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Giant Kerr nonlinearity in an open V-type system with spontaneously generated coherence
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Giant Kerr nonlinearity in an open V-type system with spontaneously generated coherence Yanfeng Bai a,∗ , Taigang Liu b , Xiaoqiang Yu a a b
Department of Physics, Southeast University, Nanjing 210096, China Department of Life Science and Technology, Xinxiang Medical University, Xinxiang 453003, China
a r t i c l e
i n f o
Article history: Received 24 August 2011 Accepted 23 December 2011 PACS: 42.50.Gy 42.65.-k
a b s t r a c t We investigate the Kerr nonlinearity in an open three-level V-type system with spontaneously generated coherence (SGC). It is shown that the giant Kerr nonlinearity with vanishing linear absorption can be obtained with the presence of the SGC effect. While the Kerr nonlinearity is smaller than that in a general closed system due to the existence of the additional term from the open system. The method to enhance the Kerr nonlinearity in the open system is also discussed. © 2012 Elsevier GmbH. All rights reserved.
Keywords: Kerr nonlinearity Spontaneously generated coherence
1. Introduction Quantum coherence and interference in an atomic system have led to a number of interesting phenomena, such as coherent population trapping (CPT) [1], electromagnetically induced transparency (EIT) [2–4], lasing without inversion (LWI) [5], subluminal and superluminal light [6–8], and the enhanced nonlinear optics [9], and so on. The generic EIT (reducing absorption by atomic coherence) consists of three-level atoms, then the people extended their interest to multilevel atomic systems, which opened up a completely new route to achieving large nonlinear optical processes at very low light intensities, or even at energies of a few photons per atomic cross-section [10–12]. Recently, many schemes of EIT-enhanced nonlinear phenomena have been investigated, both experimentally and theoretically [13–22]. Nakajima showed the third-order nonlinearity can be enhanced with vanishing absorption via autoionizing resonance [15]. Matsko et al. proposed a new method of resonant enhancement of optical Kerr nonlinearity that uses multilevel atomic coherence [17]. A scheme for giant enhancement of the Kerr nonlinearity in a four-level system with double dark resonances was proposed [19]. Very recently, Niu and Gong showed that the giant Kerr nonlinearity with zero absorption can be obtained via the SGC effect in a V-type system [21].
∗ Corresponding author. E-mail address: [email protected] (Y. Bai).
Recently, some research groups investigated the quantum coherence and interference in the open atomic systems [23–26]. However, all of the work analyzed only the properties of the linear susceptibility, no third-order susceptibility is mentioned. In this paper, the third-order Kerr nonlinearity is investigated theoretically in an open three-level V-type system. We show that it is possible to obtain the giant Kerr nonlinearity with vanishing linear absorption via the SGC effect. While the Kerr nonlinearity is decreased in comparison with that in a general closed system due to the additional term from the open system. In addition, the Kerr nonlinearity can be enhanced by changing the ratio of the atomic injection rates.
2. Theoretical model We consider an open V-type three-level system (see Fig. 1) with the lower state |1 and the excited states |2 and |3. The transition |1 ↔ |2 with frequency ω21 is driven by a strong coherent coupling )/(2¯h). A field of frequency ωc with a Rabi frequency G = (ε c · d 12 weak, coherent probe field of frequency ωp with Rabi frequency )/(2¯h) is applied to the transition |1 ↔ |3, where ε p and g = (ε p · d 13 ε c represent the amplitudes of two coherent fields, respectively. The spontaneous decay rates from the states |2 and |3 to the state |1 are 2 2 and 2 3 , respectively. The atomic exit rate from the cavity is r0 , and the atomic injection rates for levels |1 and |2 are W1 and W2 , respectively.
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Please cite this article in press as: Y. Bai, et al., Giant Kerr nonlinearity in an open V-type system with spontaneously generated coherence, Optik - Int. J. Light Electron Opt. (2012), doi:10.1016/j.ijleo.2011.12.038
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2
Under the rotating-wave approximation, the density matrix equation of motion can be written as ˙ 22
=
˙ 33
=
˙ 11
=
˙ 31
=
˙ 21
=
˙ 23
=
√ −(22 + r0 )22 + iG(12 − 21 ) − p 2 3 (23 + 32 ) + W2 , √ −(23 + r0 )33 + ig(13 − 31 ) − p 2 3 (23 + 32 ), √ 22 22 + 23 33 − r0 11 − iG(12 − 21 ) − ig(13 − 31 ) + 2p 2 3 (23 + 32 ) + W1 , √ −(3 − i3 )31 + ig(11 − 33 ) − iG32 − p 2 3 21 , √ −(2 − i2 )21 + iG(11 − 22 ) − ig23 − p 2 3 31 , √ −[2 + 3 + i(3 − 2 )]23 + iG13 − ig21 − p 2 3 (22 + 33 ),
the above equations are constrained by 11 + 22 + 33 = 1 and ij = ji∗ . In the following discussion, we always set W1 + W2 = r0 to keep
the total number of atoms as a constant. Here 2 = ω21 − ωc and 3 = ω31 − ωp are the detunings of the probe and driving fields, respectively. The effect of SGC is very sensitive to the orientations and d . Here the parameter p of the atomic dipole moments d 12 13 denotes the alignment of the two dipole moments and is defined ·d /|d ·d | = cos with being the angle between as p = d 12 13 12 13 √ the two dipole moments. The terms including p 2 3 represent the SGC effect. Noted that when p = / 0, the SGC effect presents; otherwise p = 0, the SGC effect is absent. With the restriction that each field acts only on one transition, the Rabi frequencies g and 0 G are connected to the angle and represented by g = g sin =
g 0 1 − p2 and G = G0 sin = G0 1 − p2 . When W1 = W2 = r0 = 0, Eqs. (1) reduce to the equations for a closed three-level V-type system. It is known that the response of the atomic medium to the probe field is governed by its polarization P = ε0 (Ep + Ep∗ ∗ )/2, where is the susceptibility of the atomic medium. By performing a quantum average of the dipole moment over an ensemble of N atoms, P = N( 31 13 + 13 31 ). To derive the equations for the linear and nonlinear susceptibilities, we need to give the steady state solutions for the density matrix Eq. (1). In the present approach, an iterative method is used to achieve increasingly accurate approximations to the matrix elements. The density matrix elements can be written as mn = (1) (2) (3) (0) mn + mn + mn + mn + · · ·, where each successive approximation is calculated using the matrix elements of one order less than the one being calculated. Under the condition that the probe field is very small as compared with the coupling field, the zeroth-order (0) solution will be 11 = 1 and other elements are equal to zero. Under the weak-probe approximation, we can get the matrix elements
⎢3〉
Δ3 r0 2γ3
⎢2〉
Δ2 r0
g G
W2
2γ2 W1
⎢1〉
r0 Fig. 1. Schematic diagram of an open three-level V-type.
(1)
(1)
(3)
31 and 31 . By combining with = (1) + 3|εp |2 (3) , we can get the linear and the third-order nonlinear susceptibilities: Im[(1) ] = − Re[(3) ]
=
−
31 |2 2N| (1) Im[31 ], ε0h ¯g 31 |4 2N| 3
3ε0 h ¯ g3
(1)
(2) (1)
(1)
(1)
(1)
{AIm[31 ] + B21 + C23 + D[11 − 33 ] + K},
(3)
where N is the atom density, and parameters A, B, C, D, and K are the functions of G, 3 , r0 , W1 , and W2 , their expressions are so tedious that we will not present them here. From Eq. (3), the Kerr (1) (1) (1) (1) (1) nonlinearity is related to Im[31 ], 21 , 23 , 11 − 33 , and K. Here K represents the additional term from the open system (r0 , W1 , and W2 ), and this term is absent for a closed atomic system. Obviously, the additional term can affect the third-order Kerr nonlinearity. 3. The numerical results In this section, we give the numerical simulations to demonstrate the above results. Based on Eqs. (2) and (3), we give the linear absorption (dash line) and the refractive part of the third-order susceptibility (solid line) as a function of the probe detuning 3 in Fig. 2. From this figure, it is shown that when p = 0, which means no SGC effect, the general linear absorption and Kerr nonlinearity curves occur. While the real part of the third-order nonlinear susceptibility is accompanied by strong linear absorption, which is undesirable for application of low-intensity nonlinear optics. With SGC present, the maximal Kerr nonlinearity is enhanced by about ten times and also the EIT window narrows. we also note that the refractive part of (3) has a large value whereas the linear absorption is zero for a certain probe detuning. In other words, the giant Kerr nonlinearity with vanishing linear absorption can be obtained due to the presence of the SGC effect. Based on the conclusion in Ref. [21] and the difference shown in Fig. 2(a) and (b), we attribute the enhancement of Kerr nonlinearity to the presence of an extra atomic coherence induced by SGC. Next we want to enquire: the difference between the Kerr nonlinearity obtained in the open system and in the closed scheme. To answer the question, we depict in Fig. 3 the corresponding results in the closed V-type system. In comparison with the open system, it is seen that the variation trend is quite similar. If no SGC is considered, the Kerr nonlinearity with strong linear absorption is presented. When the optimal SGC is taken into account, the refractive part of the third-order susceptibility is enhanced by even 30 times and also enters the transparency window. So in this case, the giant Kerr nonlinearity with vanishing linear absorption in the closed system can be obtained via the optimal SGC. By comparing Figs. 3 with 2, the Kerr nonlinearity is decreased when the system is changed from the closed to the open.
Please cite this article in press as: Y. Bai, et al., Giant Kerr nonlinearity in an open V-type system with spontaneously generated coherence, Optik - Int. J. Light Electron Opt. (2012), doi:10.1016/j.ijleo.2011.12.038
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40
2
(arb. unit)
(arb.unit)
80
(a)
Re[χ(3) ] (1) Im[χ ]
4
3
0
0
-2 -40 -4 -8
-4
0
4
Δ3
-80 -3
8
-2
-1
0
Δ3
1
2
3
40
(b)
(arb.unit)
20
Fig. 4. The Kerr nonlinearity (solid line) and the contribution from the additional term owing to the open system (dotted line) as a function of 3 . The other parameters are the same as those in Fig. 2.
0 40
-40
-4
-2
0
2
Δ3
(arb. unit)
-20
4
Fig. 2. The Kerr nonlinearity Re[(3) ] (solid curve) and linear absorption Im[(1) ] (dash curve) with different SGC. The parameters are chosen as 2 = 0, 2 = 10 3 = 1, G0 = 4.0, r0 = 0.03, W1 = 0.02, and W2 = 0.01.
We are currently discussing the physical mechanism leading to the difference mentioned above. From Eq. (3), K is an additional term from the open system, and this term does not exist for the closed system. We show the contribution from the additional term to the Kerr nonlinearity in Fig. 4, Re[(3) ] is also displayed for comparison. Obviously, the occurrence of the additional term from the open system cumbers greatly the enhancement of the Kerr nonlinearity. Finally we want to discuss the method of enhancing the Kerr nonlinearity in the open system. From Ref. [26], we know that the properties of linear susceptibility depend on the ratio of the atomic
(a)
Re[χ(3) ] (1) Im[χ ]
4
(arb. unit)
0 -2 -4 -4
0
Δ3
4
8
100
(b)
(arb. unit)
-40 -3
-2
-1
0
Δ3
1
2
3
Fig. 5. The Kerr nonlinearity under different ratios of the atomic injection rates. The other parameters are the same as those in Fig. 2.
injection rates s = W1 /W2 . Then if the parameter can affect the thirdorder susceptibility? Fig. 5 shows the refractive part of the thirdorder susceptibility for different values of s. It is shown that the Kerr nonlinearity is enhanced with a decrease of s values. In other words, we can enhance the Kerr nonlinearity by changing the ratio of the atomic injection rates. Especially, the maximal value of the Kerr nonlinearity is enhanced by about 15 times at s = 1/9 in comparison with Fig. 2(a). 4. Conclusions
Acknowledgement This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 10904015 and 11004030).
50
0
References
×20 -50
-100
0
In conclusion, we have investigated the properties of the Kerr nonlinearity in an open three-level V-type system with SGC. It is shown that the Kerr nonlinearity is enhanced greatly with vanishing linear absorption via SGC. While the maximal value of the Kerr nonlinearity is smaller as compared with that in the closed system due to the additional term from the open system. Additionally, we discuss the method to enhance the Kerr nonlinearity in the open V-type system.
2
-8
s=2 s=1 s=1/9
-4
-2
0
Δ3
2
4
Fig. 3. The same as Fig. 2, but with a closed system r0 = W1 = W2 = 0.
[1] E. Arimondo, in: E. Wolf (Ed.), Progress in Optics XXXV, North-Holland, Amsterdam, 1996, p. 257. [2] S.E. Harris, J.E. Field, A. Imamoglu, Nonlinear optical processes using electromagnetically induced transparency, Phys. Rev. Lett. 64 (1990) 1107–1110. [3] S.E. Harris, Electromagnetically induced transparency, Phys. Today 52 (1997) 36–42. [4] K.J. Boller, A. Imamoglu, S.E. Harris, Observation of electromagnetically induced transparency, Phys. Rev. Lett. 66 (1991) 2593–2596.
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[5] S.E. Harris, Lasers without inversion: interference of lifttime-broadened resonances, Phys. Rev. Lett. 62 (1989) 1033–1036. [6] L.V. Hau, S.E. Harris, Z. Dutton, C.H. Behroozi, Light speed reduction to 17 metres per second in an ultracold atomic gas, Nature (London) 397 (1999) 594–598. [7] L.J. Wang, A. Kuzmich, A. Dogariu, Gain-assisted superluminal light propagation, Nature (London) 406 (2000) 277–279. [8] C. Liu, Z. Dutton, C.H. Behroozi, L.V. Hau, Observation of coherent optical information storage in an atomic medium using halted light pulses, Nature (London) 409 (2001) 490–493. [9] R.W. Boyd, Nonlinear Optics, Academic, San Diego, 1992. [10] S.E. Harris, Y. Yamamoto, Photon switching by quantum interference, Phys. Rev. Lett. 81 (1998) 3611–3614. [11] S.E. Harris, L.V. Hau, Nonlinear optics at low light levels, Phys. Rev. Lett. 82 (1999) 4611–4614. [12] M.D. Lukin, A. Imamoglu, Nonlinear optics and quantum entanglement of ultraslow single photons, Phys. Rev. Lett. 84 (2000) 1419–1422. [13] H. Schmidt, A. Imamoglu, Giant Kerr nonlinearities obtained by electromagnetically induced transparency, Opt. Lett. 21 (1996) 1936–1938. [14] A. Imamoglu, H. Schmidt, G. Woods, M. Deutsch, Strongly interacting photons in a nonlinear cavity, Phys. Rev. Lett. 79 (1997) 1467–1470. [15] T. Nakajima, Enhanced nonlinearity and transparency via autoionizing resonance, Opt. Lett. 25 (2000) 847–849. [16] H. Wang, D. Goorskey, M. Xiao, Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system, Phys. Rev. Lett. 87 (2001) 073601-1-4.
[17] A.B. Matsko, I. Novikova, G.R. Welch, M.S. Zubairy, Enhancement of Kerr nonlinearity by multiphoton coherence, Opt. Lett. 28 (2003) 96–98. [18] A. Joshi, M. Xiao, Phase gate with a four-level inverted-Y system, Phys. Rev. A 72 (2005) 062319-1-5. [19] Y. Niu, S. Gong, R. Li, Z. Xu, X. Liang, Giant Kerr nonlinearity induced by interacting dark resonances, Opt. Lett. 30 (2005) 3371–3373. [20] C. Ottaviani, S. Rebic, D. Vitali, P. Tombesi, Cross phase modulation in a five-level atomic medium: semiclassical theory, Eur. Phys. J. D 40 (2006) 281–296. [21] Y. Niu, S. Gong, Enhancing Kerr nonlinearity via spontaneously generated coherence, Phys. Rev. A 73 (2006) 053811-1-6. [22] H. Sun, S. Gong, Y. Niu, S. Jin, R. Li, Z. Xu, Enhancing Kerr nonlinearity in an asymmetric double quantum well via Fano interference, Phys. Rev. B 74 (2006) 155314-1-5. [23] X.J. Fan, J.J. Li, S.F. Tian, N. Cui, H. Li, A.Y. Li, Z.Z. Xu, Phase control of gain and dispersion in an open V-type system, J. Mod. Opt. 52 (2005) 1757–1767. [24] A. Li, H. Ma, X. Tan, Y. Yang, D. Tong, X. Fan, Phase control of probe response in an open ladder type system with spontaneously generated coherence, Opt. Commun. 280 (2007) 397–403. [25] Z.H. Xiao, K. Kim, Comparison of transient process between open and closed ladder-type systems with spontaneously generated coherence and incoherent pumping, Opt. Commun. 282 (2009) 2547–2551. [26] X. Fan, B. Liang, Z. Wang, D. Tong, Phase modulation of transient evolution of atomic response in open lambda-type system with spontaneously generated coherence, Opt. Commun. 283 (2010) 1810–1816.
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ARTICLE IN PRESS Optik xxx (2012) xxx–xxx
Contents lists available at SciVerse ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Giant Kerr nonlinearity in an open V-type system with spontaneously generated coherence Yanfeng Bai a,∗ , Taigang Liu b , Xiaoqiang Yu a a b
Department of Physics, Southeast University, Nanjing 210096, China Department of Life Science and Technology, Xinxiang Medical University, Xinxiang 453003, China
a r t i c l e
i n f o
Article history: Received 24 August 2011 Accepted 23 December 2011 PACS: 42.50.Gy 42.65.-k
a b s t r a c t We investigate the Kerr nonlinearity in an open three-level V-type system with spontaneously generated coherence (SGC). It is shown that the giant Kerr nonlinearity with vanishing linear absorption can be obtained with the presence of the SGC effect. While the Kerr nonlinearity is smaller than that in a general closed system due to the existence of the additional term from the open system. The method to enhance the Kerr nonlinearity in the open system is also discussed. © 2012 Elsevier GmbH. All rights reserved.
Keywords: Kerr nonlinearity Spontaneously generated coherence
1. Introduction Quantum coherence and interference in an atomic system have led to a number of interesting phenomena, such as coherent population trapping (CPT) [1], electromagnetically induced transparency (EIT) [2–4], lasing without inversion (LWI) [5], subluminal and superluminal light [6–8], and the enhanced nonlinear optics [9], and so on. The generic EIT (reducing absorption by atomic coherence) consists of three-level atoms, then the people extended their interest to multilevel atomic systems, which opened up a completely new route to achieving large nonlinear optical processes at very low light intensities, or even at energies of a few photons per atomic cross-section [10–12]. Recently, many schemes of EIT-enhanced nonlinear phenomena have been investigated, both experimentally and theoretically [13–22]. Nakajima showed the third-order nonlinearity can be enhanced with vanishing absorption via autoionizing resonance [15]. Matsko et al. proposed a new method of resonant enhancement of optical Kerr nonlinearity that uses multilevel atomic coherence [17]. A scheme for giant enhancement of the Kerr nonlinearity in a four-level system with double dark resonances was proposed [19]. Very recently, Niu and Gong showed that the giant Kerr nonlinearity with zero absorption can be obtained via the SGC effect in a V-type system [21].
∗ Corresponding author. E-mail address: [email protected] (Y. Bai).
Recently, some research groups investigated the quantum coherence and interference in the open atomic systems [23–26]. However, all of the work analyzed only the properties of the linear susceptibility, no third-order susceptibility is mentioned. In this paper, the third-order Kerr nonlinearity is investigated theoretically in an open three-level V-type system. We show that it is possible to obtain the giant Kerr nonlinearity with vanishing linear absorption via the SGC effect. While the Kerr nonlinearity is decreased in comparison with that in a general closed system due to the additional term from the open system. In addition, the Kerr nonlinearity can be enhanced by changing the ratio of the atomic injection rates.
2. Theoretical model We consider an open V-type three-level system (see Fig. 1) with the lower state |1 and the excited states |2 and |3. The transition |1 ↔ |2 with frequency ω21 is driven by a strong coherent coupling )/(2¯h). A field of frequency ωc with a Rabi frequency G = (ε c · d 12 weak, coherent probe field of frequency ωp with Rabi frequency )/(2¯h) is applied to the transition |1 ↔ |3, where ε p and g = (ε p · d 13 ε c represent the amplitudes of two coherent fields, respectively. The spontaneous decay rates from the states |2 and |3 to the state |1 are 2 2 and 2 3 , respectively. The atomic exit rate from the cavity is r0 , and the atomic injection rates for levels |1 and |2 are W1 and W2 , respectively.
0030-4026/$ – see front matter © 2012 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2011.12.038
Please cite this article in press as: Y. Bai, et al., Giant Kerr nonlinearity in an open V-type system with spontaneously generated coherence, Optik - Int. J. Light Electron Opt. (2012), doi:10.1016/j.ijleo.2011.12.038
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2
Under the rotating-wave approximation, the density matrix equation of motion can be written as ˙ 22
=
˙ 33
=
˙ 11
=
˙ 31
=
˙ 21
=
˙ 23
=
√ −(22 + r0 )22 + iG(12 − 21 ) − p 2 3 (23 + 32 ) + W2 , √ −(23 + r0 )33 + ig(13 − 31 ) − p 2 3 (23 + 32 ), √ 22 22 + 23 33 − r0 11 − iG(12 − 21 ) − ig(13 − 31 ) + 2p 2 3 (23 + 32 ) + W1 , √ −(3 − i3 )31 + ig(11 − 33 ) − iG32 − p 2 3 21 , √ −(2 − i2 )21 + iG(11 − 22 ) − ig23 − p 2 3 31 , √ −[2 + 3 + i(3 − 2 )]23 + iG13 − ig21 − p 2 3 (22 + 33 ),
the above equations are constrained by 11 + 22 + 33 = 1 and ij = ji∗ . In the following discussion, we always set W1 + W2 = r0 to keep
the total number of atoms as a constant. Here 2 = ω21 − ωc and 3 = ω31 − ωp are the detunings of the probe and driving fields, respectively. The effect of SGC is very sensitive to the orientations and d . Here the parameter p of the atomic dipole moments d 12 13 denotes the alignment of the two dipole moments and is defined ·d /|d ·d | = cos with being the angle between as p = d 12 13 12 13 √ the two dipole moments. The terms including p 2 3 represent the SGC effect. Noted that when p = / 0, the SGC effect presents; otherwise p = 0, the SGC effect is absent. With the restriction that each field acts only on one transition, the Rabi frequencies g and 0 G are connected to the angle and represented by g = g sin =
g 0 1 − p2 and G = G0 sin = G0 1 − p2 . When W1 = W2 = r0 = 0, Eqs. (1) reduce to the equations for a closed three-level V-type system. It is known that the response of the atomic medium to the probe field is governed by its polarization P = ε0 (Ep + Ep∗ ∗ )/2, where is the susceptibility of the atomic medium. By performing a quantum average of the dipole moment over an ensemble of N atoms, P = N( 31 13 + 13 31 ). To derive the equations for the linear and nonlinear susceptibilities, we need to give the steady state solutions for the density matrix Eq. (1). In the present approach, an iterative method is used to achieve increasingly accurate approximations to the matrix elements. The density matrix elements can be written as mn = (1) (2) (3) (0) mn + mn + mn + mn + · · ·, where each successive approximation is calculated using the matrix elements of one order less than the one being calculated. Under the condition that the probe field is very small as compared with the coupling field, the zeroth-order (0) solution will be 11 = 1 and other elements are equal to zero. Under the weak-probe approximation, we can get the matrix elements
⎢3〉
Δ3 r0 2γ3
⎢2〉
Δ2 r0
g G
W2
2γ2 W1
⎢1〉
r0 Fig. 1. Schematic diagram of an open three-level V-type.
(1)
(1)
(3)
31 and 31 . By combining with = (1) + 3|εp |2 (3) , we can get the linear and the third-order nonlinear susceptibilities: Im[(1) ] = − Re[(3) ]
=
−
31 |2 2N| (1) Im[31 ], ε0h ¯g 31 |4 2N| 3
3ε0 h ¯ g3
(1)
(2) (1)
(1)
(1)
(1)
{AIm[31 ] + B21 + C23 + D[11 − 33 ] + K},
(3)
where N is the atom density, and parameters A, B, C, D, and K are the functions of G, 3 , r0 , W1 , and W2 , their expressions are so tedious that we will not present them here. From Eq. (3), the Kerr (1) (1) (1) (1) (1) nonlinearity is related to Im[31 ], 21 , 23 , 11 − 33 , and K. Here K represents the additional term from the open system (r0 , W1 , and W2 ), and this term is absent for a closed atomic system. Obviously, the additional term can affect the third-order Kerr nonlinearity. 3. The numerical results In this section, we give the numerical simulations to demonstrate the above results. Based on Eqs. (2) and (3), we give the linear absorption (dash line) and the refractive part of the third-order susceptibility (solid line) as a function of the probe detuning 3 in Fig. 2. From this figure, it is shown that when p = 0, which means no SGC effect, the general linear absorption and Kerr nonlinearity curves occur. While the real part of the third-order nonlinear susceptibility is accompanied by strong linear absorption, which is undesirable for application of low-intensity nonlinear optics. With SGC present, the maximal Kerr nonlinearity is enhanced by about ten times and also the EIT window narrows. we also note that the refractive part of (3) has a large value whereas the linear absorption is zero for a certain probe detuning. In other words, the giant Kerr nonlinearity with vanishing linear absorption can be obtained due to the presence of the SGC effect. Based on the conclusion in Ref. [21] and the difference shown in Fig. 2(a) and (b), we attribute the enhancement of Kerr nonlinearity to the presence of an extra atomic coherence induced by SGC. Next we want to enquire: the difference between the Kerr nonlinearity obtained in the open system and in the closed scheme. To answer the question, we depict in Fig. 3 the corresponding results in the closed V-type system. In comparison with the open system, it is seen that the variation trend is quite similar. If no SGC is considered, the Kerr nonlinearity with strong linear absorption is presented. When the optimal SGC is taken into account, the refractive part of the third-order susceptibility is enhanced by even 30 times and also enters the transparency window. So in this case, the giant Kerr nonlinearity with vanishing linear absorption in the closed system can be obtained via the optimal SGC. By comparing Figs. 3 with 2, the Kerr nonlinearity is decreased when the system is changed from the closed to the open.
Please cite this article in press as: Y. Bai, et al., Giant Kerr nonlinearity in an open V-type system with spontaneously generated coherence, Optik - Int. J. Light Electron Opt. (2012), doi:10.1016/j.ijleo.2011.12.038
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ARTICLE IN PRESS
IJLEO-52073; No. of Pages 4
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40
2
(arb. unit)
(arb.unit)
80
(a)
Re[χ(3) ] (1) Im[χ ]
4
3
0
0
-2 -40 -4 -8
-4
0
4
Δ3
-80 -3
8
-2
-1
0
Δ3
1
2
3
40
(b)
(arb.unit)
20
Fig. 4. The Kerr nonlinearity (solid line) and the contribution from the additional term owing to the open system (dotted line) as a function of 3 . The other parameters are the same as those in Fig. 2.
0 40
-40
-4
-2
0
2
Δ3
(arb. unit)
-20
4
Fig. 2. The Kerr nonlinearity Re[(3) ] (solid curve) and linear absorption Im[(1) ] (dash curve) with different SGC. The parameters are chosen as 2 = 0, 2 = 10 3 = 1, G0 = 4.0, r0 = 0.03, W1 = 0.02, and W2 = 0.01.
We are currently discussing the physical mechanism leading to the difference mentioned above. From Eq. (3), K is an additional term from the open system, and this term does not exist for the closed system. We show the contribution from the additional term to the Kerr nonlinearity in Fig. 4, Re[(3) ] is also displayed for comparison. Obviously, the occurrence of the additional term from the open system cumbers greatly the enhancement of the Kerr nonlinearity. Finally we want to discuss the method of enhancing the Kerr nonlinearity in the open system. From Ref. [26], we know that the properties of linear susceptibility depend on the ratio of the atomic
(a)
Re[χ(3) ] (1) Im[χ ]
4
(arb. unit)
0 -2 -4 -4
0
Δ3
4
8
100
(b)
(arb. unit)
-40 -3
-2
-1
0
Δ3
1
2
3
Fig. 5. The Kerr nonlinearity under different ratios of the atomic injection rates. The other parameters are the same as those in Fig. 2.
injection rates s = W1 /W2 . Then if the parameter can affect the thirdorder susceptibility? Fig. 5 shows the refractive part of the thirdorder susceptibility for different values of s. It is shown that the Kerr nonlinearity is enhanced with a decrease of s values. In other words, we can enhance the Kerr nonlinearity by changing the ratio of the atomic injection rates. Especially, the maximal value of the Kerr nonlinearity is enhanced by about 15 times at s = 1/9 in comparison with Fig. 2(a). 4. Conclusions
Acknowledgement This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 10904015 and 11004030).
50
0
References
×20 -50
-100
0
In conclusion, we have investigated the properties of the Kerr nonlinearity in an open three-level V-type system with SGC. It is shown that the Kerr nonlinearity is enhanced greatly with vanishing linear absorption via SGC. While the maximal value of the Kerr nonlinearity is smaller as compared with that in the closed system due to the additional term from the open system. Additionally, we discuss the method to enhance the Kerr nonlinearity in the open V-type system.
2
-8
s=2 s=1 s=1/9
-4
-2
0
Δ3
2
4
Fig. 3. The same as Fig. 2, but with a closed system r0 = W1 = W2 = 0.
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Please cite this article in press as: Y. Bai, et al., Giant Kerr nonlinearity in an open V-type system with spontaneously generated coherence, Optik - Int. J. Light Electron Opt. (2012), doi:10.1016/j.ijleo.2011.12.038