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In this work, we derive analytical expression for the dispersion coefficient of 85Rb atom for a weak probe laser beam induced by a strong coupling laser beams. Our results show possible ways to control dispersion coefficient by frequency detuning and of the coupling lasers.
VNU Journal of Science: Mathematics – Physics, Vol 35, No (2019) 101-107 Original Article Determine Dispersion Coefficient of 85Rb Atom in the Y–Configuration Nguyen Tien Dung* Vinh University,182 Le Duan, Vinh, Nghe An, Vietnam Received 26 March 2019 Revised 26 April 2019; Accepted 17 May 2019 Abstract: In this work, we derive analytical expression for the dispersion coefficient of 85Rb atom for a weak probe laser beam induced by a strong coupling laser beams Our results show possible ways to control dispersion coefficient by frequency detuning and of the coupling lasers The results show that a Y-configuration appears two transparent window of the dispersion coefficient for the probe laser beam The depth and width or position of these windows can be altered by changing the intensity or frequency detuning of the coupling laser fields Keywords: Electromagnetically induced transparency, dispersion coefficient Introduction The manipulation of subluminal and superluminal light propagation in optical medium has attracted many attentions due to its potential applications during the last decades, such as controllable optical delay lines, optical switching [1], telecommunication [2], interferometry, optical data storage and optical memories quantum information processing, and so on [3] The most important key to manipulate subluminal and superluminal light propagations lies in its ability to control the absorption and dispersion properties of a medium by a laser field [4, 5] As we know that coherent interaction between atom and light field can lead to interesting quantum interference effects such as electromagnetically induced transparency (EIT) [6] The EIT is a quantum interference effect between the probability amplitudes that leads to a reduction of resonant absorption for a weak probe light field propagating through a medium induced by a strong coupling light field Basic configurations of the EIT effect are three-level atomic systems including the -Ladder and V Corresponding author Email address: tiendungunivinh@gmail.com https//doi.org/ 10.25073/2588-1124/vnumap.4352 101 102 N.T Dung / VNU Journal of Science: Mathematics – Physics, Vol 35, No (2019) 101-107 type configurations In each configuration, the EIT efficiency is different, in which the -type configuration is the best, whereas the V-type configuration is the worst [7], therefore, the manipulation of light in each configuration are also different To increase the applicability of this effect, scientists have paid attention to creating many transparent windows One proposed option is to add coupling laser fields to further stimulate the states involved in the interference process This suggests that we choose to use the analytical model to determine the dispersion coefficient for the Y configuration of the 85Rb atomic system [8] The density matrix equation We consider a Y-configuration of 85Rb atom as shown in Fig State is the ground states of the level 5S1/2 (F=3) The , and states are excited states of the levels 5P3/2 (F’=3), 5D5/2 (F”=4) and 5D5/2 (F”=3) [8] Fig Four-level excitation of the Y- configuration Put this Y-configuration into three laser beams atomic frequency and intensity appropriate: a week probe laser Lp has intensity Ep with frequency p applies the transition and the Rabi frequencies of the probe p 42 E p ; two strong coupling laser Lc1 and Lc2 couple the transition and the Rabi frequencies of the two coupling fields c1 21 Ec1 and c 32 Ec , where ij is the electric dipole matrix element i j The evolution of the system, which is represented the density operator is determined by the following Liouville equation [2]: N.T Dung / VNU Journal of Science: Mathematics – Physics, Vol 35, No (2019) 101-107 103 i (1) H , t where, H represents the total Hamiltonian and Λ represents the decay part Hamilton of the systerm can be written by matrix form: H 1 1 2 2 3 3 4 4 HI p 4 2e i p t 4e i p t (2.2) c1 eic1t e ic1t (2.3) c ic t ic 2t 2e 3e In the framework of the semiclassical theory, the density matrix equations can be written as: (3.1) , H 44 p ei pt 42 ei pt 24 43 44 (3.2) , H 41 c1 eic1t 42 p ei pt 21 1 4 41 4141 2 , H 42 c1 eic1t 41 c eic 2t 43 p ei pt ( 44 22 ) 2 2 4 42 42 42 (3.3) p i p t c ic 2t e 42 e 23 3 4 43 43 43 2 , H 33 c eic 2t 32 eic 2t 23 43 44 32 33 , H 31 c1 eic1t 32 c eic 2t 21 1 3 31 3131 2 , H 32 c1 eic1t 31 c eic 2t 33 22 p ei pt 34 2 , H 43 2 3 32 32 32 , H 34 p e c ic 2t e 24 4 3 34 43 34 c ic 2t eic1t 21 eic1t 12 e 23 eic 2t 32 i p t 1 2 21 21 21 (3.5) (3.6) (3.7) 32 , H 22 c1 p i pt i t e 24 e p 42 32 33 21 22 , H 21 c1 eic1t 22 11 c eic 2t 31 p ei pt 41 2 (3.4) (3.8) (3.9) (3.10) 104 N.T Dung / VNU Journal of Science: Mathematics – Physics, Vol 35, No (2019) 101-107 , H 23 p i p t c ic 2t c1 ic1t e 22 33 e 13 e 43 2 3 2 23 32 23 , H 24 p e i p t ( 22 44 ) (3.11) c1 ic1t e 14 c eic 2t 34 2 4 2 24 42 24 , H 11 , H 12 (3.12) c1 ic1t e 12 21 21 22 (3.13) p i p t c1 ic1t c ic 2t e 11 22 e 13 e 14 2 2 1 12 21 12 c ic 2t e 12 p i p t e 12 (3.14) , H 13 c1 ic1t e 23 3 1 13 3113 (3.15) , H 14 c1 ic1t e 24 4 1 14 4114 (3.16) In addition, we suppose the initial atomic system is at a level therefore, 11 33 44 0, 22 and solve the density matrix equations under the steady-state condition by setting the time derivatives to zero: d (4) 0 dt i ( ) t i t We consider the slow variation and put: 43 43e p c , 42 42 e p , i t 41 41e p c1 , 32 32eic 2t , 31 31eic1 c t , 21 21eic1t Therefore, the equations (3.2), (3.3) and (3.4) are rewriten: i p i c1 42 21 [i( c1 p ) 41 ] 41 (5.1) 2 i p i i c1 41 c 43 ( 44 22 ) (i p 42 ) 42 (5.2) 2 i p i c 42 23 [i( p c ) 43 ]43 (5.3) 2 where, the frequency detuning of the probe and Lc1, Lc2 coupling lasers from the relevant atomic transitions are respectively determined by p p 42 , c1 c1 21 Because of p