- Email: [email protected]
Comparison of transient process between open and closed ladder-type systems with spontaneously generated coherence and incoherent pumping
Optics Communications 282 (2009) 2547–2551
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Comparison of transient process between open and closed ladder-type systems with spontaneously generated coherence and incoherent pumping Zhi-Hong Xiao *, Kisik Kim * Department of Physics, Inha University, Incheon 402-751, Republic of Korea
a r t i c l e
i n f o
Article history: Received 29 September 2008 Received in revised form 15 March 2009 Accepted 15 March 2009
PACS: 42.50.Gy 42.50.Hz
a b s t r a c t We theoretically investigated transient response of open and closed three-level ladder-type atomic system with or without the spontaneously generated coherence (SGC) which could be satisfied with the help of an incoherent pumping. The existence of the SGC effect makes the open and closed system to be distinguished. We compared transient response of weak probe between open and closed system and found that transient properties exhibit different features by adjusting some related parameters, such as the relative phase between probe and coupling fields, the angle between two dipole moments. Crown Copyright Ó 2009 Published by Elsevier B.V. All rights reserved.
Keywords: Transient response Spontaneously generated coherence Open and closed atomic system Incoherent pumping Equispaced levels
1. Introduction Recently, many works have focused on the transient research related to spontaneously generated coherence (SGC) [1–10], which is also known as vacuum-induced coherence (VIC). Many important physical phenomena have been accomplished in the theoretical scheme with the SGC, such as coherence population trapping [11,12], Kerr nonlinearity [13–15], group velocity [16], dark states [8], optical bistability and multistability [17], and so on. Also, experimental investigations to stimulate the SGC have been reported through electromagnetically induced transparency features in a four-level ^-type system [18]. Most of previous works with respect to the SGC focused on the steady-state response of the medium. While Xu et al. have made investigations of the SGC effect in transient process in closed three-level ^-type and _-type systems with near-degenerate levels in the case of a weak probe [19,20]. In this paper, we investigate the SGC effect in transient process in open and closed three-level ladder-type systems with neardegenerated levels and incoherent pumping. In this kind of model, the high excited state of atoms with same transition frequency and non-orthogonal dipole moment is more easily obtained than in ^-type and _-type using external static electronic field and * Corresponding authors. Tel.: +82 32 860 7650; fax: +82 32 872 7562. E-mail addresses: [email protected] (Z.-H. Xiao), [email protected] (K. Kim).
external static magnetic field. It is organized as follows: in Section 2, the model and density matrix equations are presented; in Section 3, we analyze and discuss numerical results using different parameters with or without the SGC; finally, some useful conclusions are achieved in Section 4. 2. The model and density matrix equations An open three-level ladder-type system is considered as shown in Fig. 1. A coupling field is applied on transition j2i $ j3i with frequency (xc) and Rabi frequency (Gc). Rabi frequencies are denoted lab ec =h and Gp ¼ ~ lac ep =h, respectively. A probe field as Gc ¼ ~ with frequency (xp) and Rabi frequency (Gp) is applied on transition j1i $ j2i. An incoherent pumping with rate (2R) acts on transition j1i $ j3i. c1(c2) is the spontaneous emission rate from state j2iðj3iÞ to state j1iðj2iÞ. J1 and J2 are atomic injection rates of levels j1iandj2i, respectively. r0 is atomic exit rate from the cavity. Let J1 + J2 = r0 for keeping the total number of atoms as a constant (this system will be a closed one when J1 = J2 = r0 = 0). The detunings of corresponding fields are d1 = x12 xp, d2 = x23 xc. In the case of near equispaced levels, the SGC could occur in the ladder-type sysl12 and ~ l23 is defined tem. The alignment of two dipole moments ~ lac ~ lab Þ=ðj~ lab j j~ lac jÞ, where h is the angle between to be cos h ¼ ~ two dipole moments. Then, the density matrix of the ladder-type system involving the SGC is written as:
0030-4018/$ - see front matter Crown Copyright Ó 2009 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.03.032
2548
Z.-H. Xiao, K. Kim / Optics Communications 282 (2009) 2547–2551
r0
γ2
J2
GC R
where Gc is replaced by gc, Gp is replaced by gp gc and gp are treated pffiffiffiffiffiffiffiffiffiffi as real parameters. g is replaced by gu exp (iU). Term 2 c1 c2 g/ expðiUÞ cosðhÞ represents quantum interference. According to pffiffiffiffiffiffiffiffiffiffi term 2 c1 c2 g/ expðiUÞ cosðhÞ we know that the SGC depends on the spontaneous emission coherence in this ladder-type system. While in the case of the weak probe field, there is no population distribution (no spontaneous emission) in the highest level j3i. In the case of using incoherent pumping, the population of ground state j1i could be pumped to the highest level j3i, so the SGC condition could be realized with an incoherent pumping.
3
δ2
δ1
2
γ1
r0 Gp
J1
3. Numerical results and analysis
1
r0 Fig. 1. The schematic of an open three-level ladder-type atomic system. The system will be a closed system if r0 = J1 = J2 = 0.
@ q11 ¼ ð2R þ r 0 Þq11 þ 2c1 q22 @t þ iGp q21 iGp q12 þ J 1 @ q22 ¼ 2c2 q33 ð2c1 þ r 0 Þq22 @t iGp q21 þ iGp q12 iGc q23 þ iGc q32 þ J 2 @ q23 ¼ ðc2 þ c1 þ id2 Þq23 þ @t iGc ðq33 q22 Þ þ iGp q13 @ q12 ¼ ðR þ c1 þ id1 Þq12 þ iGp ðq22 q11 Þ @t pffiffiffiffiffiffiffiffiffiffi iGc q13 þ 2 c1 c2 g cosðhÞq23 @ q13 ¼ ½c2 þ R þ iðd1 þ d2 Þq13 þ iGp q23 @t iGc q12 3 X
ð1Þ
qii ¼ 1; qij ¼ qji ði; j ¼ 1; 2; 3; i–jÞ
i¼1
The existence of the SGC effect makes system to be extremely sensitive to the phase of probe and coupling fields. So Rabi frequencies as complex parameters are considered. up and uc are defined as probe field phase and coupling field phase. There are Gc = gc exp (iuc), Gp = gp exp (iup). Phase difference between probe and coupling field is U = up uc. Redefining density matrix elements in Eq. (1) as rii ¼ qii r12 ¼ q12 expði/p Þ, r13 = q13 exp (iu), r23 = q23 exp (iuc), u = up + uc. The density matrix of the ladder-type system involving the SGC is eventually given as:
As well known, gain-absorption and refractive index of the probe field on transition j1i $ j2i are proportional to imaginary and real part of r12, which can be obtained from Eq. (2). If (Im(r12) > 0), the probe laser will be amplified. In Fig. 2, with c1 = 1.2, c2 = 1, R = 1.05, d1 = 5, d2 = 0, gp = 0.1 sin (h), gc = 5 sin (h), h = p/4, U = p/3, we depicted time evolution of gain-absorption in open system (r0 = 0.6, J2/J1 = 5) and closed system (r0 = J1 = J2 = 0) with the SGC gu = 1 or without the SGC (gu = 0). If gu = 0, the probe field exhibits periodic absorption and oscillation bellow zero-absorption line. The oscillatory behavior reaches steady-state eventually (Im(r12) 0) (absorption disappears). However, when the SGC effect is put into consideration (gu = 1), the transient property is greatly changed. First of all, the amplitude of oscillation is much larger than that of gu = 0, which means that much larger transient gain-absorption is obtained. In open system, it shows periodic amplification or absorption and leaves transient absorption bellow zero-absorption line reaching nonzero steadystate value (Im(r12) – 0, absorption exits) in the end. In closed system, transient absorption exists in a short time at the beginning, only leaves transient gain above zero-absorption line reaching positive nonzero steady-state value in the end. According to Eq. (2), the physical phenomena of open and closed system are affected by same physical parameters (c1, c2, d1, d2, R, gp) when the SGC is absent. So without the SGC, we almost could not distinguish the open system from the closed system. These physical parameters r0, J1, J2 make physical phenomena different. At last, we could use them to make correct judgment when the SGC effect is existence. In Fig. 3a and b, we plot time evolution of gain-absorption in open system (r0 = 0.6, J2/J1 = 5) and closed system (r0 = J1 = J2 = 0) for different values of U. Other parameters are the same as those
@ r11 ¼ ð2R þ r 0 Þr11 þ 2c1 r22 @t þ ig p ðr21 r12 Þ þ J 1 @ r22 ¼ 2c2 r33 ð2c1 þ r0 Þr22 @t ig p ðr21 r12 Þ ig c ðr23 r32 Þ þ J 2 @ r23 ¼ ðc2 þ c1 þ id2 Þr23 @t þ ig c ðr33 r22 Þ þ ig p r13 @ r12 ¼ ðR þ c1 þ id1 Þr12 @t pffiffiffiffiffiffiffiffiffiffi þ ig p ðr22 r11 Þ ig c r13 þ 2 c1 c2 g/ expðiUÞ cosðhÞr23 @ r13 ¼ ½c2 þ R þ iðd1 þ d2 Þr13 @t þ ig p r23 ig c r12 ð2Þ
Fig. 2. Time evolution of the gain-absorption Im(r12) in open system (r0 = 0.6, J2/ J1 = 5) and closed system (r0 = J1 = J2 = 0) with the SGC (gu = 0) or without the SGC (gu = 1). Other parameters: c1 = 1.2, c2 = 1, R = 1.05, d1 = 5, d2 = 0, h = p/4, U = p/3, gp = 0.1 sin (h), gc = 5 sin (h).
Z.-H. Xiao, K. Kim / Optics Communications 282 (2009) 2547–2551
Fig. 3. Time evolution of the gain-absorption Im(r12) in open system (r0 = 0.6, J2/ J1 = 5) and closed system (r0 = J1 = J2 = 0) for different values of relative phase U. Other parameters: gu = 1 (included the SGC), c1 = 1.2, c2 = 1, d1 = 5, R = 1.05, d2 = 0, h = p/4, gp = 0.1 sin (h), gc = 5 sin (h).
used in Fig. 2. We find that transient property exhibits many different features for different relative phase U. In closed system, the probe field no longer exhibits periodic amplification and absorption, as shown in Fig. 3a. When U = 0 and U = p/2, transient absorption exist a short time, only leaves transient gain above zero-absorption line reaching positive steady-state value in the end; when U = p, transient gain almost disappears and transient absorption eventually reaches negative steady-state value; when U = 3p/2, transient gain can be completely eliminated, probe field always exhibits absorption without any amplification in whole transient process and eventually reaches negative steady-state value around the zero-absorption line. However, in open system (see Fig. 3b), the probe field exhibits both periodic amplification and absorption. When U = 0 and U = p/2, transient gain finally disappears, only leaves transient absorption bellow zero-absorption line and reaches negative steady-state value; when U = p and U = 3p/2, transient absorption disappears, only leaves transient gain above zero-absorption line and reaches positive steady-state value. Comparing Fig. 3a with Fig. 3b, different transient phenomena could be found in open and closed system for different relative phase U. In Fig. 4, we depict time evolution of gain-absorption in open system (r0 = 0.6, J2/J1 = 5) and closed system (r0 = J1 = J2 = 0) for different values of h. Other parameters are the same as those used
2549
Fig. 4. Time evolution of the gain-absorption Im(r12) in open system (r0 = 0.6, J2/ J1 = 5) and closed system (r0 = J1 = J2 = 0) for different values of h. Other parameters: gu = 1 (included the SGC), c1 = 1.2, c2 = 1, d1 = 5, R = 1.05, d2 = 0, U = p/3, gp = 0.1 sin (h), gc = 5 sin (h).
in Fig. 2. Different physical features are shown. In closed system (see Fig. 4a), when h = p /9 and h = p/6, transient absorption can be almost eliminated, transient gain eventually goes into steadystate above zero-absorption line; when h = p/3, transient absorption exists a short time, only leaves transient gain to reach positive steady-state value. In open system (see Fig. 4b), it reveal that transient gain can be completely eliminated while transient absorption still exists reaching to steady-state bellow zero-absorption line. We also find that the oscillation amplitude decreases when the value of h increases in both systems. In Fig. 5, time evolution (s) of gain-absorption and refractive index of open system (r0 = = 0.6, J2/J1 = 5) and closed system (r0 = J1 = J2 = 0) is plotted for different values of U. Other parameters are the same as those used in Fig. 3. When U = p/3 and U = p/6, high refractive index without absorption can be achieved in open system, as shown in Fig. 5a. While high refractive index without absorption can not be obtained in closed system. Fig. 6 presents time evolution (s) of population distribution for different relative phase U in open system and closed system with h = p/ 4, gu = 1. In open system, it shows that population distribution curves are completely superposed for different values of U (relative phase U cannot affect population distribution) and finally reaches the same positive steady-state values ðr11 r22 r33 0:333Þ, as shown in Fig. 6a. In closed system, relative phase U
2550
Z.-H. Xiao, K. Kim / Optics Communications 282 (2009) 2547–2551
Fig. 5. Time evolution (s) of the gain-absorption Im(r12) and refractive index Re(r12) in open system (r0 = 0.6, J2/J1 = 5) and closed system (r0 = J1 = J2 = 0) for different values of U. Other parameters: gu = 1 (included the SGC), c1 = 1.2, c2 = 1, R = 1.05, d1 = 5, d2 = 0, h = p /4, gp = 0.1 sin (h), gc = 5 sin (h).
Fig. 6. Time evolution (s) of population distribution in open system (r0 = 0.6, J2/J1 = 5) and closed system (r0 = J1 = J2 = 0) for different values of U. Other parameters: gu = 1 (included the SGC), c1 = 1.2, c2 = 1, R = 1.05, d1 = 5, d2 = 0, h = p/4, gp = 0.1 sin (h), gc = 5 sin (h).
has few effects on population distribution of level j1i (r11), level j2iðr22 Þ and level j3i (r33). Eventually, the population distribution (r11, r22, r33) of these three levels reaches positive steady-state (r11, r33, r22), as shown in Fig. 6b. 4. Conclusions In the paper, we have investigated transient response of the weak probe field in open and closed ladder-type systems with or without the SGC effect. We find that the transient properties of
these two systems are greatly affected by the SGC effect. The gainabsorption of both systems is extremely sensitive to the relative phase U between probe and coupling fields. Transient gain-absorption exhibits different features for different relative phase U. In open system, high refractive index without absorption could be obtained by modulating the relative phase U; however, in the closed system, the result cannot be achieved. Population distribution of closed system in transient process is also affected by the relative phase U; while in open system, population distribution is insensitive to relative phase U. And the angle h between two dipole
Z.-H. Xiao, K. Kim / Optics Communications 282 (2009) 2547–2551
moments also affects the transient gain-absorption. Different h leads to different features of transient gain-absorption. The existence of the SGC makes both systems to be distinguishable in transient process. And the SGC condition could be satisfied in the case of using an incoherent pumping. References [1] [2] [3] [4] [5]
J.-H. Wu, J.-Y. Gao, Phys. Rev. A 65 (2002) 063807. J.-H. Wu, W.-H. Xu, H.-F. Zhang, J.-Y. Gao, Opt. Commun. 206 (2002) 135. J.-H. Wu, H.-F. Zhang, J.-Y. Gao, Opt. Lett. 28 (2003) 654. H.-M. Ma, S.-Q. Gong, et al., Opt. Commun. 223 (2003) 97. C.-P. Liu, S.-Q. Gong, X.-J. Fan, Z.-Z. Xu, Opt. Commun. 231 (2004) 289.
[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
Y.-F. Bai, H. Guo, et al., Phys. Rev. A 69 (2004) 043814. A.-J. Li, J.-Y. Gao, et al., J. Phys. B 38 (2005) 3815. P.R. Berman, Phys. Rev. A 72 (2005) 035801. C.-P. Liu, S.-Q. Gong, et al., Opt. Commun. 254 (2005) 368. A.-J. Li, X.-L. Song, et al., Phys. Rev. A 77 (2008) 053806. S.Y. Zhu, R.C.F. Chan, C.P. Lee, Phys. Rev. A. 52 (1995) 710. S.Y. Zhu, M.O. Scully, Phys. Rev. Lett. 76 (1996) 388. Y.P. Niu, S.Q. Gong, Phys. Rev. A 73 (2006) 053811. X.-A. Y, L.-Q. Wang, et al., Phys. Lett. A 372 (2008) 6456. W.-J. Jiang, X.-A. Yang, et al., Opt. Commun. 282 (2009) 101. S. Dutta, K.R. Dastidar, J. Phys. B 40 (2007) 4287. D.-C. Chen, C.-P. Liu, S.-Q. Gong, Phys. Lett. A 332 (2004) 244. C.-L. Wang, A.-J. Li, et al., Opt. Lett. 33 (2008) 687. W.H. Xu, J.H. Wu, J.Y. Gao, Opt. Commun. 215 (2003) 345. W.H. Xu, J.H. Wu, J.Y. Gao, Phys. Rev. A 66 (2002) 063812.
2551
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Comparison of transient process between open and closed ladder-type systems with spontaneously generated coherence and incoherent pumping Zhi-Hong Xiao *, Kisik Kim * Department of Physics, Inha University, Incheon 402-751, Republic of Korea
a r t i c l e
i n f o
Article history: Received 29 September 2008 Received in revised form 15 March 2009 Accepted 15 March 2009
PACS: 42.50.Gy 42.50.Hz
a b s t r a c t We theoretically investigated transient response of open and closed three-level ladder-type atomic system with or without the spontaneously generated coherence (SGC) which could be satisfied with the help of an incoherent pumping. The existence of the SGC effect makes the open and closed system to be distinguished. We compared transient response of weak probe between open and closed system and found that transient properties exhibit different features by adjusting some related parameters, such as the relative phase between probe and coupling fields, the angle between two dipole moments. Crown Copyright Ó 2009 Published by Elsevier B.V. All rights reserved.
Keywords: Transient response Spontaneously generated coherence Open and closed atomic system Incoherent pumping Equispaced levels
1. Introduction Recently, many works have focused on the transient research related to spontaneously generated coherence (SGC) [1–10], which is also known as vacuum-induced coherence (VIC). Many important physical phenomena have been accomplished in the theoretical scheme with the SGC, such as coherence population trapping [11,12], Kerr nonlinearity [13–15], group velocity [16], dark states [8], optical bistability and multistability [17], and so on. Also, experimental investigations to stimulate the SGC have been reported through electromagnetically induced transparency features in a four-level ^-type system [18]. Most of previous works with respect to the SGC focused on the steady-state response of the medium. While Xu et al. have made investigations of the SGC effect in transient process in closed three-level ^-type and _-type systems with near-degenerate levels in the case of a weak probe [19,20]. In this paper, we investigate the SGC effect in transient process in open and closed three-level ladder-type systems with neardegenerated levels and incoherent pumping. In this kind of model, the high excited state of atoms with same transition frequency and non-orthogonal dipole moment is more easily obtained than in ^-type and _-type using external static electronic field and * Corresponding authors. Tel.: +82 32 860 7650; fax: +82 32 872 7562. E-mail addresses: [email protected] (Z.-H. Xiao), [email protected] (K. Kim).
external static magnetic field. It is organized as follows: in Section 2, the model and density matrix equations are presented; in Section 3, we analyze and discuss numerical results using different parameters with or without the SGC; finally, some useful conclusions are achieved in Section 4. 2. The model and density matrix equations An open three-level ladder-type system is considered as shown in Fig. 1. A coupling field is applied on transition j2i $ j3i with frequency (xc) and Rabi frequency (Gc). Rabi frequencies are denoted lab ec =h and Gp ¼ ~ lac ep =h, respectively. A probe field as Gc ¼ ~ with frequency (xp) and Rabi frequency (Gp) is applied on transition j1i $ j2i. An incoherent pumping with rate (2R) acts on transition j1i $ j3i. c1(c2) is the spontaneous emission rate from state j2iðj3iÞ to state j1iðj2iÞ. J1 and J2 are atomic injection rates of levels j1iandj2i, respectively. r0 is atomic exit rate from the cavity. Let J1 + J2 = r0 for keeping the total number of atoms as a constant (this system will be a closed one when J1 = J2 = r0 = 0). The detunings of corresponding fields are d1 = x12 xp, d2 = x23 xc. In the case of near equispaced levels, the SGC could occur in the ladder-type sysl12 and ~ l23 is defined tem. The alignment of two dipole moments ~ lac ~ lab Þ=ðj~ lab j j~ lac jÞ, where h is the angle between to be cos h ¼ ~ two dipole moments. Then, the density matrix of the ladder-type system involving the SGC is written as:
0030-4018/$ - see front matter Crown Copyright Ó 2009 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.03.032
2548
Z.-H. Xiao, K. Kim / Optics Communications 282 (2009) 2547–2551
r0
γ2
J2
GC R
where Gc is replaced by gc, Gp is replaced by gp gc and gp are treated pffiffiffiffiffiffiffiffiffiffi as real parameters. g is replaced by gu exp (iU). Term 2 c1 c2 g/ expðiUÞ cosðhÞ represents quantum interference. According to pffiffiffiffiffiffiffiffiffiffi term 2 c1 c2 g/ expðiUÞ cosðhÞ we know that the SGC depends on the spontaneous emission coherence in this ladder-type system. While in the case of the weak probe field, there is no population distribution (no spontaneous emission) in the highest level j3i. In the case of using incoherent pumping, the population of ground state j1i could be pumped to the highest level j3i, so the SGC condition could be realized with an incoherent pumping.
3
δ2
δ1
2
γ1
r0 Gp
J1
3. Numerical results and analysis
1
r0 Fig. 1. The schematic of an open three-level ladder-type atomic system. The system will be a closed system if r0 = J1 = J2 = 0.
@ q11 ¼ ð2R þ r 0 Þq11 þ 2c1 q22 @t þ iGp q21 iGp q12 þ J 1 @ q22 ¼ 2c2 q33 ð2c1 þ r 0 Þq22 @t iGp q21 þ iGp q12 iGc q23 þ iGc q32 þ J 2 @ q23 ¼ ðc2 þ c1 þ id2 Þq23 þ @t iGc ðq33 q22 Þ þ iGp q13 @ q12 ¼ ðR þ c1 þ id1 Þq12 þ iGp ðq22 q11 Þ @t pffiffiffiffiffiffiffiffiffiffi iGc q13 þ 2 c1 c2 g cosðhÞq23 @ q13 ¼ ½c2 þ R þ iðd1 þ d2 Þq13 þ iGp q23 @t iGc q12 3 X
ð1Þ
qii ¼ 1; qij ¼ qji ði; j ¼ 1; 2; 3; i–jÞ
i¼1
The existence of the SGC effect makes system to be extremely sensitive to the phase of probe and coupling fields. So Rabi frequencies as complex parameters are considered. up and uc are defined as probe field phase and coupling field phase. There are Gc = gc exp (iuc), Gp = gp exp (iup). Phase difference between probe and coupling field is U = up uc. Redefining density matrix elements in Eq. (1) as rii ¼ qii r12 ¼ q12 expði/p Þ, r13 = q13 exp (iu), r23 = q23 exp (iuc), u = up + uc. The density matrix of the ladder-type system involving the SGC is eventually given as:
As well known, gain-absorption and refractive index of the probe field on transition j1i $ j2i are proportional to imaginary and real part of r12, which can be obtained from Eq. (2). If (Im(r12) > 0), the probe laser will be amplified. In Fig. 2, with c1 = 1.2, c2 = 1, R = 1.05, d1 = 5, d2 = 0, gp = 0.1 sin (h), gc = 5 sin (h), h = p/4, U = p/3, we depicted time evolution of gain-absorption in open system (r0 = 0.6, J2/J1 = 5) and closed system (r0 = J1 = J2 = 0) with the SGC gu = 1 or without the SGC (gu = 0). If gu = 0, the probe field exhibits periodic absorption and oscillation bellow zero-absorption line. The oscillatory behavior reaches steady-state eventually (Im(r12) 0) (absorption disappears). However, when the SGC effect is put into consideration (gu = 1), the transient property is greatly changed. First of all, the amplitude of oscillation is much larger than that of gu = 0, which means that much larger transient gain-absorption is obtained. In open system, it shows periodic amplification or absorption and leaves transient absorption bellow zero-absorption line reaching nonzero steadystate value (Im(r12) – 0, absorption exits) in the end. In closed system, transient absorption exists in a short time at the beginning, only leaves transient gain above zero-absorption line reaching positive nonzero steady-state value in the end. According to Eq. (2), the physical phenomena of open and closed system are affected by same physical parameters (c1, c2, d1, d2, R, gp) when the SGC is absent. So without the SGC, we almost could not distinguish the open system from the closed system. These physical parameters r0, J1, J2 make physical phenomena different. At last, we could use them to make correct judgment when the SGC effect is existence. In Fig. 3a and b, we plot time evolution of gain-absorption in open system (r0 = 0.6, J2/J1 = 5) and closed system (r0 = J1 = J2 = 0) for different values of U. Other parameters are the same as those
@ r11 ¼ ð2R þ r 0 Þr11 þ 2c1 r22 @t þ ig p ðr21 r12 Þ þ J 1 @ r22 ¼ 2c2 r33 ð2c1 þ r0 Þr22 @t ig p ðr21 r12 Þ ig c ðr23 r32 Þ þ J 2 @ r23 ¼ ðc2 þ c1 þ id2 Þr23 @t þ ig c ðr33 r22 Þ þ ig p r13 @ r12 ¼ ðR þ c1 þ id1 Þr12 @t pffiffiffiffiffiffiffiffiffiffi þ ig p ðr22 r11 Þ ig c r13 þ 2 c1 c2 g/ expðiUÞ cosðhÞr23 @ r13 ¼ ½c2 þ R þ iðd1 þ d2 Þr13 @t þ ig p r23 ig c r12 ð2Þ
Fig. 2. Time evolution of the gain-absorption Im(r12) in open system (r0 = 0.6, J2/ J1 = 5) and closed system (r0 = J1 = J2 = 0) with the SGC (gu = 0) or without the SGC (gu = 1). Other parameters: c1 = 1.2, c2 = 1, R = 1.05, d1 = 5, d2 = 0, h = p/4, U = p/3, gp = 0.1 sin (h), gc = 5 sin (h).
Z.-H. Xiao, K. Kim / Optics Communications 282 (2009) 2547–2551
Fig. 3. Time evolution of the gain-absorption Im(r12) in open system (r0 = 0.6, J2/ J1 = 5) and closed system (r0 = J1 = J2 = 0) for different values of relative phase U. Other parameters: gu = 1 (included the SGC), c1 = 1.2, c2 = 1, d1 = 5, R = 1.05, d2 = 0, h = p/4, gp = 0.1 sin (h), gc = 5 sin (h).
used in Fig. 2. We find that transient property exhibits many different features for different relative phase U. In closed system, the probe field no longer exhibits periodic amplification and absorption, as shown in Fig. 3a. When U = 0 and U = p/2, transient absorption exist a short time, only leaves transient gain above zero-absorption line reaching positive steady-state value in the end; when U = p, transient gain almost disappears and transient absorption eventually reaches negative steady-state value; when U = 3p/2, transient gain can be completely eliminated, probe field always exhibits absorption without any amplification in whole transient process and eventually reaches negative steady-state value around the zero-absorption line. However, in open system (see Fig. 3b), the probe field exhibits both periodic amplification and absorption. When U = 0 and U = p/2, transient gain finally disappears, only leaves transient absorption bellow zero-absorption line and reaches negative steady-state value; when U = p and U = 3p/2, transient absorption disappears, only leaves transient gain above zero-absorption line and reaches positive steady-state value. Comparing Fig. 3a with Fig. 3b, different transient phenomena could be found in open and closed system for different relative phase U. In Fig. 4, we depict time evolution of gain-absorption in open system (r0 = 0.6, J2/J1 = 5) and closed system (r0 = J1 = J2 = 0) for different values of h. Other parameters are the same as those used
2549
Fig. 4. Time evolution of the gain-absorption Im(r12) in open system (r0 = 0.6, J2/ J1 = 5) and closed system (r0 = J1 = J2 = 0) for different values of h. Other parameters: gu = 1 (included the SGC), c1 = 1.2, c2 = 1, d1 = 5, R = 1.05, d2 = 0, U = p/3, gp = 0.1 sin (h), gc = 5 sin (h).
in Fig. 2. Different physical features are shown. In closed system (see Fig. 4a), when h = p /9 and h = p/6, transient absorption can be almost eliminated, transient gain eventually goes into steadystate above zero-absorption line; when h = p/3, transient absorption exists a short time, only leaves transient gain to reach positive steady-state value. In open system (see Fig. 4b), it reveal that transient gain can be completely eliminated while transient absorption still exists reaching to steady-state bellow zero-absorption line. We also find that the oscillation amplitude decreases when the value of h increases in both systems. In Fig. 5, time evolution (s) of gain-absorption and refractive index of open system (r0 = = 0.6, J2/J1 = 5) and closed system (r0 = J1 = J2 = 0) is plotted for different values of U. Other parameters are the same as those used in Fig. 3. When U = p/3 and U = p/6, high refractive index without absorption can be achieved in open system, as shown in Fig. 5a. While high refractive index without absorption can not be obtained in closed system. Fig. 6 presents time evolution (s) of population distribution for different relative phase U in open system and closed system with h = p/ 4, gu = 1. In open system, it shows that population distribution curves are completely superposed for different values of U (relative phase U cannot affect population distribution) and finally reaches the same positive steady-state values ðr11 r22 r33 0:333Þ, as shown in Fig. 6a. In closed system, relative phase U
2550
Z.-H. Xiao, K. Kim / Optics Communications 282 (2009) 2547–2551
Fig. 5. Time evolution (s) of the gain-absorption Im(r12) and refractive index Re(r12) in open system (r0 = 0.6, J2/J1 = 5) and closed system (r0 = J1 = J2 = 0) for different values of U. Other parameters: gu = 1 (included the SGC), c1 = 1.2, c2 = 1, R = 1.05, d1 = 5, d2 = 0, h = p /4, gp = 0.1 sin (h), gc = 5 sin (h).
Fig. 6. Time evolution (s) of population distribution in open system (r0 = 0.6, J2/J1 = 5) and closed system (r0 = J1 = J2 = 0) for different values of U. Other parameters: gu = 1 (included the SGC), c1 = 1.2, c2 = 1, R = 1.05, d1 = 5, d2 = 0, h = p/4, gp = 0.1 sin (h), gc = 5 sin (h).
has few effects on population distribution of level j1i (r11), level j2iðr22 Þ and level j3i (r33). Eventually, the population distribution (r11, r22, r33) of these three levels reaches positive steady-state (r11, r33, r22), as shown in Fig. 6b. 4. Conclusions In the paper, we have investigated transient response of the weak probe field in open and closed ladder-type systems with or without the SGC effect. We find that the transient properties of
these two systems are greatly affected by the SGC effect. The gainabsorption of both systems is extremely sensitive to the relative phase U between probe and coupling fields. Transient gain-absorption exhibits different features for different relative phase U. In open system, high refractive index without absorption could be obtained by modulating the relative phase U; however, in the closed system, the result cannot be achieved. Population distribution of closed system in transient process is also affected by the relative phase U; while in open system, population distribution is insensitive to relative phase U. And the angle h between two dipole
Z.-H. Xiao, K. Kim / Optics Communications 282 (2009) 2547–2551
moments also affects the transient gain-absorption. Different h leads to different features of transient gain-absorption. The existence of the SGC makes both systems to be distinguishable in transient process. And the SGC condition could be satisfied in the case of using an incoherent pumping. References [1] [2] [3] [4] [5]
J.-H. Wu, J.-Y. Gao, Phys. Rev. A 65 (2002) 063807. J.-H. Wu, W.-H. Xu, H.-F. Zhang, J.-Y. Gao, Opt. Commun. 206 (2002) 135. J.-H. Wu, H.-F. Zhang, J.-Y. Gao, Opt. Lett. 28 (2003) 654. H.-M. Ma, S.-Q. Gong, et al., Opt. Commun. 223 (2003) 97. C.-P. Liu, S.-Q. Gong, X.-J. Fan, Z.-Z. Xu, Opt. Commun. 231 (2004) 289.
[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
Y.-F. Bai, H. Guo, et al., Phys. Rev. A 69 (2004) 043814. A.-J. Li, J.-Y. Gao, et al., J. Phys. B 38 (2005) 3815. P.R. Berman, Phys. Rev. A 72 (2005) 035801. C.-P. Liu, S.-Q. Gong, et al., Opt. Commun. 254 (2005) 368. A.-J. Li, X.-L. Song, et al., Phys. Rev. A 77 (2008) 053806. S.Y. Zhu, R.C.F. Chan, C.P. Lee, Phys. Rev. A. 52 (1995) 710. S.Y. Zhu, M.O. Scully, Phys. Rev. Lett. 76 (1996) 388. Y.P. Niu, S.Q. Gong, Phys. Rev. A 73 (2006) 053811. X.-A. Y, L.-Q. Wang, et al., Phys. Lett. A 372 (2008) 6456. W.-J. Jiang, X.-A. Yang, et al., Opt. Commun. 282 (2009) 101. S. Dutta, K.R. Dastidar, J. Phys. B 40 (2007) 4287. D.-C. Chen, C.-P. Liu, S.-Q. Gong, Phys. Lett. A 332 (2004) 244. C.-L. Wang, A.-J. Li, et al., Opt. Lett. 33 (2008) 687. W.H. Xu, J.H. Wu, J.Y. Gao, Opt. Commun. 215 (2003) 345. W.H. Xu, J.H. Wu, J.Y. Gao, Phys. Rev. A 66 (2002) 063812.
2551