Roles of Atomic Injection Rate and External Magnetic Field on Optical Properties of Elliptical Polarized Probe Light

Commun. Theor. Phys. 65 (2016) 57 65 Vol. 65, No. 1, January 1, 2016 Roles of Atomic Injection Rate and External Magnetic Field on Optical Properties of Elliptical Polarized Probe Light R. Karimi, S.H. Asadpour, S. Batebi, and H. Rahimpour Soleimani Department of Physics, University of Guilan, Rasht, Iran (Received March 16, 2015; revised manuscript received April 23, 2015) Abstract In this paper we investigate the optical properties of an open four-level tripod atomic system driven by an elliptically polarized probe field in the presence of the external magnetic field and compare its properties with the corresponding closed system. Our result reveals that absorption, dispersion and group velocity of probe field can be manipulated by adjusting the phase difference between the two circularly polarized components of a single coherent field, magnetic field and cavity parameters i.e. the atomic exit rate from cavity and atomic injection rates. We show that the system can exhibit multiple electromagnetically induced transparency windows in the presence of the external magnetic field. The numerical result shows that the probe field in the open system can be amplified by appropriate choice of cavity parameters, while in the closed system with introduce appropriate phase difference between fields the probe field can be enhanced. Also it is shown that the group velocity of light pulse can be controlled by external magnetic field, relative phase of applied fields and cavity parameters. By changing the parameters the group velocity of light pulse changes from subluminal to superluminal light propagation and vice versa. PACS numbers: 42.50.Gy Key words: elliptically polarized field, magnetic field, phase difference 1 Introduction Electromagnetically induced transparency (EIT) based on atomic coherence and quantum interference is discovered by Harris et al. [1] This phenomenon has led to many interesting phenomena such as lasing without inversion (LWI), [2] enhanced Kerr nonlinearity, [3] optical solitons, [4] optical bistability and multi stability, [5 7] slow light and fast light [8 9] and so on. [10 16] The control of light speed is highly desirable for many practical applications in all optical communication system and quantum information processing and etc. It is well known that the group velocity of a light pulse can exceed the speed of light in vacuum (c), leading to the superluminal light propagation or it can become slow down, leading to the subluminal light propagation. The subluminal light and superluminal light propagation has been investigated extensively. Recently, many studies have been done on the switch from subluminal to superluminal light propagation and vice versa in an atomic medium. Switching from subluminal to superluminal light has been studied in a two-level atomic system in degenerate and non-degenerate cases. [10] It is shown that the switching from subluminal to superluminal light propagation can be achieved with a relative phase between two weak probe fields. [17] The effects of spontaneously generated coherence (SGC) [18] on the absorption and dispersion profile have extensively been investigated. [19 22] In the general case, in EIT phenomena a single transparency window appears at line center. Nevertheless, one E-mail: s.hosein.asadpour@gmail.com Corresponding author, E-mail: S Batebi@guilan.ac.ir c 2016 Chinese Physical Society and IOP Publishing Ltd can find schemes in which additional transparency windows can appear. Such models can be potentially applied for slowing down of light pulses at various frequencies. [9] The purpose of the present study is to control the absorption, dispersion and group velocity of the probe field in the closed and open system. We investigate the optical response of an open four-level tripod atomic system with two degenerate sublevels driven by an elliptically polarized probe field and a linearly polarized control field. It was demonstrated that switching from subluminal to superluminal pulse propagation can be controlled by adjusting the parameters for example: cavity parameters of system (open system), magnetic field and the relative phase of applied fields. It is shown that in the open system the probe field can be amplified and the super luminal light propagation can be occurred in the absence of magnetic field and phase difference. To the best of our knowledge, no related theoretical or experimental work has been carried out to study the influence of magnetic field and phase difference on the optical properties of open tripod system. In the following section, we present the model and density matrix equation of motion. The results are discussed in Sec. 3, and the conclusion can be found in Sec. 4. 2 Model We consider an open four-level atomic system in configuration as depicted in Fig. 1. In this system the ground state 1 is 5S 1/2, F = 1, m = 0, the states 2 and 3 are 5S 1/2, F = 2, m = 1 and 5S 1/2, F = 2, m = 1 degenerate Zeeman sublevels respectively and the excited http://www.iopscience.iop.org/ctp http://ctp.itp.ac.cn

58 Communications in Theoretical Physics Vol. 65 state 4 is the Zeeman sublevels corresponding to 5P 3/2, F = 2, m = 0. Due to external magnetic field the degenerate Zeeman sublevels corresponding to the magnetic quantum numbers m ± 1 can shift by ± B. The Zeeman shift is given by B = µ B m s g s B/, where g s is the Lande factor, µ B is the Bohr magneton, and m s is magnetic quantum number of the sublevels of the system. If magnetic field is zero B = 0, B is also zero. An elliptically polarized probe field with frequency ω p is used to create electric dipole transitions from 4 to 2 and 4 to 3 simultaneously. A probe field with electric field amplitude E 0 after passing through the quarter-wave plate (QWP) becomes elliptically polarized that has been rotated by an angle θ. An elliptically polarized field can be decomposed into two mutually polarized components as: E p = E p + σ + +Ep σ, where E p + = E 0 / 2(cos θ+sin θ) e iθ and Ep = E 0 / 2(cos θ sin θ) e iθ. Here, σ + and σ are the unit vectors of the right-hand circularly and the lefthand circularly polarized basis, respectively. The strength of the electric field components and phase difference between them can be changed by QWP. Then the Rabi frequency for right-circularly polarized component become Ω p+ = Ω p (cos θ + sin θ) e iθ and for left-circularly polarized component is Ω p = Ω p (cos θ sin θ) e iθ. Where Ω p = µe 0 / 2, here we assume that µ 42 = µ 43 = µ. Electric dipole transition 4 1 is driven by a linearly polarized control laser with carrier frequency ω c and Rabi frequency Ω c = µ 41 E c /2, where µ ij denotes the dipole moment for atomic transition between levels i and j. J 1, J 2 and J 3 are the atomic injection rates for levels 1, 2 and 3, respectively. r 0 = J 1 + J 2 + J 3 is the exit rate from the cavity. Fig. 1 Four level atomic system in a tripod-type configuration in the presence of a magnetic field. By applying an external magnetic field, the degeneracy among the ground states 3, 2 is lifted, B indicates the Zeeman shift. J 1, J 2 and J 3 are the atomic injection rates for levels 1, 2 and 3, respectively. r 0 = J 1 + J 2 + J 3 is the exit rate from the cavity. The equations of motion for the density matrix elements for the tripod system under the rotating wave and electric dipole approximations become: ρ 44 = (Γ 41 + Γ 42 + Γ 43 )ρ 44 + iω c ρ 14 iω cρ 41 + iω + p ρ 24 iω + p ρ 42 + iω p ρ 34 iω p ρ 43 r 0 ρ 44, ρ 33 = (Γ 31 + Γ 32 )ρ 33 + Γ 43 ρ 44 iω p ρ 34 + iω p ρ 43 r 0 ρ 33 + J 3, ρ 22 = Γ 42 ρ 44 + Γ 32 ρ 33 Γ 21 ρ 22 iω + p ρ 24 + iω + p ρ 42 r 0 ρ 22 + J 2, ρ 11 = Γ 41 ρ 44 + Γ 31 ρ 33 + Γ 21 ρ 22 iω c ρ 14 + iω cρ 41 r 0 ρ 11 + J 1, ρ 14 = (γ 41 i c )ρ 14 + iω c(ρ 44 ρ 11 ) iω + p ρ 12 iω p ρ 13, ρ 13 = (γ 31 i( c ( p + B )))ρ 13 + iω cρ 43 iω p ρ 14, ρ 12 = (γ 21 i( c ( p B )))ρ 12 + iω cρ 42 iω + p ρ 14, ρ 23 = (γ 32 + 2i B )ρ 23 + iω + p ρ 43 iω p ρ 24, ρ 24 = (γ 42 i( p B ))ρ 24 + iω + p (ρ 44 ρ 22 ) iω cρ 21 iω p ρ 23, ρ 34 = (γ 43 i( p + B ))ρ 34 + iω p (ρ 44 ρ 33 ) iω cρ 31 iω + p ρ 32, ρ ij = ρ ji, (1) in the above equations, p = ω 43 ω p B = ω 42 ω p + B and c = ω 41 ω c are the frequency detuning of the probe and control fields, where ω ij is the frequency deference between levels i and j. Γ ij is the spontaneous decay rate from level i and level j, γ ij is the de-phasing rate γ ij = 1/2 l Γ il + 1/2 l Γ jl. If J 1 = J 2 = J 3 = r 0 = 0, Eq. (1) changes to that for a closed tripod system. The set of equations can be used to calculate the response of the medium to the applied fields, by calculating the susceptibility of the probe field, which is defined as χ p = N µ 2 (ρ 42 + ρ 43 ) 2 ε 0 Ω p, (2) where, N is the atomic density number in the medium. The imaginary and real part of χ p denote the absorption and dispersion coefficients for the probe field respectively. If Im(ρ 42 + ρ 43 ) < 0, the probe field will be amplified and if Im(ρ 42 + ρ 43 ) > 0 the probe filed will be attenuated. The group velocity of the probe field is given by: c v g = 1 + (1/2)χ (ω p ) + (ω p /2)[ χ (ω p )/ ω p ], (3) where, c is the speed of light in the vacuum and χ (ω p ) is the real part of χ p. The above equation implies that, the steep positive dispersion can significantly reduce the group velocity. Moreover, the group velocity can be increased via a strong negative dispersion. If c/v g 1 < 0, the group velocity of the radiation is larger than c or it becomes negative, thus the propagation of radiation is superluminal and the region c/v g 1 > 0 corresponds to the subluminal propagation. In this paper all parameters are scaled by γ, that should be in the order of MHz for rubidium atoms. We

No. 1 Communications in Theoretical Physics 59 assume Γ 41 = Γ 42 = Γ 43 = γ and Γ 32 = 0, Γ 31 = Γ 21 = 0.001γ and all the other parameters are scaled with γ where γ = 6 MHz. 3 Result and Discussion 3.1 Closed System In this section, we numerically simulate the effect of the external magnetic field ( B ) and the phase difference between the two circularly polarized components of a single coherent field on the absorption, dispersion spectrum and group velocity of the probe field when the system is closed (J 1 = J 2 = J 3 = r 0 = 0). It is shown that optical properties of the system are very sensitive to magnetic field and relative phase between applied fields. First, we fix the phase difference θ, while scanning the Zeeman shift B. In Fig. 2, we plot the absorption coefficient of the probe field (Im(ρ 42 +ρ 43 )) versus probe field detuning and Zeeman shift for the case that θ = 0. As seen in Fig. 3(a), at zero magnetic field, we observe a single EIT window at the line center and the atomic medium become transparent at resonance region. This configuration, at zero magnetic fields, is equivalent to the degenerate two-level configuration. [27] Fig. 2 (a) The effect of magnetic field on absorption spectra of probe field, (b) the absorption of probe field versus probe detuning and Zeeman shift in the closed system J 1 = J 2 = J 3 = r 0 = 0. The absorption and dispersion under four typical values of Zeeman shift have been shown in Fig. 3. The parameters: θ = 0, c = 0, Ω c = 4, and Ω p = 0.01. Fig. 3 The absorption (solid line) and dispersion (dashed line) in the closed system when B = 0, 2γ, 4γ, 8γ respectively. Other parameters are the same as those in Fig. 2.

60 Communications in Theoretical Physics Vol. 65 Fig. 4 The group velocity of probe field in the closed system when B = 0, 2γ, 4γ, 8γ respectively. Other parameters are the same as those in Fig. 2. Fig. 5 (a) The absorption and (b) the dispersion of probe field versus probe detuning and Zeeman shift (c) the effect of magnetic field on absorption spectra of probe field in the closed system when θ = π/5. The absorption and dispersion under four typical values of Zeeman shift have been shown in Fig. 6. Other parameters are the same as those in Fig. 2. When we apply a weak magnetic field, the absorption peaks in previous case convert to transparency window, so in this case three EIT windows appear: one at line center P = 0 and others at P = ±2 B. The system behaves as a combination of two three-level systems. When the magnetic field is increased ( B = Ω c ), an absorption peak appears in resonance region, which is the signature of an interference phenomenon between the EIT windows. In this case three absorption peaks appear in the absorption profile which the amount of absorption at line center is greater than that at other peaks. With further increase of the magnetic field B > Ω c the absorption peak in line center again convert to EIT window and two other windows shift to P = ± B. The dispersion properties of the probe field are also changed by external magnetic field. The dispersion curve is plotted with dashed line in Figs. 3(a) 3(d) for B = 0, 2, 4, 8γ respectively. For the system considered here one can observe some additional region of normal and anomalous dispersion, superluminal and subluminal propagation related to the presence of the additional transparency windows. The probe field, depending on the positive or negative slope of dispersion, will propagate in subluminal or superluminal region. Note that the presence of magnetic field in the system can lead to nontrivial results, for example in simultaneous the probe field will propagate in superluminal or subluminal region at various frequencies. As shown in Figs. 4(a) 4(d), in the absence of the magnetic field ( B = 0), at line center c/v g 1 > 0 corresponding to the subluminal light propagation by changing the Zeeman shift to B = 4γ the slope of dispersion at line center becomes negative (Fig. 3(c)) corresponding to the superluminal propagation (c/v g 1 < 0).

No. 1 Communications in Theoretical Physics 61 Fig. 6 The absorption (solid line) and dispersion (dashed line) in the closed system for B = 0, 2γ, 4γ, 8γ respectively. Other parameters are the same as those in Fig. 5. Fig. 7 The group velocity of probe field in the closed system for B = 0, 2γ, 4γ, 8γ respectively. Other parameters are the same as those in Fig. 5. It is found that with introduce a phase difference between electric field components, the absorption and dispersion profiles drastically change. In Figs. 5 7 we fix the phase difference on θ = π/5 and investigate the effect of magnetic field on the properties of the system in this case. Figures 5(a) and 5(b) show a three-dimensional plot of the absorption and dispersion of the probe field respectively. As is evident from the Fig. 5(a), with increasing magnetic field ( B ) the absorption dip shift toward the positive probe detuning. In order to further illustrate ex-

62 Communications in Theoretical Physics Vol. 65 plicitly the dependence of the absorption spectra of the probe field on the magnetic field, two-dimensional plot of the probe field absorption as a function of probe detuning for B = 0, 2γ, 4γ, 8γ are plotted in Figs. 6(a) 6(d) respectively. From the figures, we observe that the phase difference leading to negative absorption corresponding to amplification of probe field (solid lines in Fig. 6). So the probe field can be amplified for appropriate values of phase difference without applying an incoherent pumping field to the system. Note that in this case, unlike the previous case (Fig. 3), the absorption and dispersion profiles are asymmetric. The slope of dispersion curve is also changed by phase difference (dashed line in Fig. 6). The group velocity of probe field is shown in Fig. 7. Like the previous case, by changing the magnetic field we can switch the slope of dispersion curve from positive to negative and vice versa. Then, by changing the magnetic field, the transparent subluminal light propagation switches to the transparent superluminal light propagation and vice versa. 3.2 Open System Fig. 8 (Color online) (a) The absorption and (b) the dispersion of probe field versus probe detuning and Zeeman shift in the open system r 0 = 1, j 1 = j 2 = j 3, when θ = 0, the absorption and dispersion under four typical values of Zeeman shift have been shown in Fig. 9. Other parameters are the same as those in Fig. 2. Fig. 9 The absorption (solid line) and dispersion (dashed line) in the open system for B = 0, 2γ, 4γ, 8γ respectively. Other parameters are the same as those in Fig. 8.

No. 1 Communications in Theoretical Physics 63 Fig. 10 The group velocity of probe field in the open system for B = 0, 2γ, 4γ, 8γ respectively. Other parameters are the same as those in Fig. 8. Fig. 11 (a) The absorption and (b) the dispersion of probe field versus probe detuning and Zeeman shift in the open system r 0 = 1, j 1 = j 2 = j 3, when θ = π/5, the absorption and dispersion under four typical values of Zeeman shift have been shown in Fig. 12. Other parameters are the same as those in Fig. 2. In the following, in Figs. 8 13, we present a few numerical results for steady state spectral characteristics of the probe absorption and dispersion in the open system. We analyze how the cavity parameters, magnetic field and phase difference modify the optical properties of the system. In the following discussion, the exit rate from cavity is r 0 = 1 and j 1 = j 2 = j 3 = 1/3. When θ = 0, the Rabi frequency for right and left-circularly polarized components are equal to (Ω p ) + = Ω p = Ω p. The absorption and dispersion spectra for this case are shown in Figs. 8(a) and 8(b) respectively. The absorption and dispersion under four typical values of magnetic field ( B ) are shown in Figs. 9(a) 9(d). By comparing Figs. 9 and 3, we can observe that in addition to the absorption of the probe field has been changed to negative values; the slope of the dispersion curve is also varied, so the probe field can be amplified in the open system even without phase difference or incoherent pumping field and also a switch from subluminal light propagation to superluminal light propagation also can occur in this system. Note that in Fig. 9(a), at zero magnetic field, the slope of dispersion curve is negative corresponding to superluminal light propagation. Thus, in the open system to create a group velocity greater than the speed of light no external magnetic field or phase difference is required. We plot the numerically results for the group velocity of probe field versus probe detuning in the open system in Figs. 10(a) 10(d) for magnetic field of B = 0, 2γ, 4γ, 8γ respectively. Our numerical simulations show that the optical properties of the system can be manipulated by appropriate choice of the cavity parameters.

64 Communications in Theoretical Physics Vol. 65 Fig. 12 The absorption (solid line) and dispersion (dashed line) in the open system for B = 0, 2γ, 4γ, 8γ respectively. Other parameters are the same as those in Fig. 11. Fig. 13 The group velocity of probe field in the open system for B = 0, 2γ, 4γ, 8γ respectively. Other parameters are the same as those in Fig. 11. When we introduce a phase difference between electric field components in the open system, θ 0, the Rabi frequency for right- and left-circularly polarized components are not equal (Ω p ) + Ω p. In Figs. 11 13 we investigate the effect of magnetic field on the properties of the open system for θ = π/5. Figures 11(a) and 11(b) show a threedimensional plot of the absorption and dispersion of the probe field respectively. From the figures, we observe that

No. 1 Communications in Theoretical Physics 65 the phase difference change the amount of negative absorption (solid lines in Fig. 11), so that affect the amount of gain in the open system. The slope of dispersion curve is also changed by phase difference (dashed line in Fig. 12). The group velocity of probe field is shown in Fig. 13. Like the previous case, by changing the magnetic field we can switch the slope of dispersion curve from positive to negative and vice versa in some frequency region. Then, by introduce phase difference, the absorption and dispersion profiles drastically change, the symmetry on the system breaks and the absorption dip shift toward the positive probe detuning with magnetic field. 4 Conclusion In this paper, we investigate the influence of the external magnetic field and phase difference between the two circularly polarized components of a single coherent field on the optical properties of open and corresponding closed system. We find from the numerical results of equations (1): (i) When phase difference is zero, θ = 0, the closed system can exhibit multiple electromagnetically induced transparency windows in the presence of the external magnetic field. The location and the number of EIT windows can be manipulated by magnetic field. In the open system the probe field amplify and probe gain can be occurred, with increasing the magnetic field ( B ) the location of absorption dip change. (ii) With introduce a phase difference between the two circularly polarized components of a single coherent probe field, the probe field in closed system amplify. Comparing the close system with the corresponding open system, we find that the linear gain in the closed system is smaller than that in the open system. (iii) In the absence of magnetic field, the slope of dispersion curve in the closed system is positive corresponding to subluminal light propagation while that in the open system is negative corresponding to superluminal light propagation. In the closed system for group velocity greeter than the speed of light in the vacuum an external magnetic field is required. Theoretical results show that, the magnetic field and cavity parameters could play important roles in obtaining the optimal gain and dispersion. Note that in the open system, probe gain is much larger than that in the corresponding closed system can be obtained by choosing proper values of cavity parameters. References [1] Y. Wu and X. Yang, Phys. Rev. A 71 (2005) 053806. [2] M.O. Scully, S.Y. Zhu, and A. Gavrielides, Phys. Rev. Lett. 62 (1989) 2813. [3] Y. Wu and X. Yang, Appl. Phys. Lett. 91 (2007) 094104. [4] Y. Wu and L. Deng, Phys. Rev. Lett. 93 (2004) 143904. [5] S.H. Asadpour and A. Eslami-Majd, J. Lumin. 132 (2012) 1477. [6] X.Y. Lu, J.H. Li, J.B. Liu, and J.M. Luo, J. Phys. B 39 (2006) 5161. [7] Y. Wu and X. Yang, Phys. Rev A 70 (2004) 053818. [8] L.V. Hau, S.E. Harris, Z. Dutton, and C.H. Behroozi, Nature (London) 397 (1999) 594. [9] L.J. Wang, A. Kuzmich, and A. Dogariu, Nature (London) 406 (2000) 277. [10] W.X. Yang, A.X. Chen, R.K. Lee, and Y. Wu, Phys. Rev. A 84 (2011) 013835. [11] Z. Wang and B. Yu, Laser Phys. Lett. 11 (2014) 035201. [12] P. Maboodi, S. Hemmatzadeh, S.H. Asadpour, and H. Rahimpour Soleimani, Commun. Theor. Phys. 62 (2014) 864. [13] L.G. Si, W.X. Yang, and X. Yang, J. Opt. Soc. Am. B 26 (2009) 478. [14] T. Naseri, S.H. Asadpour, and R. Sadighi-Bonabi, J. Opt. Soc. Am. B 30 (2013) 641. [15] D.A. Han, et al., Commun. Theor. Phys. 46 (2006) 731. [16] A. Joshi, W. Yang, and M. Xiao, Phys. Rev. A 68 (2003) 015806. [17] H. Sun, H. Guo, Y. Bai, D. Han, S. Fan, and X. Chen, Phys. Lett. A 335 (2005) 68; D. Bortman-Arbiv, A.D. Wilson-Grodon, and H. Friedmann, Phys. Rev. A 63 (2001) 043818. [18] J. Javanainen, Europhys. Lett. 17 (1992) 407. [19] S. Menon and G.S. Agarwal, Phys. Rev. A 57 (1998) 4014. [20] P.B. Hou, S.J. Wang, W.L. Yu, and W.L. Sun, Phys. Rev. A 69 (2004) 053805. [21] W.H. Xu, J.H. Wu, and J.Y. Gao, Phys. Rev. A 66 (2002) 063812. [22] W.H. Xu and H.F. Zhang, J. Opt. Soc. Am. B 20 (2003) 2377. [23] Z. Wang and M. Xu, Opt. Commun. 282 (2009) 1574. [24] Z. Wang, A.X. Chen, Y. Bai, W.X. Yang, and R.K. Lee, J. Opt. Soc. Am. B 29 (2012) 2891. [25] Jia-Hua Li, Xin-You Lü, Jing-Min Luo, and Qiu-Jun Huang, Phys. Rev. A 74 (2006) 035801. [26] A.T. Rosenberger, L.A. Orozco, and H.J. Kimble, Phys. Rev. A 28 (1983) 2529. [27] F. Goldfarb, J. Ghosh, M. David, et al., Europhys. Lett. 82 (2008) 54002.