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Transient properties in a resonant ladder-type atomic system with vacuum-induced coherence and incoherent pumping
Optik 124 (2013) 973–976
Contents lists available at SciVerse ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Transient properties in a resonant ladder-type atomic system with vacuum-induced coherence and incoherent pumping Zhiqiang Zeng a,b,∗ , Yuping Wang c , Bangpin Hou d , Zenghui Gao a,b a
College of Physics and Electronic Engineering, Yibin University, Yibin 644000, China Computational Physics Key Laboratory of Sichuan Province, Yibin University, Yibin 644000, China College of Mining and Safety Engineering, Yibin University, Yibin 644000, China d College of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610066, China b c
a r t i c l e
i n f o
Article history: Received 28 September 2012 Accepted 30 January 2013
PACS: 42.50.Gy 42.50.Hz
a b s t r a c t The transient absorption and population in a resonant ladder-type atomic system with or without an incoherent pumping field are theoretically investigated. We find that the vacuum-induced coherence and the relative phase between probe and coupling fields can clearly affect the transient absorption (gain) properties, but the steady-state value of the probe absorption (gain) is remarkably dependent on the incoherent pumping. And the steady population distribution is sensitive to the incoherent pumping and insensitive to the relative phase. Additionally, it is shown that the lasing with inversion can be reached with the incoherent pumping. © 2013 Elsevier GmbH. All rights reserved.
Keywords: Transient properties Vacuum-induced coherence Incoherent pumping Relative phase
1. Introduction The interference induced by a strong resonant coupling field makes an opaque atomic system transparent for a probe field. This phenomenon, which is termed as electromagnetically induced transparency (EIT), has been studied in all kinds of three-level atomic systems with ∧,∨ and ladder configurations [1,2]. Another way of generating quantum interference connects with relaxation processes such as spontaneous emission, i.e., vacuum-induced coherence (VIC), which is also known as spontaneously generated coherence (SGC). It is created by the interference of spontaneous emission of either a single excited level to two closely lying atomic levels (∧-type system) [3–5], or by two closely lying atomic levels to a single ground level (∨-type system) [6–8] with the same vacuum radiation field. In a ladder-type system [9–11], it can also be created in a nearly equispaced atomic levels case. A theoretical study was made of the steady-state intensity and squeezing properties of the fluorescent light from a three-level ladder system, which was subject to spontaneous emission decay to the electromagnetic-field vacuum [10]. And the quantum interference
∗ Corresponding author at: College of Physics and Electronic Engineering, Yibin University, Yibin 644000, China. E-mail address: zhiqiang [email protected] (Z. Zeng). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.01.031
in two-photon excitation of a three-level ladder system interacting with squeezed vacuum was considered [12]. Ma et al. found that the population inversion can be enhanced on one of the optical transitions due to the VIC effect [13]. Fan et al. shown that lasing without inversion can be realized in an open ladder system without incoherent pumping [14]. Recently, Kumar and Singh presented a general theory of EIT in inhomogeneously broadened ladder system in which Doppler broadening of both one- and two-photon transitions can occur [15]. On the other hand, experimental investigations have been reported through EIT features in a three-level ladder system [16–20]. Besides, many theoretical and experimental works have focused on the transient research in order to understand the dynamical process [21–32], which is important due to its novel application in the optical switch. For example, Xu et al. discussed the effect of VIC on the transient process in a three-level ∧-type system [25]. With the nonresonant probe field, Xiao and Kim mainly compared the transient response between open and closed systems [29]. The dynamical behaviors of the dispersion and absorption in a three-level ∧-type system were investigated [31]. In the previous works, our group studied the effects of VIC on the stationary-state response in a multilevel atomic system [33–36]. In this paper, we investigate the transient absorption and population in a resonant ladder-type atomic system with VIC and incoherent pumping. It is well displayed that due to the effects of VIC and incoherent pumping, there
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Fig. 1. The energy scheme of the three-level ladder-type atomic system.
appear some interesting transient properties in the ladder-type system. 2. Model and equations A three-level ladder-type system is shown in Fig. 1, which has been experimentally studied [18] by the 87 Rb atom with 5S1/2 , 5P3/2 and 4D5/2 behaving as the |1, |2 and |3 state labels, respectively. A weak probe field with frequency ωpandRabi frequency Gp = ៝ 12 · ε៝ p / is applied on the transition 1 ↔ 2 , and a strong coherent coupling field with frequency ωc and Rabi frequency Gc = ៝ 23 · ε៝ c / drives the transition 2 ↔ 3 . An incoherent pumping field with
rate 2R interacts with the transition 1 ↔ 3 . 21 and 32 are the spontaneous decay rates from the level |2> to the level |1> and from the level |3> to the level |2>, respectively. In the interaction picture the density–matrix equations of motion in the dipole and rotating-wave approximations can be written as
˙ 11 = −2R11 + 221 22 + iGp (21 − 12 )
(1)
˙ 33 = 2R11 − 232 33 + iGc (23 − 32 )
(2)
˙ 12 = −(R + 21 + ip )12 + iGp (22 − 11 ) − iGc 13 √ +2 32 21 · exp(i˚) · cos · 23
(3)
˙ 23 = −(32 + 21 + ic )23 + iGc (33 − 22 ) + iGp 13
(4)
˙ 13 = −[R + 32 + i(p + c )]13 − iGc 12 + iGp 23
(5)
The above density–matrix elements additionally obey the nor3 malization and Hermitian condition = 1 and ij = ji∗ . i=1 ii The detunings of the probe and coupling lasers are defined as p = ω21 − ωp and c = ω32 − ωc , respectively. In the case of nearly √ equispaced levels, an additional term 2 32 21 · exp(i˚) · cos · 23 of the optical Bloch equations represents the effect of VIC. Where ˚ = ϕp − ϕc is the phase difference between the probe and the coupling fields, and is the angle between the two induced ៝ 12 and ៝ 23 . It is obviously that when = 1, the VIC dipole moments effect is included and its strength will vary with the value of ; otherwise = 0, the VIC effect is not included. As is well known, in the limit of a weak probe, coefficient for the probe the gain-absorption laser on transition 1 ↔ 2 is proportional to the imaginary part of 12 , which can be obtained from Eqs. (1) to (5). In our notation, the system exhibits gain for the probe laser if Im(12 ) > 0. Considering the case of two resonant transitions (p = c = 0), with the initial conditions 11 (0) = 22 (0) = 0.5 and other ij (0) = 0(i,j = 1,2,3), the following discussions will be deployed based on the timedependent numerical solutions of Eqs. (1)–(5).
Fig. 2. Time evolution of the absorption coefficient Im(12 ), where parameter values are 21 = 1, 32 = 1, Gp = 0.1 sin , Gc = 5 sin , p = 0, c = 0, ˚ = 0. (a) and (c) R = 0 (solid curve), R = 1 (dashed curve), R = 3 (dotted curve); (b) and (d) = /20 (solid curve), = /4 (dashed curve), = 9/20 (dotted curve).
Z. Zeng et al. / Optik 124 (2013) 973–976
3. The numerical analysis of transient process In Fig. 2 with the parameters given by 21 = 32 = 1, Gp = 0.4 sin , Gc = 5 sin , c = 0, p = 0, ˚ = 0, we plot the time evolution of the absorption coefficient Im(12 ). First, we consider the influence of incoherent pumping field without the VIC ( = 0), which is shown in Fig. 2(a). When the incoherent pumping is not included in this system (seen solid curve in Fig. 2(a)), the probe absorption displays oscillatory behavior versus time, but it always exhibits considerable large absorption without any gain during the total response process, including the transient and steady process. When considering the incoherent pumping, the atomic dynamical behavior is shown by the dashed curve (R = 1) and dotted curve (R = 3) in Fig. 2(a). It can be seen that the oscillatory amplitude of the transient absorption affected by incoherent pumping becomes much smaller than that in the absence of incoherent pumping. And the response time for the atomic medium from oscillatory region to the steady-state becomes shorter with increasing of R. Besides, in the presence of incoherent pumping, it is clear that the probe field can be amplified. Especially, with R = 3, the transient absorption disappears, only leaves the transient gain to reach positive steady-state value. So the probe field can be changed from absorption to gain due to incoherent pumping. In Fig. 2(b) without incoherent pumping, for small value of the angle , such as = /20 (seen solid curve in Fig. 2(b)), it also always exhibits considerable large absorption without any gain during the total response process. It is similar to the case where the VIC is not included, except that the oscillatory frequency becomes smaller, which is caused by the angle . However, the probe laser exhibits different features for different values of . It is shown that the transient gain can be appeared, and the oscillatory amplitude of
Fig. 3. Time evolution of the absorption coefficient Im(12 ) for different values of ˚: ˚ = 0 (or 2) (solid curve), = /2 (or 3/2) (dashed curve), and ˚ = (dotted curve), where the other parameter values are 21 = 1, 32 = 1, Gp = 0.1 sin , Gc = 5 sin , p = 0, c = 0, = 1, = /4. (a) R = 0; (b) R = 1.
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the transient absorption decreases with increasing of . Moreover, the absolute value of steady-state absorption also decreases with increasing of , but there is not positive steady-state value. Thus, without incoherent pumping, the probe field cannot be amplified in the presence of VIC, which is in good agreement with Ref. [37]. Fig. 2(c) and (d) depict the situations when both the VIC and incoherent pumping are present. From Fig. 2(c) with = /4, the phenomenon that the steady-state value from negative to positive and weakening of oscillatory behavior with R increasing is similar to the situation of = 0. Comparing Fig. 2(a) with (c), it can be found it is impossible to only appear transient gain or transient absorption when the VIC effect is existence and = /4. However, it is different the case of = /20, as shown the solid curve in Fig. 2(d), transient gain can be almost eliminated, and transient absorption rapidly goes into the steady-state value Im(12 ) ≈ 0. Moreover, Fig. 2(d) shows that with incoherent pumping, the transient response is different for the different values of , but the steady-state value of Im(12 ) is kept at the same value, which is different to the result without incoherent pumping in Fig. 2(b). Above all, it is clear that the steady-state value of the probe absorption coefficient is related to incoherent pumping, whether the VIC effect is included or not. In the following, let us investigate the effect of the relative phase ˚ between the probe and the coupling fields on the atomic absorption. It is noted that, there is no effect of the relative phase ˚ on the atomic optical properties when = 0. This is because that the relative phase appears in the Eq. (3) proportional to exp(i ). So we should study the influence of the relative phase ˚ under the condition of = / 0. In Fig. 3 with = 1 and = /4, we plot the time evolution of the absorption coefficient Im(12 ) for different values of ˚. It is apparently shown that the transient property is greatly altered with the relative phase and exhibits periodic absorption
Fig. 4. Time evolution of the population distribution, where parameter values are 21 = 1, 32 = 1, Gp = 0.1 sin , Gc = 5 sin , p = 0, c = 0, = /4, = 0, = 0 (or = 1). (a) R = 0; (b) R = 3.
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and amplification. The curve of ˚ = 3/2 and the curve of ˚ = /2 are completely superposed. The solid and the dotted curves corresponding to Fig. 3(a) and (b) are symmetric about Im(12 ) = −0.005 and Im(12 ) = 0, respectively. And it is the equivalent steady-state value for the different values of ˚ no matter whether there exists the incoherent pumping or not. In a word, the steady absorption (or gain) is independent on the relative phase. It is easy to distinguish from Ref. [29] where Xiao and Kim considered the case of nonresonant probe field (p = / 0). Now we consider the effect of the incoherent pumping on the transient population distribution. In Fig. 4 we plot the time evolution of the population distribution without VIC. When excluding the incoherent pumping (R = 0), both 22 and 33 appear damped-oscillation, and almost all of the atomic populations are accumulated in 11 , which is caused by the strong coupling and weak probe fields. However, it is seen in Fig. 4(b) that the population distribution is clearly altered in the presence of incoherent pumping. After a short time oscillating, over 80% of the atomic populations are averagely distributed in the levels |2> and |3>, but the steady population in ground level |1> only keep about 14% with R = 3. Apparently a lot of populations are pumped from the ground level |1> to the upper level |3> interacting with incoherent pumping. As a result, there is a population inversion on the transition between the upper level and the ground level. When the VIC is included ( = 1), the same interesting phenomena can also be seen for different values of ˚, which is not shown by the figures due to the length limit of the paper. Besides, numerical calculation result shows that the relative phase ˚ cannot affect the population distribution in the presence of VIC. 4. Conclusions In summary, we have investigated the effects of VIC and relative phase on the transient absorption coefficient in a ladder-type three-level atomic system whether the incoherent pumping field is present or absent. It is found that the VIC and the relative phase ˚ can clearly affect the transient absorption (gain) properties, but the steady-state value of the probe absorption (gain) is related to incoherent pumping. We have additionally considered the influence of incoherent pumping on the transient population distribution. The steady population distribution is sensitive to the incoherent pumping field and insensitive to the relative phase ˚. And with the incoherent pumping, the lasing with inversion can be reached whether there exists the VIC effect or not. Acknowledgements This work was supported in part by the National Natural Science Foundation of China under grants no. 10647007, by the Education Foundation of Sichuan Province, China under grant no. 07ZA086, and by the Youth Foundation of Sichuan Province, China under grant no. 09ZQ026 – 008. References [1] K.J. Boller, A. Imamoglu, S.E. Harris, Observation of electromagnetically induced transparency, Phys. Rev. Lett. 66 (1991) 2593–2596. [2] S.E. Harris, Electromagnetically induced transparency, Phys. Today 50 (1997) 36–42. [3] J. Javanainen, Effect of state superpositions created by spontaneous Emission on laser-driven transitions, Europhys. Lett. 17 (1992) 407–412. [4] S. Menon, G.S. Agarwal, Effects of spontaneously generated coherence on the pump-probe response of a system, Phys. Rev. A 57 (1998) 4014–4018.
[5] J.H. Wu, J.Y. Gao, Phase control of light amplification without inversion in a
system with spontaneously generated coherence, Phys. Rev. A 65 (2002) 063807. [6] P. Zhou, S. Swain, Ultranarrow spectral lines via quantum interference, Phys. Rev. Lett. 77 (1996) 3995–3998. [7] P. Zhou, S. Swain, Quantum interference in probe absorption: narrow resonances, transparency, and gain without population inversion, Phys. Rev. Lett. 78 (1997) 832–835. [8] E. Paspalakis, S.Q. Gong, P.L. Knight, Spontaneous emission-induced coherent effects in absorption and dispersion of a V-type three-level atom, Opt. Commun. 152 (1998) 293–298. [9] R.M. Whitley, C.R. Stroud, Double optical resonance, Phys. Rev. A 14 (1976) 1498–1513. [10] Z. Ficek, B.J. Dalton, P.L. Knight, Fluorescence intensity and squeezing in a driven three-level atom: ladder case, Phys. Rev. A 51 (1995) 4062–4077. [11] X.J. Fan, A.Y. Li, Y.L. Yang, D.M. Tong, Effect of spontaneously generated coherence on probe response in an open system, Optik 121 (2010) 33–38. [12] N.Ph. Georgiades, E.S. Polzik, H.J. Kimble, Quantum interference in two-photon excitation with squeezed and coherent fields, Phys. Rev. A 59 (1997) 676–690. [13] H.M. Ma, S.Q. Gong, C.P. Liu, et al., Effects of spontaneous emission-induced coherence on population inversion in a ladder-type atomic system, Opt. Commun. 223 (2003) 97–101. [14] X.J. Fan, A.Y. Li, S.F. Tian, et al., Effects of relative phase in an open ladder system without incoherent pumping, Eur. Phys. J. D 42 (2007) 483–488. [15] M.A. Kumar, S. Singh, Electromagnetically induced transparency and slow light in three-level ladder systems: effect of velocity-changing and dephasing collisions, Phys. Rev. A 79 (2009) 063821. [16] M. Xiao, Y.Q. Li, S.Z. Jin, J. Gea-Banacloche, Measurement of dispersive properties of electromagnetically induced transparency in rubidium atoms, Phys. Rev. Lett. 74 (1995) 666–669. [17] G.B. Serapiglia, E. Paspalakis, C. Sirtori, et al., Laser-induced quantum coherence in a semiconductor quantum well, Phys. Rev. Lett. 84 (2000) 1019–1022. [18] H.A. Camp, M.H. Shah, M.L. Trachy, et al., Numerical exploration of coherent excitation in three-level systems, Phys. Rev. A 71 (2005) 053401. [19] B.D. Yang, J. Gao, T.C. Zhang, J.M. Wang, Electromagnetically induced transparency without a Doppler background in a multilevel ladder-type cesium atomic system, Phys. Rev. A 83 (2011) 013818. [20] H.R. Noh, H.S. Moon, Transmittance signal in real ladder-type atoms, Phys. Rev. A 85 (2012) 033817. [21] V.R. Blok, G.M. Krochik, Lasing without inversion: theory of transient behavior, Phys. Rev. A 44 (1991) 2036–2039. [22] Y. Zhu, Lasing without inversion in a V-type system: transient and steady-state analysis, Phys. Rev. A 53 (1996) 2742–2747. [23] Y. Zhu, Light amplification mechanisms in a coherently coupled atomic system, Phys. Rev. A 55 (1997) 4568–4575. [24] S.R. de Echaniz, A.D. Greentree, A.V. Durrant, et al., Observation of transient gain without population inversion in a laser-cooled rubidium system, Phys. Rev. A 64 (2001) 055801. [25] W.H. Xu, J.H. Wu, J.Y. Gao, Effects of spontaneously generated coherence on transient process in a system, Phys. Rev. A 66 (2002) 063812. [26] D. Braunstein, R. Shuker, X-ray laser without inversion in a three-level ladder system, Phys. Rev. A 68 (2003) 013812. [27] P.R. Berman, Spontaneously generated coherence and dark states, Phys. Rev. A 72 (2005) 035801. [28] A.J. Li, X.L. Song, X.G. Wei, et al., Effects of spontaneously generated coherence in a microwave-driven four-level atomic system, Phys. Rev. A 77 (2008) 053806. [29] 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. [30] M. Sahrai, A. Maleki, R. Hemmati, M. Mahmoud, Transient dispersion and absorption in a V-shaped atomic system, Eur. Phys. J. D 56 (2010) 105–112. [31] Z.Q. Zeng, B.P. Hou, Effects of squeezed vacuum on transient optical response in a system, Optik 122 (2011) 1231–1235. [32] Z.Q. Zeng, B.P. Hou, Z.H. Gao, Y.P. Yang, Transient response in a three-level system with the squeezed vacuum, Opt. Rev. 19 (2012) 45–49. [33] B.P. Hou, S.J. Wang, W.L. Yu, W.L. Sun, Effect of vacuum-induced coherence on single- and two-photon absorption in a four-level Y-type atomic system, Phys. Rev. A 69 (2004) 053805. [34] B.P. Hou, S.J. Wang, W.L. Yu, W.L. Sun, Effects of vacuum-induced coherence on dispersion and absorption properties in a tripod-scheme atomic system, J. Phys. B 39 (2006) 2335–2347. [35] B.P. Hou, L.F. Wei, G.L. Long, S.J. Wang, Large cross-phase-modulation between two slow pulses by coupled double dark resonances, Phys. Rev. A 79 (2009) 033813. [36] Z.Q. Zeng, B.P. Hou, Effects of vacuum-induced coherence on the single and two-photon transparency in a four-level ladder atomic system, Acta Opt. Sin. 30 (2010) 251–256 [in Chinese]. [37] Y. Zhu, Light amplification by coherence in a three-level laser, Opt. Commun. 105 (1994) 253–262.
Contents lists available at SciVerse ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Transient properties in a resonant ladder-type atomic system with vacuum-induced coherence and incoherent pumping Zhiqiang Zeng a,b,∗ , Yuping Wang c , Bangpin Hou d , Zenghui Gao a,b a
College of Physics and Electronic Engineering, Yibin University, Yibin 644000, China Computational Physics Key Laboratory of Sichuan Province, Yibin University, Yibin 644000, China College of Mining and Safety Engineering, Yibin University, Yibin 644000, China d College of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610066, China b c
a r t i c l e
i n f o
Article history: Received 28 September 2012 Accepted 30 January 2013
PACS: 42.50.Gy 42.50.Hz
a b s t r a c t The transient absorption and population in a resonant ladder-type atomic system with or without an incoherent pumping field are theoretically investigated. We find that the vacuum-induced coherence and the relative phase between probe and coupling fields can clearly affect the transient absorption (gain) properties, but the steady-state value of the probe absorption (gain) is remarkably dependent on the incoherent pumping. And the steady population distribution is sensitive to the incoherent pumping and insensitive to the relative phase. Additionally, it is shown that the lasing with inversion can be reached with the incoherent pumping. © 2013 Elsevier GmbH. All rights reserved.
Keywords: Transient properties Vacuum-induced coherence Incoherent pumping Relative phase
1. Introduction The interference induced by a strong resonant coupling field makes an opaque atomic system transparent for a probe field. This phenomenon, which is termed as electromagnetically induced transparency (EIT), has been studied in all kinds of three-level atomic systems with ∧,∨ and ladder configurations [1,2]. Another way of generating quantum interference connects with relaxation processes such as spontaneous emission, i.e., vacuum-induced coherence (VIC), which is also known as spontaneously generated coherence (SGC). It is created by the interference of spontaneous emission of either a single excited level to two closely lying atomic levels (∧-type system) [3–5], or by two closely lying atomic levels to a single ground level (∨-type system) [6–8] with the same vacuum radiation field. In a ladder-type system [9–11], it can also be created in a nearly equispaced atomic levels case. A theoretical study was made of the steady-state intensity and squeezing properties of the fluorescent light from a three-level ladder system, which was subject to spontaneous emission decay to the electromagnetic-field vacuum [10]. And the quantum interference
∗ Corresponding author at: College of Physics and Electronic Engineering, Yibin University, Yibin 644000, China. E-mail address: zhiqiang [email protected] (Z. Zeng). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.01.031
in two-photon excitation of a three-level ladder system interacting with squeezed vacuum was considered [12]. Ma et al. found that the population inversion can be enhanced on one of the optical transitions due to the VIC effect [13]. Fan et al. shown that lasing without inversion can be realized in an open ladder system without incoherent pumping [14]. Recently, Kumar and Singh presented a general theory of EIT in inhomogeneously broadened ladder system in which Doppler broadening of both one- and two-photon transitions can occur [15]. On the other hand, experimental investigations have been reported through EIT features in a three-level ladder system [16–20]. Besides, many theoretical and experimental works have focused on the transient research in order to understand the dynamical process [21–32], which is important due to its novel application in the optical switch. For example, Xu et al. discussed the effect of VIC on the transient process in a three-level ∧-type system [25]. With the nonresonant probe field, Xiao and Kim mainly compared the transient response between open and closed systems [29]. The dynamical behaviors of the dispersion and absorption in a three-level ∧-type system were investigated [31]. In the previous works, our group studied the effects of VIC on the stationary-state response in a multilevel atomic system [33–36]. In this paper, we investigate the transient absorption and population in a resonant ladder-type atomic system with VIC and incoherent pumping. It is well displayed that due to the effects of VIC and incoherent pumping, there
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Fig. 1. The energy scheme of the three-level ladder-type atomic system.
appear some interesting transient properties in the ladder-type system. 2. Model and equations A three-level ladder-type system is shown in Fig. 1, which has been experimentally studied [18] by the 87 Rb atom with 5S1/2 , 5P3/2 and 4D5/2 behaving as the |1, |2 and |3 state labels, respectively. A weak probe field with frequency ωpandRabi frequency Gp = ៝ 12 · ε៝ p / is applied on the transition 1 ↔ 2 , and a strong coherent coupling field with frequency ωc and Rabi frequency Gc = ៝ 23 · ε៝ c / drives the transition 2 ↔ 3 . An incoherent pumping field with
rate 2R interacts with the transition 1 ↔ 3 . 21 and 32 are the spontaneous decay rates from the level |2> to the level |1> and from the level |3> to the level |2>, respectively. In the interaction picture the density–matrix equations of motion in the dipole and rotating-wave approximations can be written as
˙ 11 = −2R11 + 221 22 + iGp (21 − 12 )
(1)
˙ 33 = 2R11 − 232 33 + iGc (23 − 32 )
(2)
˙ 12 = −(R + 21 + ip )12 + iGp (22 − 11 ) − iGc 13 √ +2 32 21 · exp(i˚) · cos · 23
(3)
˙ 23 = −(32 + 21 + ic )23 + iGc (33 − 22 ) + iGp 13
(4)
˙ 13 = −[R + 32 + i(p + c )]13 − iGc 12 + iGp 23
(5)
The above density–matrix elements additionally obey the nor3 malization and Hermitian condition = 1 and ij = ji∗ . i=1 ii The detunings of the probe and coupling lasers are defined as p = ω21 − ωp and c = ω32 − ωc , respectively. In the case of nearly √ equispaced levels, an additional term 2 32 21 · exp(i˚) · cos · 23 of the optical Bloch equations represents the effect of VIC. Where ˚ = ϕp − ϕc is the phase difference between the probe and the coupling fields, and is the angle between the two induced ៝ 12 and ៝ 23 . It is obviously that when = 1, the VIC dipole moments effect is included and its strength will vary with the value of ; otherwise = 0, the VIC effect is not included. As is well known, in the limit of a weak probe, coefficient for the probe the gain-absorption laser on transition 1 ↔ 2 is proportional to the imaginary part of 12 , which can be obtained from Eqs. (1) to (5). In our notation, the system exhibits gain for the probe laser if Im(12 ) > 0. Considering the case of two resonant transitions (p = c = 0), with the initial conditions 11 (0) = 22 (0) = 0.5 and other ij (0) = 0(i,j = 1,2,3), the following discussions will be deployed based on the timedependent numerical solutions of Eqs. (1)–(5).
Fig. 2. Time evolution of the absorption coefficient Im(12 ), where parameter values are 21 = 1, 32 = 1, Gp = 0.1 sin , Gc = 5 sin , p = 0, c = 0, ˚ = 0. (a) and (c) R = 0 (solid curve), R = 1 (dashed curve), R = 3 (dotted curve); (b) and (d) = /20 (solid curve), = /4 (dashed curve), = 9/20 (dotted curve).
Z. Zeng et al. / Optik 124 (2013) 973–976
3. The numerical analysis of transient process In Fig. 2 with the parameters given by 21 = 32 = 1, Gp = 0.4 sin , Gc = 5 sin , c = 0, p = 0, ˚ = 0, we plot the time evolution of the absorption coefficient Im(12 ). First, we consider the influence of incoherent pumping field without the VIC ( = 0), which is shown in Fig. 2(a). When the incoherent pumping is not included in this system (seen solid curve in Fig. 2(a)), the probe absorption displays oscillatory behavior versus time, but it always exhibits considerable large absorption without any gain during the total response process, including the transient and steady process. When considering the incoherent pumping, the atomic dynamical behavior is shown by the dashed curve (R = 1) and dotted curve (R = 3) in Fig. 2(a). It can be seen that the oscillatory amplitude of the transient absorption affected by incoherent pumping becomes much smaller than that in the absence of incoherent pumping. And the response time for the atomic medium from oscillatory region to the steady-state becomes shorter with increasing of R. Besides, in the presence of incoherent pumping, it is clear that the probe field can be amplified. Especially, with R = 3, the transient absorption disappears, only leaves the transient gain to reach positive steady-state value. So the probe field can be changed from absorption to gain due to incoherent pumping. In Fig. 2(b) without incoherent pumping, for small value of the angle , such as = /20 (seen solid curve in Fig. 2(b)), it also always exhibits considerable large absorption without any gain during the total response process. It is similar to the case where the VIC is not included, except that the oscillatory frequency becomes smaller, which is caused by the angle . However, the probe laser exhibits different features for different values of . It is shown that the transient gain can be appeared, and the oscillatory amplitude of
Fig. 3. Time evolution of the absorption coefficient Im(12 ) for different values of ˚: ˚ = 0 (or 2) (solid curve), = /2 (or 3/2) (dashed curve), and ˚ = (dotted curve), where the other parameter values are 21 = 1, 32 = 1, Gp = 0.1 sin , Gc = 5 sin , p = 0, c = 0, = 1, = /4. (a) R = 0; (b) R = 1.
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the transient absorption decreases with increasing of . Moreover, the absolute value of steady-state absorption also decreases with increasing of , but there is not positive steady-state value. Thus, without incoherent pumping, the probe field cannot be amplified in the presence of VIC, which is in good agreement with Ref. [37]. Fig. 2(c) and (d) depict the situations when both the VIC and incoherent pumping are present. From Fig. 2(c) with = /4, the phenomenon that the steady-state value from negative to positive and weakening of oscillatory behavior with R increasing is similar to the situation of = 0. Comparing Fig. 2(a) with (c), it can be found it is impossible to only appear transient gain or transient absorption when the VIC effect is existence and = /4. However, it is different the case of = /20, as shown the solid curve in Fig. 2(d), transient gain can be almost eliminated, and transient absorption rapidly goes into the steady-state value Im(12 ) ≈ 0. Moreover, Fig. 2(d) shows that with incoherent pumping, the transient response is different for the different values of , but the steady-state value of Im(12 ) is kept at the same value, which is different to the result without incoherent pumping in Fig. 2(b). Above all, it is clear that the steady-state value of the probe absorption coefficient is related to incoherent pumping, whether the VIC effect is included or not. In the following, let us investigate the effect of the relative phase ˚ between the probe and the coupling fields on the atomic absorption. It is noted that, there is no effect of the relative phase ˚ on the atomic optical properties when = 0. This is because that the relative phase appears in the Eq. (3) proportional to exp(i ). So we should study the influence of the relative phase ˚ under the condition of = / 0. In Fig. 3 with = 1 and = /4, we plot the time evolution of the absorption coefficient Im(12 ) for different values of ˚. It is apparently shown that the transient property is greatly altered with the relative phase and exhibits periodic absorption
Fig. 4. Time evolution of the population distribution, where parameter values are 21 = 1, 32 = 1, Gp = 0.1 sin , Gc = 5 sin , p = 0, c = 0, = /4, = 0, = 0 (or = 1). (a) R = 0; (b) R = 3.
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and amplification. The curve of ˚ = 3/2 and the curve of ˚ = /2 are completely superposed. The solid and the dotted curves corresponding to Fig. 3(a) and (b) are symmetric about Im(12 ) = −0.005 and Im(12 ) = 0, respectively. And it is the equivalent steady-state value for the different values of ˚ no matter whether there exists the incoherent pumping or not. In a word, the steady absorption (or gain) is independent on the relative phase. It is easy to distinguish from Ref. [29] where Xiao and Kim considered the case of nonresonant probe field (p = / 0). Now we consider the effect of the incoherent pumping on the transient population distribution. In Fig. 4 we plot the time evolution of the population distribution without VIC. When excluding the incoherent pumping (R = 0), both 22 and 33 appear damped-oscillation, and almost all of the atomic populations are accumulated in 11 , which is caused by the strong coupling and weak probe fields. However, it is seen in Fig. 4(b) that the population distribution is clearly altered in the presence of incoherent pumping. After a short time oscillating, over 80% of the atomic populations are averagely distributed in the levels |2> and |3>, but the steady population in ground level |1> only keep about 14% with R = 3. Apparently a lot of populations are pumped from the ground level |1> to the upper level |3> interacting with incoherent pumping. As a result, there is a population inversion on the transition between the upper level and the ground level. When the VIC is included ( = 1), the same interesting phenomena can also be seen for different values of ˚, which is not shown by the figures due to the length limit of the paper. Besides, numerical calculation result shows that the relative phase ˚ cannot affect the population distribution in the presence of VIC. 4. Conclusions In summary, we have investigated the effects of VIC and relative phase on the transient absorption coefficient in a ladder-type three-level atomic system whether the incoherent pumping field is present or absent. It is found that the VIC and the relative phase ˚ can clearly affect the transient absorption (gain) properties, but the steady-state value of the probe absorption (gain) is related to incoherent pumping. We have additionally considered the influence of incoherent pumping on the transient population distribution. The steady population distribution is sensitive to the incoherent pumping field and insensitive to the relative phase ˚. And with the incoherent pumping, the lasing with inversion can be reached whether there exists the VIC effect or not. Acknowledgements This work was supported in part by the National Natural Science Foundation of China under grants no. 10647007, by the Education Foundation of Sichuan Province, China under grant no. 07ZA086, and by the Youth Foundation of Sichuan Province, China under grant no. 09ZQ026 – 008. References [1] K.J. Boller, A. Imamoglu, S.E. Harris, Observation of electromagnetically induced transparency, Phys. Rev. Lett. 66 (1991) 2593–2596. [2] S.E. Harris, Electromagnetically induced transparency, Phys. Today 50 (1997) 36–42. [3] J. Javanainen, Effect of state superpositions created by spontaneous Emission on laser-driven transitions, Europhys. Lett. 17 (1992) 407–412. [4] S. Menon, G.S. Agarwal, Effects of spontaneously generated coherence on the pump-probe response of a system, Phys. Rev. A 57 (1998) 4014–4018.
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