffiffi 1 1 pffiffiffi pffiffiffi pffiffiffi pffiffiffi 1 2 cos 2kt1 þ 1 cos 2kt2 þ 1 ji; 0; 0; 0; 0i cos 2kt2 þ 1 je; 0; 0i cos 2kt2 1 jW2 ðtÞi ¼ N 1 2 2 2 2 # ) pffiffiffi pffiffiffi pffiffiffi pffiffiffi i i 2 0 cos 2kt2 1 ji; 1; 1i pffiffiffi sin 2kt2 jg; 1; 0i pffiffiffi sin 2kt2 jg ; 1; 0i þ 2 2 2 ( 2 1 pffiffiffi pffiffiffi pffiffiffi pffiffiffi 1 1 1 þ cos 2kt1 1 2 cos 4kt2 þ 2 j1; 1i: cos 2kt2 þ 1 ji; 1; 1i þ cos 2kt2 1 je; 0; 0i 2 4 2 2 " pffiffiffi pffiffiffi pffiffiffi pffiffiffi 6 i 1 pffiffiffi sin 2kt2 jg; 1; 0i cos 4kt2 1 j2; 2i 2 cos 4kt2 þ 2 je; 1; 1i 4 4 2 # pffiffiffi )) pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi i 2 6 2 sin 4kt2 jg; 2; 1i cos 4kt2 1 ji; 2; 2i i sin 4kt2 jg0 ; 2; 1; 1; 1i : ð17Þ þ 4 4 4 If the atom is detected in the state jii, the cavity field collapses onto the state 2 pffiffiffi pffiffiffi pffiffiffi pffiffiffi 1 1 cos 2kt1 þ 1 cos 2kt2 þ 1 j0; 0; 0; 0i þ cos 2kt2 1 jW2f i ¼ N 2 cos 2kt1 þ 1 4 4 2 pffiffiffi pffiffiffi pffiffiffi 1 þ cos 4kt2 þ 1 cos 2kt2 þ 1 cos 2kt1 1 j1; 1; 1; 1i 8 2 2 pffiffiffi pffiffiffi 3 cos 4kt2 1 cos 2kt1 1 j2; 2; 2; 2i ; þ 16 where N2 is given by ( 2 1 2 pffiffiffi pffiffiffi pffiffiffi pffiffiffi 1 cos 2kt1 þ 1 cos 2kt2 þ 1 þ cos 2kt2 1 N2 ¼ cos 2kt1 þ 1 4 4 ) 2 2 3 2 2 2 pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 1 þ cos 4kt2 þ 1 cos 2kt2 þ 1 cos 2kt1 1 cos 4kt2 1 þ cos 2kt1 1 : 8 16
ð18Þ
ð19Þ
Suppose we repeat the procedure N times, and each time we detect the atom in the same state jii. In this way, we can convert an initial two-mode vacuum state of the cavity field into correlated four-mode states. After the detection of the (N 1)th atom in the state jii, the cavity field is in the state jWfN 1 i ¼ N N 1
N 1 X
C KN 1 jK; K; K; Ki:
ð20Þ
K¼0
Suppose the Nth atom is initially prepared in the state jWNa i ¼ jei;
ð21Þ
After it interacts with the field for an interaction time tn and is detected in the state jii, the field state reads jWNf i ¼ N N
N X
C NK jK; K; K; Ki;
K¼0
where NN is a normalization factor given by
ð22Þ
Z.-R. Zhong / Optics Communications 277 (2007) 103–108
" NN ¼
N X N X
107
#1=2 C NJ ðC NK Þ
;
ð23Þ
J ¼0 K¼0
where pffiffiffi 1 cos 2ktN þ 1 C 0N 1 ; ðC 00 ¼ 1Þ; 2 2 pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi K þ 1 1 N N 1 cos 2K ktN 1 C K1 CK ¼ cos 2K ktN þ 1 cos 2kt0N þ 1 C KN 1 ; 4K 4 pffiffiffiffiffiffi 0 pffiffiffiffiffiffiffi N þ 1 N 1 cos 2N ktN 1 cos 2K ktN 1 C NN 1 CN ¼ : N
C N0 ¼
ð24Þ ð25Þ ð26Þ
In order to generate the state jWNf i with the desired coefficients C NK , we have to obtain the coefficients C NK 1 for the state jWfN 1 i firstly. According to Eqs. (24)–(26), we can express the unknown coefficients C KN 1 in terms of C NK by pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi
2
4KðK þ 1Þ cos 2K ktN 1 C NK1 4K 2 cos 2K ktN þ 1 cos 2ktN þ 1 C NK C KN 1 ¼ : ð27Þ pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi
2 h
2 i 2 K cos 2K ktN þ 1 cos 2ktN þ 1 þ ðK þ 1Þ cos 2K ktN 1 We take jWfN 1 i as a new desired state which can be obtained by sending N 1 atoms through the cavity. For the state jWfN 1 i we do the same calculations and obtain N 1 coefficients C KN 2 for the state jWfN 2 i. We repeat the process until the cavity field is in two-mode vacuum state, we get a series of parameters t1, t2, . . ., tN which determine the interaction time as to obtain the desired state jWNf i. The probability of finding all N atoms in the jii state is given by P¼
N Y
Pn ¼
n¼1
1 ðN n Þ2
;
ð28Þ
where Pn is the probability to find the nth atom in the jii state. As an example, we show how to four-mode multiphoton maximally entangled states. According to Eq. (18), if pffiffigenerate ffi p ffiffiffi the interaction time is chosen as 2kt1 ¼ p5 ; 2kt2 ¼ p2, the state reads 1 jWi ¼ pffiffiffi ½j0; 0; 0; 0i þ j1; 1; 1; 1i þ j2; 2; 2; 2i; 3
ð29Þ
here the state jWi is the four-mode multiphoton maximally entangled states. The success probability to prepare this state is 0.33. It is necessary to give a brief discussion on the experimental realization of the proposed scheme. To generate four-mode multiphoton maximally entangled states, we may consider a four-level atom with states jei, jgi, jii, and jg0 i, of which the radiative lifetimes are of the order of Tat = 30 ms. In the experiment reported in Ref. [20], the photon lifetime in the cavity Q is T c ¼ 2pv 130 ms, thus the lifetime for the two-photon state is T 2p ¼ T2c ¼ 75 ms. By using a velocity selector and applying a Stark field adjustment, we can make the atom resonant with the cavity field for the right amount of time. Setting k = 2p · 25 kHz, we have t1 ¼ 5ppffiffi2k ¼ 3 ls and t2 ¼ 2ppffiffi2k ¼ 7 ls in our scheme. Considering the travelling time of the atom [21], we obtain that the total time required to complete the process is about 100 ls, which is much shorter than Tat and T2P. Therefore, there is sufficient time to complete the process to generate four-mode multiphoton maximally entangled states. In summary, we have proposed a scheme for generating four-mode correlated states. It is based on the injection of a sequence of four-level atoms into two bimodal cavities one by one and the detection of them are in the ground state. The distinct advantage of our scheme is that the interaction time can be greatly shortened, which is important in view of decoherence. Furthermore, we show how to generate four-mode multiphoton maximally entangled states. Acknowledgements This work supported by National Natural Science Foundation of China Under Grant No. 10674025, and Funds from Key Lab of Quantum Information, University of Science and Technology of China, and FuJian Department of Education under Grant No. JB06043. References [1] M. Brune, S. Haroche, J.M. Raimond, L. Davidovich, N. Zagury, Phys. Rev. A 45 (1992) 5193. [2] B.M. Garraway, B. Sherman, H. Moya-Cessa, P.L. Knight, G. Kurizki, Phys. Rev. A 49 (1994) 535. [3] C.C. Gerry, Phys. Rev. A 53 (1996) 3818.
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