Absorption-Amplification Response with or Without Spontaneously Generated Coherence in a Coherent Four-Level Atomic Medium

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1 Commun. Theor. Phys. (Beijing, China) 42 (2004) pp c International Academic Publishers Vol. 42, No. 3, September 15, 2004 Absorption-Amplification Response with or Without Spontaneously Generated Coherence in a Coherent Four-Level Atomic Medium LI Jia-Hua, 1, YANG Wen-Xing, 1,2 and PENG Ju-Cun 3 1 Department of Physics, Huazhong University of Science and Technology, Wuhan , China 2 Institute of Physics and Mathematics, the Chinese Academy of Sciences, Wuhan , China 3 Department of Physics, Xiaogan Normal University, Xiaogan , China (Received December 31, 2003) Abstract We discuss and analyze the absorption-amplification properties of a weak probe field in a typical fourlevel atomic system in the presence of an additional coherence term, the spontaneously generated coherence term. The influences of the spontaneously generated coherence and a coherent pump field on the probe absorption (amplification) are investigated in detail. We show that the absorption of such a weak probe field can be dramatically enhanced due to the presence of the spontaneously generated coherence. At the same time, the probe-absorption profile exhibits the double-peak structure and the probe-absorption peak gradually decreases as the pump intensity increases. On the contrary, the amplification of such a weak probe field near the line center of the probe transition can be achieved by adjusting the coherent pump field intensity in the absence of the spontaneously generated coherence. PACS numbers: Gy, Hz Key words: spontaneously generated coherence, probe absorption, probe amplification 1 Introduction Quantum coherent and interference have led to the observation of many new effects and techniques in quantum optics and atomic physics, such as non-absorption resonances and electromagnetically induced transparency (EIT), etc. There have been many theoretical and experimental studies in the literatures dealing with the absorption and amplification of light. [1 24] Agarwal et al. have shown that in a four-level ladder-type atomic system, two-photon absorption can be selectively suppressed or enhanced. [5] In a recent experiment, Yan et al. demonstrate that electromagnetically induced transparency in a standard -type configuration can be used to suppress both single-photon and two-photon absorptions simultaneously. [6,7] Harris et al. have proposed the use of the quantum interference to suppress the absorption of the short-wavelength light generated in a four-wave mixing (FWM) scheme and shown that the FWM efficiency can be greatly enhanced. [8] Quite recently, Deng et al. have reported the first theoretical investigation of optical coherent four-wave mixing (OCFWM) with a weak probe wave based on electromagnetically induced transparency and shown that such a scheme can lead to many orders of magnitude enhancement in the amplitude of the generated wave in a typical four-level atomic system. [13] Later on, on the basis of the proposal of Deng et al., Wu et al. analyzed and discussed a four-wavemixing (FWM) scheme in a five-level atomic system and hyper-raman scattering (HRS) in resonant coherent media by use of EIT, which leads to suppressing both twophoton and three-photon absorptions in both FWM and HRS schemes and enabling the four-wave mixing to proceed through real, resonant intermediate states without absorption loss. [17,18] On the other hand, the effects of the spontaneously generated coherence on absorption or spontaneous emission spectra have been extensively investigated recently. Menon and Agarwal have reported the influences of the spontaneously generated coherence on the pump-probe response of a -type atomic system and shown that such coherence can preserve both electromagnetically induced transparency and coherence population trapping phenomena. [23] Later, Xu et al. investigated the effects of this spontaneously generated coherence on the transient-absorption process and found that the transient gain (absorption) properties can be greatly affected by the spontaneously generated coherence. [24] In particular, with the advent of Bose Einstein condensate in an atomic gas there has been much interest in studying super-low light-speed propagation with EIT, [2] coherent optical information storage based on EIT, [3] and four-wave mixing with matter wave both experimentally and theoretically in the framework of nonlinear atomic optics. [25 33] In this paper, we analyze and discuss the absorptionamplification properties of the probe field in a typical fourlevel atomic system in the presence of the spontaneously generated coherence. This paper is organized as follows. In Sec. 2, the model is given and the density-matrix equations of motion describing the atom-field interaction for The project supported in part by National Natural Science Foundation of China under Grant Nos and Correspondence author, huajia li@163.com

2 426 LI Jia-Hua, YANG Wen-Xing, and PENG Ju-Cun Vol. 42 the system under consideration are derived when a coherent coupling field, a coherent pump field, and a weak coherent probe field are applied. In Sec. 3, we investigate numerically the absorption-amplification properties of the weak probe field in the steady state with or without the spontaneously generated coherence via a simple Mathematica code. Section 4 is a brief summary of our main results. 2 Theoretical Model and Equations of Motion Consider a closed four-level atomic system coupled with three laser fields as depicted in Fig. 1. The transition 2 3 of frequency ω 32 is driven by a strong coherent coupling laser of frequency ω c with Rabi frequency Ω c. A weak probe laser of frequency ω p with Rabi frequency Ω p is applied to the transition 1 3 of frequency ω 31. A pump laser of frequency ω s with Rabi frequency Ω s drives the transition 1 4 and controls the population inversion between the states 1 and 3, which determines the probe absorption or amplification. In the present analysis we use the following assumptions: (i) Here the Rabi frequencies are real. (ii) We assume that the dipole moments µ 31 and µ 32 are not orthogonal, between which the angle is θ 1. As such, µ 41 and µ 42 also are not orthogonal, between which the angle is θ 2. These conditions are necessary for the existence of the spontaneously generated coherence effect. (iii) As usual, we will ignore the dephasing rate γ 21 between levels 2 and 1. In the Schrödinger picture and in the dipole and rotating-wave approximations, the semiclassical Hamiltonian describing the atom-field interaction for the system under study can be written as Fig. 1 Schematic diagram of the four-level atomic system under consideration. H = 4 j=1 E (a) j j j h(ω p e iωpt Ω c e iωct Ω s e iωst c.c.), (1) where Ω n (n = c, p, s) stands for one-half Rabi frequency for the respective transition, i.e., Ω c = µ 32 E c /(2 h), Ω p = µ 31 E p /(2 h), and Ω s = µ 41 E s /(2 h) with µ ij denoting the dipole moment for the transition between levels i and j, and E (a) j = hω j is the energy of the atomic state j. For simplicity of analysis, in what follows we will take E (a) 1 = hω 1 as the energy origin for the ground state 1. Turning to the interaction picture, the Hamiltonian can be rewritten in the form (assuming h = 1) H 0 = ω p (ω p ω c ) ω s 4 4, (2) H int = p ( p c ) s 4 4 (Ω p Ω c Ω s c.c.), (3) where c = ω 32 ω c, p = ω 31 ω p, and s = ω 41 ω s are the corresponding coupling, probe, and pump detunings, as shown in Fig. 1. Under the dipole approximation and the rotating-wave approximation, by the standard approach, we can easily obtain the following density matrix equations of motion, ρ 11 = γ 31 ρ 33 + γ 41 ρ 44 + iω p (ρ 31 ρ 13 ) + iω s (ρ 41 ρ 14 ), ρ 22 = γ 32 ρ 33 + γ 42 ρ 44 + iω c (ρ 32 ρ 23 ), ρ 33 = (γ 31 + γ 32 )ρ 33 + iω p (ρ 13 ρ 31 ) + iω c (ρ 23 ρ 32 ), ρ 44 = (γ 41 + γ 42 )ρ 44 + iω s (ρ 14 ρ 41 ), ρ 12 = i( p c )ρ 12 + iω p ρ 32 + iω s ρ 42 iω c ρ 13 + γ 31 γ 32 cos(θ 1 )η 1 ρ 33 + γ 41 γ 42 cos(θ 2 )η 2 ρ 44, ρ 13 = [(γ 31 + γ 32 )/2 i p ]ρ 13 + iω s ρ 43 iω c ρ 12 + iω p (ρ 33 ρ 11 ), ρ 14 = [(γ 41 + γ 42 )/2 i s ]ρ 14 + iω p ρ 34 + iω s (ρ 44 ρ 11 ), ρ 23 = [(γ 31 + γ 32 )/2 i c ]ρ 23 iω p ρ 21 + iω c (ρ 33 ρ 22 ), (4a) (4b) (4c) (4d) (4e) (4f) (4g) (4h)

3 No. 3 Absorption-Amplification Response with or Without Spontaneously Generated Coherence in 427 ρ 24 = [(γ 41 + γ 42 )/2 + i( p c s )]ρ 24 + iω c ρ 34 iω s ρ 21, ρ 34 = [(γ 31 + γ 32 + γ 41 + γ 42 )/2 + i( p s )]ρ 34 + iω p ρ 14 + iω c ρ 24 iω s ρ 31, (4i) (4j) where γ ij in the above equations denotes the spontaneous decay rate from level i to level j. γ31 γ 32 cos(θ 1 )η 1 ρ 33 ( γ 41 γ 42 cos(θ 2 )η 2 ρ 44 ) represents the quantum interference effect resulting from the cross coupling between spontaneous emissions 3 1 ( 4 1 ), and 3 2 ( 4 2 ), i.e., the spontaneously generated coherence effect (SGC). If levels 1 and 2 lie so closely that the SGC effect has to be taken into account, then η 1 (η 2 ) = 1, otherwise η 1 (η 2 ) = 0. It is worth while to point out that only for small energy spacing between the two lower levels 1 and 2 is the SGC effect remarkable; as for large energy spacing, the rapid oscillation in ρ 12 will average out such an effect. Closure of this atomic system requires that ρ ij = ρ ji and ρ 11 + ρ 22 + ρ 33 + ρ 44 = 1. By a straightforward semiclassical analysis, the above matrix element can be used to calculate the total linear complex susceptibility χ of the probe transition of the SGC effect, the amplification of the probe field can be achieved by using strong coherent pump field, and it increases with increasing the coherent pump field intensity. As such, the probe-amplification profile exhibits the two-peak structure, as shown in Fig. 2(b). In view of the above two cases, the double-peak structure can be clearly understood in terms of the dressed states. When Ω s /Ω c is large, i.e., Ω s is much stronger than Ω c, the double peaks are located at p ±Ω s corresponding to the energy separation of the dressed states + = 1/ 2 ( ) and = 1/ 2 ( 4 1 ) due to the dynamic Autler Townes splitting. For comparison, figure 3 presents the schematic plots of the absorption-amplification Im (ρ 13 ) of the probe field versus the probe detuning p with or without the SGC effect for the different pump field intensities. χ = N 0 µ 31 2 ρ 13 2 hε 0 Ω p, (5) where N 0 is the atomic number density. Therefore, the absorption-amplification coefficient for the probe laser coupled to the transition 1 3 is proportional to Im ρ 13. If Im ρ 13 < 0, the probe field will be absorbed. On the contrary, the probe field will be amplified. In the following, we begin with investigating the absorption properties of such a weak probe field by numerically solving the above density matrix equations (4a) and (4b) in the steady state with or without the SGC effect via a simple MATHEMATICA code. Note that, in this paper, parameters Ω c,p,s, c,p,s, and γ ij are in units of γ. 3 Steady State Analysis with or Without SGC Effect In the following analysis, for the coherent coupling and pump fields we will only consider the case of resonant excitations, i.e., c = s = 0. We numerically solve Eq. (4) in the steady state under a variety of conditions. In Figs. 2(a) and 2(b), we plot the absorption-amplification coefficient Im(ρ 13 ) of the probe field versus the probe detuning p based on Eq. (4) for the different pump field intensities Ω s. The calculations show that for the fixed Ω c and Ω s values, in the presence of the SGC effect the probe field will be dramatically absorbed. For this case, as the coherent pump intensity increases, the probe-absorption peak gradually decreases, but the probe field cannot be amplified near the line center of the probe transition and the probe-absorption lineshape takes the two-peak structure, as shown in Fig. 2(a). In contrast, in the absence Fig. 2 Absorption-amplification coefficient Im (ρ 13) of the probe field as a function of the probe detuning p for the different pump field intensity Ω s. Other fixed parameters used are Ω c = 0.5γ, Ω p = 0.01γ, c = s = 0, γ 41 = γ 42 = 1.2γ, γ 31 = γ 32 = γ, and θ 1 = θ 2 = π/4.

4 428 LI Jia-Hua, YANG Wen-Xing, and PENG Ju-Cun Vol. 42 In order to show explicitly the dependence of the probe absorption and amplification on the pump field intensity, we plot in Figs. 4(a) and 4(b) the absorption-amplification coefficient Im(ρ 13 ) of the probe field versus the coherent pump field intensity Ω s for the two different coupling field intensities Ω c = 0.5γ and Ω c = γ. We find that in the presence of the SGC effect, the probe absorption at the line center p = 0 approaches a maximum value at a suitable pump field intensity Ω s due to the quantum constructive interference between the two dressed states. As the coherent pump field intensity continues to increase, the absorption of the probe field can be considerably suppressed and reaches a saturation value. Specifically, for the two cases of η 1 = 1, η 2 = 0 and η 1 = η 2 = 1, the absorption profiles are distinct. For the former, the magnitude of the probe absorption is small compared to the latter. Moreover, to approach the saturation value for the former is faster than for the latter, as shown in Figs. 4(a) and 4(b). It should be noted that when the intensity of the coupling field increases to Ω c = γ, the probe field for the latter can be amplified with the assistance of sufficiently strong pump field. In the absence of the SGC effect, the probe field will be amplified. First, the probe amplification increases monotonically, then decreases gradually with increasing the coherent pump field intensity while the population inversion approaches the saturation value. On the basis of the above analysis, it is clear that in order to achieve the amplification of the probe field, we should eliminate the existence of the SGC effect. Figure 4(c) gives the population inversion ρ 33 ρ 11 of the probe field versus the coherent pump field intensity Ω s for the coupling field intensity ranging from Ω c = 0.5γ to Ω c = γ with or without the SGC effect. The results show that all the curves of the population inversion completely overlap under the given conditions and when Ω s > 2.7γ is satisfied, population inversion arises (ρ 33 > ρ 11 ). This means that under the condition of strong coupling field Ω c, the SGC effect cannot affect the population inversion of the probe field. Fig. 3 Absorption-amplification coefficient Im(ρ 13) of the probe field as a function of the probe detuning p for the different pump field intensity Ω s with or without the SGC effect. Other fixed parameters used are Ω c = 0.5γ, Ω p = 0.01γ, c = s = 0, γ 41 = γ 42 = 1.2γ, γ 31 = γ 32 = γ, and θ 1 = θ 2 = π/4.

5 No. 3 Absorption-Amplification Response with or Without Spontaneously Generated Coherence in 429 Fig. 4 Absorption-amplification coefficient Im(ρ 13) of the probe field as a function of the coherent pump field intensity Ω s at the line center of the probe transition p = 0 with or without the SGC effect. (c) Population inversion ρ 33 ρ 11 as a function of the coherent pump field intensity Ω s at the line center p = 0. Other fixed parameters used are Ω p = 0.01γ, c = s = 0, γ 41 = γ 42 = 1.2γ, γ 31 = γ 32 = γ, and θ 1 = θ 2 = π/4. 4 Conclusions In summary, in the present work we discuss and analyze the absorption-amplification response of the probe field in a typical four-level atomic system with or without the spontaneously coherence effect. Our results show that in the absence of the spontaneously coherence effect, the amplification of the probe field can be achieved by adjusting the coherent pump field intensity. In contrast, in the presence of the spontaneously coherence effect, the probe field will be dramatically absorbed under the condition of the same pump field intensity. As the coherent pump field intensity increases, the probe-absorption line shape exhibits the two-peak structure and the probe-absorption peak gradually decreases. Acknowledgments The authors would like to thank Dr. Wu Ying for many stimulating discussions. References [1] E. Arimondo, Progress in Optics, ed. E. Wolf, Elsevier Science, Amsterdam (1996) pp [2] L.V. Hau, et al., Nature 397 (1999) 594. [3] C. Liu, et al., Nature 409 (2001) 490. [4] Y. Li and M. Xiao, Opt. Lett. 21 (1996) [5] G.S. Agarwal, et al., Phys. Rev. Lett. 77 (1996) [6] M. Yan, E. Rickey, and Y. Zhu, Phys. Rev. A64 (2001) [7] M. Yan, E. Rickey, and Y. Zhu, Opt. Lett. 26 (2001) 548. [8] S.E. Harris, et al., Phys. Rev. Lett. 64 (1990) [9] Y. Wu, Phys. Rev. A61 (2000)

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