Physics Letters A. Effect of spontaneously generated coherence on Kerr nonlinearity in a four-level atomic system

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1 Physics Letters A 372 (2008) Contents lists available at ScienceDirect Physics Letters A Effect of spontaneously generated coherence on Kerr nonlinearity in a four-level atomic system Xiang-an Yan a,b, Li-qiang Wang c, Bao-yin Yin a, Wen-juan Jiang a, Huai-bin Zheng a, Jian-ping Song a,, Yan-peng Zhang a a Key Laboratory for Physical Electronics and Devices of the Ministry of Education, Xi an Jiaotong University, Xi an , China b School of Science, Xi an Polytechnic University, Xi an , China c Department of Applied Mathematics and Applied Physics, Xi an Post and Telecommunications College, Xi an , China article info abstract Article history: Received 16 June 2008 Received in revised form 26 August 2008 Accepted 27 August 2008 Available online 2 September 2008 Communicated by P.R. Holland PACS: Gy k Md We propose a scheme for giant enhancement of the Kerr nonlinearity in a four-level atomic system in which spontaneously generated coherence is present. The physics mechanism of the enhancement of Kerr nonlinearity is mainly based on the presence of an extra atomic coherence induced by the spontaneously generated coherence. Numerical values obtained by solving the density matrix equations agree well with these exact analytical values Elsevier B.V. All rights reserved. Keywords: Nonlinear optics Electromagnetically induced transparency Quantum interference Spontaneously generated coherence Double dark resonance Kerr nonlinearity 1. Introduction Quantum coherence and interference in an atomic system have shown a number of important phenomena, such as lasing without inversion (LWI) [1 4], cancellation of spontaneous emission [5], subluminal and superluminal light [6 9], the enhancement of the refractive index [10 12], coherent population trapping (CPT) [13], the electromagnetically induced transparency (EIT) [14 17], and so on. There are many ways to generate quantum coherence and interference in an atomic system. Generally, they are produced by a coherent driving field or by initial coherence. It is now well known that another kind of coherence, spontaneously generated coherence (SGC), can be created by the interference of spontaneous emission channels. Such coherence can be created when a single excited state decays to a closely spaced lower doublet ( -type atom) [18,19] or a closely spaced excited doublet decays to a single ground state (V-type atom) [20,21], and it can also be created in * Corresponding author. addresses: jpsong@mail.xju.edu.cn, yxa22357@yahoo.com.cn (J.-p. Song). the equispaced-atomic-level case (ladder-type atom) [22,23]. The existence of such coherence effects depends on the nonorthogonality of the two dipole matrix elements. The study of SGC has attracted considerable attention [24 38]. At present the effects of SGC on gain, dispersion and populations in an atomic system have been intensively investigated [19,25,39,40]. All the interesting features due to SGC could have useful applications in laser physics and other areas of quantum optics. Little work, however, has been reported on the effect of SGC on the Kerr nonlinearity. The Kerr nonlinearity corresponds to the refractive part of the third-order susceptibility in optical media. Since optical nonlinear susceptibilities play important roles in alloptical switch, all-optical logic operation, polarization phase gates, and so on, it is desirable to have large nonlinear susceptibilities with small absorption under conditions of low light powers. In a four-level double dark resonance system without SGC, Niu et al. proposed that the interacting double-dark resonance can be effective in enhancing the Kerr nonlinearity by proper tuning of the coherent control field [41]. The generic four-level -type system consists of a single excited state decaying to three closely spaced lower levels, which can be treated as two -type system, the com /$ see front matter 2008 Elsevier B.V. All rights reserved. doi: /j.physleta

2 X.-a. Yan et al. / Physics Letters A 372 (2008) ρ 34 = iω c ρ 24 + iω d (ρ 44 ρ 33 ). (1) Fig. 1. Generic four-level -type system with a twofold level structure. mon excited state is coupled by the same vacuum modes to two lower closely spaced levels leading to spontaneously generated coherence effect. So, in this Letter, we explore the main effect of SGC on the Kerr nonlinearity in the same four-level atomic system with double dark resonances. We find that the enhancement of Kerr nonlinearity can be obtained even without the condition of proper detuning of the coherent control field. Moreover, in the case of optimal SGC, a giant Kerr nonlinearity accompanied by vanishing absorption can be realized near zero probe detuning. We also present the exact analytical explain, which is agreed well with the numerical results obtained by solving the density matrix equations. The Letter is organized as follows. In Section 2 we describe the model and present the density-matrix equations of motion for the system. In Section 3 we discuss the effects of SGC on the Kerr nonlinearity and nonlinear absorption. In Section 4 we develop an analysis which explains the numerical results of Section 3. Finally, some conclusions are shown in Section Atomic model and density-matrix equations We consider a generic four-level -type atomic system as shown in Fig. 1 [41,42]. It has a single excited state 2 and three closely spaced lower levels 1, 3 and 4, the spontaneous emission from excited state 2 to the lower state 1 can strongly affect the neighbouring transition 2 3 ( 2 4 ), and interference due to spontaneous emission can arise. A resonant coupling field and a weak probe field with Rabi-frequencies Ω c and Ω p, respectively, couple two lower meta-stable states 1 and 3 with upper level 2, formingasimple -configuration. Meanwhile, the resulting dark state is coherently coupled to another meta-stable state 4 by a microwave or quasi-static coherent field with Rabifrequency Ω d.2γ 1,2γ 3,2γ 4 denote the spontaneous decay rates from level 2 to levels 1, 3 and 4, respectively. ω p, ω c and ω d are carrier frequencies of the corresponding fields, and Δ 1 = ω 21 ω p and Δ 3 = ω 34 ω d are the frequency detunings of the probe field and the coherent driving field, respectively. Under the rotating-wave approximation, the systematic density matrix in the interaction picture involving the SGC can be written as ρ 11 = 2γ 1 ρ 22 iω p (ρ 12 ρ 21 ), ρ 12 = (γ 1 + γ 3 + γ 4 iδ 1 )ρ 12 iω c ρ 13 + iω p (ρ 22 ρ 11 ), ρ 13 = iδ 1 ρ 13 iω c ρ 12 iω d ρ 14 + iω p ρ η γ 1 γ 3 ρ 22, ρ 14 = iδ 1 ρ 14 iω d ρ 13 + iω p ρ η γ 1 γ 4 ρ 22, ρ 22 = (2γ 1 + 2γ 3 + 2γ 4 )ρ 22 + iω c (ρ 32 ρ 23 ) + iω p (ρ 12 ρ 21 ), ρ 23 = (γ 1 + γ 3 + γ 4 )ρ 23 + iω c (ρ 33 ρ 22 ) iω d ρ 24 + iω p ρ 13, ρ 24 = (γ 1 + γ 3 + γ 4 )ρ 24 + iω c ρ 34 iω d ρ 23 + iω p ρ 13, ρ 33 = 2γ 3 ρ 22 + iω c (ρ 23 ρ 32 ) + iω d (ρ 43 ρ 34 ), The above equations are constrained by ρ 11 + ρ 22 + ρ 33 + ρ 44 = 1 and ρ ji = ρ ij. In this Letter, in order to distinguish the condition of Ref. [41], we assume the coherent control field is resonant, i.e., Δ 3 = 0 and the other parameters are the same. The term 2η γ 1 γ 3 represents the quantum interference effect from the cross-coupling between the spontaneous emissions 2 1 and 2 3, and the term 2η γ 1 γ 4 represents the quantum interference effect between the spontaneous emission pathways from 2 1 and 2 4. Theparameterη denotes the alignment of the dipole matrix elements ( μ 21 and μ 2 j ) and is defined as η μ 21 μ 2 j / μ 21 μ 2 j =cos θ(j = 3, 4), and θ represent the angle between the two dipole moments μ 21 and μ 2 j.ifthematrix elements μ 21 and μ 2 j are orthogonal to each other, i.e., η = 0, the SGC disappears. So the existence of the SGC depends on the nonorthogonality of the matrix elements μ 21 and μ 2 j. Here we only consider the case where each field acts only on one transition and the polarization direction of the driving field is parallel to the polarization direction of the coupling field. Then the Rabi frequencies Ω p, Ω c and Ω d are connected to the angle θ and represented by Ω l = Ω (0) sin θ = Ω (0) 1 η l l 2 (l = p, c, d). To gain the remarkable SGC effect on optical properties of the system we assume that three lower levels 1, 3 and 4 are nearly degenerate, i.e., ω 13 0 and ω 14 0orelsethetermswithη and accompanied by an exponential factor exp(±ω 13 t) and exp(±ω 14 t), which are not shown here, have been averaged out to zero and thereby the SGC effect vanishes [20,30,43]. As is known, the response of the atomic medium to the probe field is governed by its polarization P = ε 0 (E p χ + E pχ )/2, with χ being the susceptibility of the atomic medium. By performing a quantum average of the dipole moment over an ensemble of N atoms, we find P = N(μ 42 ρ 24 + μ 24 ρ 42 ). In order to derive the linear and nonlinear susceptibilities, we need to obtain the steady-state solution of the density matrix equation. In the present approach, an iterative method is used and the density matrix elements are expressed as ρ mn = ρ (0) mn +ρ (1) mn +ρ (2) mn + mn +. Assuming that the coupling field and the driving field are much stronger than the probe field, the zeroth order solution will be ρ (0) 11 = 1 and other elements are equal to zero. Under the weakprobe approximation, we can get the matrix element ρ 12 up to the third order: ρ (1) 21 = (Δ 2 1 Ω2 d )Ω p i(γ 1 + γ 2 + γ 3 )(Δ 2 1 Ω2 d ) + Δ 1(Δ 2 1 Ω2 c Ω 2 d ), ρ (2) (2a) 12 = (η γ 1 γ 3 Δ 1 + η γ 1 γ 4 Ω d )(ρ (1) 12 ρ(1) 21 )Ω cω p i(γ 1 + γ 3 + γ 4 )(Δ 2 1 Ω2 d ) + Δ 1(Ωc 2 + Ω 2 d ), (2b) Δ = i(η γ 1 γ 3 Δ 1 + η γ 1 γ 4 Ω d )(ρ (2) 12 ρ(2) 21 )Ω cω p iδ 1 Ωc 2 + (γ 1 + γ 3 + γ 4 iδ 1 )(Ω 2 d Δ2 1 ) i(ρ(2) 11 ρ(2) 22 )(Ω2 d Δ2 1 )Ω p + i(δ 1 ρ (2) 23 + Ω dρ (2) 24 )Ω cω p iδ 1 Ωc 2 + (γ 1 + γ 3 + γ 4 iδ 1 )(Ω 2 d Δ2 1 ). (2c) Therefore, the first-and third-order susceptibility χ (1) and χ (3) should be χ (1) = 2N μ 12 2 ε 0 hω p ρ (1) 12, (3) χ (3) = 2N μ ε 0 h 3 Ωp 3 12, (4) and χ is be defined as χ = χ (1) + 3 E p 2 χ (3). (5)

3 6458 X.-a. Yan et al. / Physics Letters A 372 (2008) Fig. 2. The Kerr nonlinearity Re[χ (3) ] (solid curve), linear absorption Im[χ (1) ] (dotted curve), and nonlinear absorption Im[χ (3) ] (dash-dotted curve) of the four-level -type system with different SGC. Parameters are Ω (0) c = 1.0γ 1, Ω (0) = 0.2γ d 1, γ 3 = γ 4 = γ Discussion and results In this part, we will focus on the dependence of the thirdorder susceptibility on the SGC. According to the expressions (3) and (4), we show in Fig. 2 the linear absorption (dotted curve) and the refractive part of the third-order susceptibility (solid curve) as a function of the probe detuning. From Eq. (2a), we can find the linear absorption is independent of the SGC (i.e. η), the nonlinear absorption given by the imaginary part of χ (3), which is dependent of η from Eq. (2c), should be considered, so we also plot the imaginary part of the third-order susceptibility (dashed curve) in Fig. 2. For simplicity, we scale all the parameters by the decay rate γ 1, setting Ω c = Ω (0) c sin θ = Ω (0) c 1 η 2, Ω d = Ω (0) sin θ = Ω (0) 1 η d d 2 and the parameters Ω (0) c = 1.0γ 1, Ω (0) = 0.2γ d 1, γ 3 = γ 4 = γ 1. From this figure, we can see that when η = 0, which means no interference between spontaneous emission channels, a couple of general linear absorption and Kerr nonlinearity curves occur [39,44]. In other words, if the SGC is absent, the maximal Kerr nonlinearity is accompanied by strong linear and nonlinear absorptions, as can be seen from Fig. 2(a), implying that, in this case, it is not desirable for applications of all-optical switch because the accompanying thermal effect of the devices is not negligible. In order to eliminate the thermal effect, Ref. [41] proposed a scheme for giant enhancement of the Kerr nonlinearity by proper tuning of the coherent driving field in the same four-level -type system without SGC. Here, when the SGC is taken into account, as Fig. 3. Variation of Re[F 1 (η)] versus the probe detuning Δ 1. Parameters are the same as in Fig. 2. shown in Figs. 2(b) and (c), a giant Kerr nonlinearity with vanishing absorption also can be obtained under the condition of keeping the coherent driving field resonant. Moreover, the Kerr nonlinearity is enhanced gradually as the SGC changes from η = 0to0.9. Compared with the case of η = 0, the maximal value of the Kerr nonlinearity is enhanced by about five times at η = 0.9, which is different from the scheme proposed by Ref. [41], in which the

4 X.-a. Yan et al. / Physics Letters A 372 (2008) Fig. 4. Variation of Re[χ (3) ] (solid curve) and Re[F 1 (η)] (dotted curve) versus the probe detuning Δ 1. Parameters are the same as in Fig. 2; (a)η = 0.8, (b) η = 0.9. maximal value of Re[χ (3) ] is invariable when the coherent driving field is detuned. On the other hand, we find, in the case of η = 0.9, that the enhanced Kerr nonlinearity is located in the nonlinear gain region and the linear absorption is zero near zero probe detuning. So in the case, we achieve giant Kerr nonlinearity accompanied by zero linear absorption and negative nonlinear absorption via SGC. 4. Qualitative explanation of above numerical results In this part, we will provide a qualitative explanation for the above numerical results. Here, we mainly discuss the effect of SGC on the Kerr nonlinearity. In Eq. (2), since the matrix elements ρ (2) 11, ρ (2) 22, ρ(2) 23, ρ(2) 24 are independent of η, we do not present the analytical expressions. It can be seen from Eq. (2b) that the density matrix element ρ (2) 12 now is an extra term introduced by SGC. This term according makes the third-order susceptibility acquire an additional term associated with SGC. As the analytical expression for 12 shows, the first term is the product of the coupling field and the additional term. That is to say, in the present generic four-level -type atomic system, the spontaneous decays from the common excited state to the lower closely spaced levels interfere and give rise to an extra coherence between the lower two levels [36,45, 46]. Thereby the coupling field interacts with this extra coherence and consequently the Kerr nonlinearity is clearly enhanced. Here, we designate the first term of Eq. (4) as F 1 (η) and another term as F 2,thenχ (3) = F 1 (η) + F 2.WeplotRe[F 1 (η)] as a function of Δ 1 in Fig. 3 with different SGC. Obviously, the stronger the SGC is, the more remarkable the coupling field interacts with this extra coherence. Hence, we attribute the enhancement of Kerr nonlinearity to the extra coherence between the lower two levels induced by SGC. Another a question arises, that is why the peak value of the enhanced Kerr nonlinearity enters the electromagnetically induced transparency and whether the position variation of Re[χ (3) ] is related to SGC. To answer this question, we plot Re[F 1 (η)] and Re[χ (3) ] as a function of Δ 1 in Fig. 4 with different SGC. Obviously, one can see that the peak position of the enhanced Re[χ (3) ] is approximately coincident with that of Re[F 1 (η)]. Hence, when the optimal SGC effect is taken into account, a giant Kerr nonlinearity accompanied by vanishing absorption can be realized near zero probe detuning even without the condition of proper detuning of the coherent control field. 5. Conclusion We have investigated the effect of SGC on the Kerr nonlinearity in a generic four-level -type atomic system. The results show that the spontaneously generated coherence is capable of enhancing the Kerr nonlinearity even without the condition of Ref. [41]. In the case of optimal SGC, both enhanced Kerr nonlinearity and negligible linear and nonlinear absorption can be realized simultaneously. Compared with the scheme that offered by Ref. [41], the enhancement of Kerr nonlinearity can be enhanced by some times even without the condition of proper detuning of the coherent control field. Close inspection of the analytical expression undoubtedly shows that the spontaneously emission interference induces an extra coherence term that is responsible for the large enhancement of the Kerr nonlinearity, suggesting the application of the above-mentioned results in all-optical switch, frequency conversion, polarization phase gates and relevant fields appear to be promising. Acknowledgements This work is supported by the National Natural Science Key Foundation of China (Grant No ), the foundation for Key program of Ministry of Education, China (Grant No ), the For Ying-Tong Education Foundation for Young Teachers in the Institutions of Higher Education of China (Grant No ) and the Specialized Research Fund for Doctoral Program of Higher Education of China (Grant No ). References [1] J. Mompart, R. Corbalan, J. Opt. B: Quantum Semiclass. Opt. 2 (2000) R7. [2] C.R. Li, Y.C. Li, F.K. Men, C.H. Pao, Y.C. Tsai, J.F. Wang, Appl. Phys. Lett. 86 (2005) [3] A.E. Allahverdyan, R.S. Gracia, T.M. Nieuwenhuizen, Phys. Rev. E 71 (2005) [4]X.J.Fan,N.Cui,H.Ma,A.Y.Li,H.Li,Eur.Phys.J.D37(2006)129. [5] H.R. Xia, C.Y. Ye, S.Y. Zhu, Phys. Rev. Lett. 77 (1996) 1032.

5 6460 X.-a. Yan et al. / Physics Letters A 372 (2008) [6] C.H. Keitel, Phys. Rev. A 57 (1998) [7] L.J. Wang, A. Kuzmich, A. Dogariu, Nature 406 (2000) 277. [8] L.V. Hau, S.E. Harris, Z. Dutton, C.H. Behroozi, Nature (London) 397 (1999) 594. [9] M.M. Kash, V.A. Sautenkov, A.S. Zibrov, L. Hollberg, G.R. Welch, M.D. Lukin, Y. Rostovtsev, E.S. Fry, M.O. Scully, Phys. Rev. Lett. 82 (1999) [10] M.O. Scully, Phys. Rev. Lett. 67 (1991) [11] Q.Y. He, T.J. Wang, J.Y. Gao, Chin. Phys. 15 (2006) [12] M. Sahrai, M. Mahmoudi, R. Kheradmand, Phys. Lett. A 367 (2007) 408. [13] E. Arimondo, in: E. Wolf (Ed.), Progress in Optics XXXV, North-Holland, Amsterdam, 1996, p [14] S.E. Harris, Phys. Today 50 (1997) 36. [15] S.E. Harris, Phys. Rev. Lett. 82 (1999) [16] A.G. Litvak, M.D. Tokman, Phys. Rev. Lett. 88 (2002) [17] Y. Wu, X.X. Yang, Phys. Rev. A 71 (2005) [18] J. Javanainen, Europhys. Lett. 17 (1992) 407. [19] S. Menon, G.S. Agarwal, Phys. Rev. A 57 (1998) [20] E. Paspalakis, S.Q. Gong, P.L. Knight, Opt. Commun. 152 (1998) 293. [21] S. Menon, G.S. Agarwal, Phys. Rev. A 61 (1999) [22] R.M. Whitley, C.R. Stround Jr., Phys. Rev. A 14 (1976) [23] Z. Ficek, B.J. Dalton, P.L. Knight, Phys. Rev. A 51 (1995) [24] P. Zhou, S. Swain, Phys. Rev. Lett. 78 (1997) 832. [25] S.Q. Gong, E. Paspalakis, P.L. Knight, J. Mod. Opt. 45 (1998) [26] F. Ghafoor, S.Y. Zhu, M.S. Zubairy, Phys. Rev. A 62 (2000) [27] H.M. Ma, S.Q. Gong, C.P. Liu, Z.R. Sun, Z.Z. Xu, Opt. Commun. 223 (2003) 97. [28] H.M. Ma, S.Q. Gong, Z.R. Sun, R.X. Li, Z.Z. Xu, Chin. Phys. 15 (2006) [29] C.P. Liu, S.Q. Gong, X.J. Fan, Z.Z. Xu, Opt. Commun. 231 (2004) 289. [30] C.P. Liu, S.Q. Gong, X.J. Fan, Z.Z. Xu, Opt. Commun. 239 (2004) 383. [31] J.H. Li, J.M. Luo, W.X. Yang, J.C. Peng, Chin. Phys. 14 (2005) 985. [32] J.H. Wu, J.Y. Gao, Phys. Rev. A 65 (2002) [33] P.R. Berman, Phys. Rev. A 72 (2005) [34] M.A. Antón, O.G. Calderón, F. Carreño, Phys. Rev. A 72 (2005) [35] I. Gonzalo, M.A. Antón, F. Carreño, O.G. Calderón, Phys. Rev. A 72 (2005) [36] Y.P. Niu, S.Q. Gong, Phys. Rev. A 73 (2006) [37] Y.F. Ban, H. Guo, D.A. Han, H. Sun, J. Opt. B 7 (2005) 35. [38] R. Arun, Phys. Rev. A 77 (2008) [39] S.Q. Gong, Y. Li, S.D. Du, Z.Z. Xu, Phys. Lett. A 59 (1999) 43. [40] W.H. Xu, J.H. Wu, J.Y. Gao, Phys. Rev. A 66 (2002) [41] Y.P. Niu, S.Q. Gong, R.X. Li, Z.Z. Xu, X.Y. Liang, Opt. Lett. 30 (2005) [42] M.D. Lukin, S.F. Yelin, M. Fleischhauer, M.O. Scully, Phys. Rev. A 60 (1999) [43] B.P. Hou, S.J. Wang, W.L. Yu, W.L. Sun, Phys. Rev. A 69 (2004) [44] H. Wang, D.J. Goorskey, M. Xiao, J. Mod. Opt. 49 (2002) 335. [45] M.O. Scully, Phys. Rev. Lett. 55 (1985) [46] J. Evers, D. Bullock, C.H. Keitel, Opt. Commun. 209 (2002) 173.