Summary of Doctoral Thesis in Physics: Influence of spontaneously generated coherence and relative phase between laser fields on optical properties of three-level atomic medium

Research purpose: to build an analytical model describing the dependence of the optical properties (absorption, dispersion, group velocity and group delay) of the three configuration energy level system Λ, Ξ and V according to the following parameters: control parameters (intensity, frequency, polarization and relative phase) of the laser fields.

MINISTRY OF EDUCATION AND TRAINING VINH UNIVERSITY -LE NGUYEN MAI ANH INFLUENCE OF SPONTANEROUSLY GENERATED COHERENCE AND RELATIVE PHASE BETWEEN LASER FIELDS ON OPTICAL PROPERTIES OF THREE-LEVEL ATOMIC MEDIUM Specialization: OPTICS Code No: 44 01 10 SUMMARY OF DOCTORAL THESIS IN PHYSICS NGHE AN, 2020 i The work is accomplished at Vinh University Supervisors: Prof Dr Nguyen Huy Bang Dr Le Van Doai Reviewer 1: Reviewer 2: Reviewer 3: The thesis is defended before the doctoral evaluation board of Vinh University at … … , …… , …… , 2020 The thesis can be found at: - Nguyen Thuc Hao Information Centre - Library of Vinh University - Viet Nam National Library PREFACE Reason to choice the investigation subject Absorption and dispersion are two basic parameters that characterize the optical properties of the atomic medium These two parameters are related in real and imaginary parts of the susceptibility In addition to the linear susceptibility, atoms also have nonlinear inductances but they are usually of very small value, so only light sources of high intensity can observe optical nonlinear phenomena Therefore, the advent of lasers has opened up many new research directions and related applications One of the most interesting research is to control the optical properties of an atom based on the quantum interference effects of the shifting probability amplitudes in the atom Among quantum interference effects, EIT (Electromagnetically Induced Transparency) is the earliest studied Accordingly, a probe and a pump laser field simultaneously excite two atomic shifts in common according to lambda (), ladder () and V (V) configurations Based on quantum theory, the above stimulation of the atomic system will lead to a superposition of the shifting probability amplitudes in the atom system, thus producing quantum interference between the shifting channels As a result, the amplitude of the total shifting probability can be either destructive (EIT) or enhanced, known as the electromagnetically Induced Absorption (EIA) So far, the EIT effect (related to absorption and dispersion) has been widely studied both theoretically and experimentally in the three configurable energy level -, - and V-type systems These studies show, as the intensity of the pump laser beam increases, the depth and width of the EIT windows also increase, while the height of the dispersion lines usually increases but the slope decreases In addition, the position of the EIT windows is also shifted to the short wavelength or to the long wavelength by varying the frequency of the pump laser beam accordingly Besides the studies of EIT effects in separate three-level configurations, the study comparing the absorption and dispersion properties in the presence of the EIT effect has also been of interest to the researchers Comparative research shows that, due to the arrangement of different energy levels between excitation configurations (,  and V), the efficiency of the quantum interference is very different and therefore the EIT efficiency is also different Specifically, the EIT effect occurs more easily for -type than for - and V-type systems In addition to the EIT and EIA effects described above, another quantum interference effect occurs between the spontaneous emission channels due to the non-orthogonal orientation of the electric dipole moments induced by the between probe and pump laser field Non-orthogonal orientation of atomic dipole moments can be achieved by polarization between laser fields The result of this interference producing atomic coherence is called the coherence created by Spontaneously Generated Coherence (SGC) The SGC effect can occur with the EIT effect in atomic systems with three configurable energy levels , , and V-type systems For the EIT effect, the intensity of the quantum interference depends on the intensity of the pump laser beam For SGC effect, the intensity of quantum interference depends on the polarization between the probe laser beam and pump laser beam In the presence of polarization between laser beams, both EIT and SGC effects can occur simultaneously in the medium Interestingly, the SGC effect also significantly changes the optical properties of the medium The studies show that the SGC effect makes the medium more transparent, but the transparent spectral width is narrowed, so the dispersion curve becomes steeper In addition, the influence of SGC makes the atomic medium asymmetric, so the response of the medium is very sensitive to the relative phase of the probe and pump laser fields Up to now, the effects of SGC and the relative phase on group velocity, lasing without population inversion, enhancement Kerr nonlinear, controlling optical bistability, controlling pulse propagation, have been widely studied Along with the studies on the influence of SGC on absorption and dispersion, there have been many studies on controlling light group velocity From there, we can shift between superluminar to subluninar and vice versa Besides, the group velocity can be reduced differently in each configuration leading to increased group delay (plays an important role in reducing the distortion of light pulses) This exciting feature could create breakthrough applications in optical communication and information processing technology Although, the effects of SGC and the relative phase between laser fields on the optical properties of the three energy atomic medium in three excitation configurations have been studied However, firstly, most of the current studies mainly use the numerical method (although there are some studies on the influence of SGC on optical properties by analytical methods, they must use using a incoherent pump laser and approximate), so investigations of the dependence of optical atomic properties on laser parameters are limited and have not shown continuous changes in properties optical according to the control parameters Furthermore, numerical studies will not favor the optimal selection of experimental parameters Second, up to now, there have been no studies to evaluate and compare the effects in the presence of SGC and without SGC in weak and strong probe fields on the optical properties of three-level -, Ξ- and V-type This comparison will be essential to see the advantages and disadvantages of each configuration and to have the appropriate selection for application purposes and experimental At the same time, we found extremely small group velocities in the ultraslow light with extremely large delays in each of the three energy configurations Facing the unresolved problems of the research field and the results achieved by the group, we chose the topic "Influence of spontaneously generated coherence and relative phase between laser fields on optical properties of three-level atomic medium” graduate thesis Chapter Basics of control of optical properties by laser 1.1 Introduction 1.2 Theoretical basis of light propagation in the medium 1.2.2 Absorption and dispersion The refractive index of the medium is determined by: (1.14) n =  = +  (1)  +  (1) In the general case, a complex refractive index is analyzed into the real and imaginary parts as follows: n = n + in (1.15) To examine the physical significance of n and n’, we examine an electromagnetic wave traveling through a medium on the z-axis, which is the solution of the wave propagation equation:   n  n   E = E0 exp  − z  exp i  z − t    c    c (1.16) Thus, the real part related to dispersion describes the wave vector change of the electromagnetic wave The imaginary part associated with the absorption wave is attenuated exponentially when propagated in a matter medium According to Bear's Law, I = − I , z (1.17) In which, I is the light intensity We find the solution of equation (1.17) in the form: I ( z ) = I exp[− z ] (1.18) By defining the absorption coefficient: = n  (1) c = c , (1.19) With is the imaginary part of linear susceptibility Accordingly, when  > 0, the electromagnetic wave is absorbed exponentially while when  < 0, the light wave is amplified when propagating in the medium 1.3 Group and phase velocities 1.3.1 Phase velocity Investigate a monochromatic flat wave with angular frequency propagating in a medium with refractive index n This wave can be described by equation: (1.20) E( z, t ) = Aei ( kz −t ) + c.c , n Inside, k = is the wave constant c We have phase velocity: vp = vp = or z , t  k = (1.23) c n (1.24) 1.3.2 Group velocity The group velocity also depends on the susceptibility: vg = c   d (  (1) ) 1+ + 2 d (1) (1.29) Thus, it is clear that the greater the angular coefficient of the normal dispersion on the dispersion shape, the greater the dispersion dn/d 1.5 Group delay of pulses Group delay of pulses propagating in the medium compared with delay when light pulses propagate in vacuum according to the formula: Tdel = L (ng − 1) c (1.43) Equation (1.43) shows the maximum time delay determined by the group refractive index ng and the Lmax value of the distance L propagation through the material medium Chapter Influences of polarization and relative phase between laser fields on absorption and dispersion 2.1 Atomic stimulation according to a -type configuration 2.1.1 Equations of density matrix Consider three energy levels excited by two laser fields according to lambda configurations shown in Figure 2.1 (a) Figure 2.1 (a) Schematics of the three-level -type atomic system and (b) - the polarization is chosen such that one field only drives one transition 2.1.3 The influence of the relative phase between laser fields When taking into account the interference between spontaneous and phase emissions between laser fields, we have the following system of density matrix equations: (2.21a) 11 = iGp ( 21 − 12 ) + 21 22 − 011 ,  22 = iG p ( 12 −  21 ) + iGc ( 32 −  23 ) −  22 22 ,  33 = iGc ( 23 −  32 ) + 22 22 − o 33 ,  21 =  21 21 + iGp (11 −  22 ) + iGc 31, 12 =  1212 − iGp (11 −  22 ) − iGc13 ,  23 =  23 23 + iGp13 + iGc ( 33 −  22 ),  31 =  31 31 − iGp 32 + iGc 21 + 221 22 ,  32 =  32 32 − iGp 31 − iGc ( 33 −  22 ),  13 =  13 13 + iGp 23 − iGc 12 + 221* 22 (2.21b) (2.21c) (2.21d) (2.21e) (2.21f) (2.21g) (2.21h) (2.21i) where 31 = 3 - 1 is the frequency difference between level |3 and level |1 If levels |1 and |3 lie so closely that the SGC effect has to be taken into account, then η = 1, otherwise η = Remark: From the equations above, we see that equations of 31 and 13 appear term 221 22 related to the interference of the relative phase and spontaneous emissions between the laser fields This term is called the coherence produced by spontaneously Generated Coherence (SGCSpontaneously Generated Coherence), where  is characteristic of the phase dependence of laser fields If the energy levels |3 and the |1 level are distributed far apart, the terms of the oscillation ei31t rotate very quickly, possibly approaching and  = (that is, the coherence of spontaneous emission or difference SCG application will not occur) So to study the interference enhancement of the optimal spontaneous emission in the configurations, we have to assume |3 and |1 two lower closely spaced levels, near the degeneration level then 31  (or the energy between |3 and |1 is quite small), so  = eit 2.2 Atomic stimulation in a -type configuration Schematic diagram of atomic system excitation of three energy levels in a ladder configuration is described in Figure 2.2(a) Figure 2.2 (a) Schematics of the three-level -type atomic system (b) The polarization is chosen such that one field only drives one transition Similar to the above, we calculate the following equations: 11 = 21 22 + iGp ( 21 − 12 ),  22 = −21 22 + 22 33 + iGp (12 −  21 ) + iGc ( 32 −  23 ),  33 = −2 2 33 + iGc ( 23 −  32 ),  21 =  21 21 − iGp ( 22 − 11 ) + iGc 31 + p 12 32 , 12 =  1212 + iGp ( 22 − 11 ) − iGc13 + p 12* 23 ,  32 =  32 32 − iGp 31 − iGc ( 33 −  22 ),  23 =  23 23 + iGp13 + iGc ( 33 −  22 ),  31 =  31 31 − iGp 32 + iGc 21 , 13 =  1313 + iGp 23 − iGc12 (2.42a) (2.42b) (2.42c) (2.42d) (2.42e) (2.42f) (2.42g) (2.42h) (2.42i) 2.3 Atomic stimulation in a V-type configuration Excitation of three energy-level atomic systems in a V-type configuration is shown in Figure 2.3 (a) Figure 2.3 (a) Schematics of the three-level V-type atomic system and (b) - the polarization is chosen such that one field only drives one transition Call  p =  p −  21 and  p =  p −  21 , respectively, the frequency offset of the probe and control beam We calculate the following system of equations:  11 = iGp ( 21 − 12 ) + iGc ( 31 − 13 ) + 21 22 + 22 33 + 221 ( 32 + * 23 ),  22 = iGp ( 12 −  21 ) − 21 22 − 21 ( 32 +  23 ), *  33 = iGc (13 −  31 ) − 22 33 − 21 ( 32 + * 23 ),  21 =  21 21 + iGp (11 −  22 ) − iGc 23 − 12 31,  12 =  12 12 + iG p ( 22 −  11 ) + iGc 32 −  21* 13 ,  23 =  23 23 + iGp13 − iGc 21 − 12 ( 33 +  22 ),  31 =  31 31 − iGp 32 + iGc (11 −  33 ) − 21 21,  32 =  32 32 − iGp 31 + iGc 12 − 21* ( 22 +  33 ), 13 =  1313 + iGp 23 + iGc ( 33 −  11 ) − 12* 12 (2.67a) (2.67b) (2.67c) (2.67d) (2.67e) (2.67f) (2.67g) (2.67h) (2.67i) 2.4 Absorption and dispersion coefficients After simpliflcation the expression for the dielectric susceptibility related to 21 for the given atom-fleld system can be written as 2 N 21 =  (2.95)  0G p 21 On the other hand, susceptibility  can be decomposed into real   and imaginary   components as follows: (2.96)  =   + i  According to the Kramer-Kronig relation, the real and imaginary parts of susceptibility are directly related to the linear dispersion coefficient and the linear absorption coefficient by:    = p , (2.97) c   n= p (2.98) 2c 2.5 Controlling absorption and dispersion coefficients To study absorption and dispersion controls according to laser parameters such as intensity, frequency, polarization and relative phase We replace the expressions for 21 found in the three configurations above and apply the calculation results to the 85Rb gas atomic medium and ignore the effects of Doppler broadening Figure 2.4 Schematics of the 85Rb atom system give a three-energy excitation configuration: (a) lambda, (b) ladder, (c) V-type configuration For the -type configuration, the energy levels |1, |2 |3 corresponding to the states 52 S1/2 F = , 52 P1/2 F = 52 S1/2 F = For the V-type configuration , energy levels |1, |2 |3 corresponding to the states 52 S1/2 F = , 52 P1/2 F = 52 P3/2 F = For the -type configuration, energy levels |1, |2 |3 corresponding to the states 52 S1/2 F = , 52 P1/2 F = 52 D5/2 F = Atomic density N = 1012 atom/cm3; dipole moments d 21 = 2.53  10−29 C.m và frequency shift D1 is  p = 3.77  1014 Hz ; spontaneous emission rates for the Λ- and V-systems 1 = 2 =  ; and systems 1 =  2 = 0.16 For simplicity, all the parameters related to frequency are given in units of  = 2×5.75 MHz 2.5.1 Influence of the SGC In this section, we examine the effect of SGC (generated by light polarization and characterized by interference parameter p) on the absorption and dispersion of the medium by fixing the intensity, the frequency and relative phase of the laser fields at Gp = 5, Gc = 10, c =  = 0,, are depicted in Figures 2.5 and 2.6 respectively In general, in all three configurations, the influence of SGC on absorption in the V-configuration is more effective This can be explained by the larger spontaneous emission rate in V-configuration due to the stronger coherence created by spontaneous emission (ie term 12* 12 ) Furthermore, comparing the density matrix equations of the three configurations, we see the term "coherence generated by spontaneous emission" appearing in all density matrix equations in the configuration, but in the lambda and ladder configurations this term appears only in the equation for 21 Figure 2.5 Variations of absorption coefficient versus probe field detuning for different values of p = 0.9 (solid line), p = 0.7 (dotted-dash line), p = (dash line): (a) , (b)  and (c) V-type Figure 2.5 Variations of dispersion coefficient versus probe field detuning for different values of p = 0.9 (solid line), p = 0.7 (dotted-dash line), p = (dash line): (a) , (b)  and (c) V-type 10 This phenomenon can be explained as follows: EIT effect is a result of destructive interference of probability amplitudes induced by laser fields or enhanced interference resulting in EIA - Electromagnetically Induced Absorption Besides, there is also the appearance of the term " Spontaneously Generated Coherence - SGC" which is a newly generated oscillating source (corresponding to a spontaneous emission shift probability) Thus, when the effects of EIT and SGC are present at the same time, inside the atom, interference of inductive probability amplitude and spontaneous emission probability will occur Interference of many such "coherent sources" will reduce the "interference pattern", ie, the spectral EIT (destructive interference) and EIA (enhanced interference) as shown in Figure 2.5 Increasing the interference parameter p is increasing the intensity of the spontaneous emission source Figure 2.7 Variations of absorption (a) and dispersion (b) coefficient versus p in Λ-type system (solid line), ladder-type system (dot-dashed line) and (c) V-type system (dashed line) To see this difference more intuitively, we compare the variation of the absorption and dispersion coefficients of the three configurations according to the interference parameter p as described in Figure 2.7 The parameters selected are  = 0, ∆c = 0, Gp = 5γ, Gc = 10γ and ∆p = 4γ corresponding to an EIT window near point From Figure 2.7 (b), the dispersion coefficient significantly changes the dispersion properties of the medium when the interference parameter p changes from 0.7 to Along with the absorption change from EIT to EIA, the dispersion was also changed from conventional to anomalous dispersion In particular, the variation of the dispersion curve with parameter p in the Vconfiguration is opposite to the lambda and ladder configuration Therefore, the light propagation properties can be changed from fast light to slow light 11 2.5.2 Influence of coupling intensity The variation of the absorption coefficient (a) and dispersion (b) in Figure 2.10, shows that when changing the intensity of the laser control from Gc = to Gc = 5, the absorption of the medium to the probe field is small in three configurations due to the distance from the resonance frequency at p = 4 At the same time, the dispersion curves have small and negative values (or anomalous dispersion) in the lambda and ladder configurations but the dispersion curves have small and positive values (or ordinary dispersion) in V-configuration In particular, when the control field strength increases from Gc = 5 to Gc = 11, the absorption on both sides of the EIT window increases rapidly due to the SGC effect, and reaches the maximum value of lambda configuration (solid line), descending for ladder configurations (dot-dashed line) and V (dashed line) Figure 2.10 Variations of absorption (a) and dispersion (b) coefficient versus Gc (intensity of the coupling field) in Λ-type system (solid line), ladder-type system (dotdashed line) and (c) V-type system (dashed line) 2.5.3 Influence of coupling detuning frequency To compare the changes of absorption and dispersion with the controlled laser frequency in the three configurations, we plot the absorption coefficient (Figure 2.13a) and dispersion coefficient (Figure 2.13b) with deturning frequency of control fields ∆c in all three configurations Other parameters selected are  = 0, p = 0.9, ∆p = 4γ, Gc = 10γ và Gp = 5γ The results in Figure 2.13a show that the low point of the EIT window is around the control field frequency at ∆c = -4γ, corresponding to the dispersion curve (Figure 2.13b) which is usually the steepest and reaches its maximum at ∆c = -4γ for lambda configurations (solid line), descending in ladder configuration (dashed line), Conversely, V-shaped configuration 12 (dashed line), small anomaly dispersion curve around frequency -4 < c < The results in this show that by adjusting the laser-controlled frequency, the amplitude and slope of the absorption curve are also changed This leads to a change in the magnitude and sign of the dispersion coefficient However, the variation of the dispersion curve in the V-configuration is in contrast to the lambda and ladder profiles due to the influence of SGC in the stronger Vconfiguration Figure 2.13 Variations of absorption (a) and dispersion (b) coefficient versus c (frequency of the coupling field) in Λ-type system (solid line), ladder-type system (dotdashed line) and (c) V-type system (dashed line) 2.5.4 Influence of relative phase between laser fields Corresponding changing the absorption coefficient, the dispersion curve also changes with period 2 between positive and negative values In each period 2π, the maximum value of the dispersion coefficient is achieved at the relative phase  = π/2, while the maximum negative value of the dispersion reaches  = 3π/2 for the lambda configuration and step In contrast, in the V-configuration the positive dispersion curve is greatest at  = π/2 Figure 2.16 Variations of absorption (a) and dispersion (b) coefficient versus  (relative phase) in Λ-type system (solid line), ladder-type system (dot-dashed line) and (c) V-type system (dashed line) 13 Thus, we see that the dispersion coefficient amplitude, sign and slope can be controlled by the relative phase between the laser fields This means that we can fully control the propagation of light in relative phases, discussed in Chapter Chapter Influences of of polarization and relative phase between laser fields on group velocity and group delay 3.1 Group index and group delay 3.1.1 Group refractive index The group refractive index of the atomic medium for the probe laser beam is related to the linear dispersion coefficient (1.28), where n is the linear dispersion coefficient determined from the real part of the susceptibility according to the following expression: N 21 =   0G p 21 (3.1) 3.1.2 Group delay The group delay of a light pulse propagating in the medium is defined as the difference between the propagation time of the pulse in the atomic medium versus the propagation time of the light pulse in a vacuum: L Tdel = (ng − 1) (3.3) c From equation (3.4), we see that for maximum delay value, the propagation length L and group ng refractive index reach maximum value The propagation length reaches the maximum value calculated by the formula: Lmax = (3.4) 2 where Tdel = (ng − 1) (3.5) 2 c In which,  is the absorption coefficient of the medium, determined by the formula (2.97) 14 3.2 Controlling group index 3.2.1 Influence of the SGC To compare the variation of the group refractive index in the three excitation configurations, we plot the group refractive index according to the interference parameter p when fixed relative phase parameters  = 0, frequency and intensity laser field at ∆c = 0, ∆p = 4γ, Gp = 5γ and Gc = 10γ Figure 3.2 Variations of ng group refractive index versus p in Λ-type system (solid line), ladder-type system (dot-dashed line) and (c) V-type system (dashed line) The results, depicted in Figure 3.2, show that the effect of SGC on group refractive index or group velocity becomes more pronounced when p> 0.7 In particular, when p increases from 0.7 to 1, the ng group refractive index changes from negative to positive (for  and -type configurations) or from positive to negative That is, you can use the interference parameter p to change the mode of light propagation from fast to slow and vice versa Depending on the excited configuration, the amplitude of the group refractive index will be different For the V-type configuration, the variation in the amplitude of the group's refractive index is largest between the fast and slow light; descending for lambda configurations and minimum for ladder configurations 3.2.2 Influence of coupling intensity Under the SGC influence, we can change the light propagation mode by varying the intensity of the controlled laser as illustrated in Figure 3.4 for (solid line), - (dotted line) and the V-type systems (dashed line) Other parameters used are  = 0, p = 0.9, p = 4γ, c = 0, Gp = 5γ It is clear that the group refractive index changes between the positive and negative values with increasing the control laser intensity around Gc = 10 In addition, the change 15 of the group refractive index according to the control field intensity in the Vtype configuration is also opposite to the other two configurations Figure 3.4 Variations of ng group refractive index versus Gc in Λ-type system (solid line), ladder-type system (dot-dashed line) and (c) V-type system (dashed line) 3.2.3 Influence of coupling detuning frequency To see more clearly the dependence of ng versus the control laser frequency, we plot the group refractive index according to the detuning frequency of the control laser as shown in Figure Other parameters used in Figure 3.6 are  = 0, p = 0.9, p = 4γ, Gp = 5γ và Gc = 10γ By varying the controlled laser frequency around the resonant frequency, the group refractive index changes between positive and negative values Similarly, the change of group ng refractive index with frequency ∆c in the V-type configuration is the opposite of the other two configurations Figure 3.6 Variations of ng group refractive index versus p in Λ-type system (solid line), ladder-type system (dot-dashed line) and (c) V-type system (dashed line) 16 3.2.4 Influence of relative phase between laser fields To see more clearly changing relative phase, we plot the group refractive index versus relative phase as shown in Figure 3.8 Here, the parameters used are p = 0.9, ∆p = ∆c = 0, Gp = 0.01γ and Gc = 5γ From the above graph we can see that the group refractive index changes in a period of 2π between the positive maximum and the negative minimum So we can also use the relative phase to convert light's propagation properties between fast and slow light In addition, we also see that in each period of 2π, the maximum positive value of the group refractive index is obtained in the relative phase  = π, while the maximum negative value of the group refractive index reaches  = and 2π in - and -type systems In contrast, for a V-type system, the largest positive group refractive index occurs at  = and 2π and the largest negative group refractive index occurs at  = π Figure 3.8 Variations of ng group refractive index versus  in Λ-type system (solid line), ladder-type system (dot-dashed line) and (c) V-type system (dashed line) 3.4 Comparing the effect of SGC and relative phase in weak and strong probe field modes In the above items, we have studied the effects of SGC and the relative phase on the group refractive index when the probe field is at the same order as the control field However, for the EIT medium, the probe beam is usually a few orders less intense than the control beam, so in this section we study the effects of SGC and the relative phase on the group refractive index and group delay in both weak and strong field At the same time, to compare with the results previously published in the weak probe mode 17 3.4.1 Influence the effects of SGC in weak and strong probe field modes We see Figure 3.13 in the - and V-type systems, the group refractive index is significantly increased in the weak probe field mode Especially in the -type system, the group refractive index increases greatly in both fast and slow light when the interference parameter increases (p = 0.99) at the coupling laser intensity is reduced by a very small about 0.0113, with group velocity (vg)min = 0.3510-4 m/s Next, the group refractive index changes sign and enhances in the fast light, having maximum magnitude with group velocity (vg)min = 2.0810-4 m/s at the coupling laser intensity Gc = 0.4 in the V-type system Besides, the group refractive index does not change significantly without SGC and has group velocity (vg)min = 590 m/s at different coupling laser intensity from Gc = 3.5 to Gc = 20 in the -type system Figure 3.13 Variations of ng group refractive index versus Gc at different values of interference parameter p = (dotted line) and p = 0.99 (red solid line) in Λ-type, (blue dotdashed line) in -type and (c) (green dashed line) in V-type corresponding to the probe field Gp = 0.01 (a,b,c) Gp =  (d,e,f) At resonance frequency p = c = 18 This shows that, for weak probe field and rather small control field intensity, the maximum quantum interference efficiency leads to the maximized group refractive index in - and V-type But in weak probe field and coupling field strength, the quantum interference efficiency leads to enhanced group refractive index in -type system 3.4.2 Influence the relative phase in weak and strong probe field modes As the results of section 3.2, when we consider the influence of SGC together with the relative phase, the group refractive index is most enhanced Therefore, we select the interference parameter p and the control field intensity Gc at the maximum group refractive index in each configuration and plot group refractive index versus relative phase in the weak field mode (Gp = 0.01) as Figure 3.14a, Figure 3.14b, Figure 3.14c and strong field mode (Gp = ) as Figure 3.14d, Figure 3.14e, Figure 3.14f Figure 3.14 Variations of ng group refractive index versus  at different values of laser coupling intensity Gc = 0.0113 (a), Gc = 20 (b), Gc = 0.4 (c) in weak field; Gc = 1.12 (d); Gc = 20 (e); Gc = 1.5 (f) in strong field for -type (red solid line), ladder (blue dashed line) and V (green dashed line) At resonance frequency p = c = 19 In the presence of SGC and relative phase, the group velocity achieves the minimum value (vg)min = 0.3510-4 m/s in -type system, followed by minimum group velocity (vg)min = 2.0710-4 m/s in V-type system and minimum group velocity (vg)min = 590 m/s in -type system For -type system, the refractive index of the enhancement group is greatest compared to the absence of SGC influence, gradually decreases in the V-type system and doesn’t change significantly in the -type system The explanation for this change: First, in the previous work, we compared the different minimum group velocities in the three configurations, which is based on the lowest coherence reduction rate 31 = 0.05 MHz (-type system) the lowest so the interference efficiency is large Most of the three configurations, simultaneously, combine the interference of the spontaneous emission - SGC and the relative phase between the laser fields, which are specifically described by the terms 221 22 in Equation (2.21g), resulting in interference maximized quantum or group velocity reaches minimum value Although in the V- system configuration, the rate of coherence degradation is (32 = MHz), but when examining the same SGC effect, we see that this coherence appears in all nine equations (2.67 ), the presence of these multiple sources leads to maximum interference efficiency in the V- system leading to a 104-fold increase in the mark-change group refractive index compared to the absence of SGC For the ladder configuration, the coherence attenuation rate is 31 = 3.5 MHz and there are only a number of terms 221 32 describing the association of SGC and the relative phase in Equation (2.42d), so this interference increases significantly Second, in order to consider the SGC effect, we need to condition that these energy levels are close together (- and V-type system) and equally spaced (-type system) Where energy levels are close to each other (21  23) then the quantum interference efficiency of the spontaneous emission between the atoms at a high level of 2 near the degradation in -type system will be greater than in the case of close together high energy levels 2 3 (21  31) in the V-type system As the -type system, the energy levels are equally spaced at the levels 2 3 (21  32) together with the decay rate from 3 to small 2 (2 = 0.16), so the interference of spontaneous emission will be the lowest compared to - and V-type system (2 = ) 20 3.4.3 Influence of relative phase on delay group in weak probe field modes In the above items, the influence of the relative phase on the group refractive index is quite sensitive and the biggest increase in the weak field From there, we plot the group delay by relative phase  at different values of the control field strength Gp when the group refractive index reaches maximum Gp = 0.0113 in a -type system (solid line), Gp = 20 in -type system (dashed line) and Gp = 0.4 in -type system (dashed line) in weak field, as shown in Figure 3.15 With the parameters of the laser field selected p = 0.99, Gp = 0.01 and p = c = From Figure 3.15, in the range from to 2, the maximum group delay (Tdel)max = 1060 s with the minimum group velocity (vg)min = 2.6310-4 m/s corresponding to the maximum group refractive index (ng)max  1.14×1012 at relative phase  =  - in ultraslow light for -type system; decreasing according to the V-type system with the largest group delay (Tdel)max = 2.9610-3 s at the relative phase  = 2; and maximum group delay (Tdel)max = 1.7674710-4 s for -type system at relative phase  = -  We see that the influence of the relative phase on the group delay increases significantly as the control laser field decreases in -type system Figure 3.15 Variations of delay group Tdel versus  in - (solid line), - (dashed) and V-type (dashed) The parameters of the laser field are Gp = 0.01, p = 0.99; Gc = 0.0113 (a); Gc = 20 (b); Gc = 0.4 (c) p = c = 21 CONCLUSIONS This thesis proposes to study the simultaneous effects of EIT and SGC on optical properties of atoms In particular, it shows the influence of SGC and relative phase on the group refractive index / group velocity and group delay in the strong and weak probe fields By using density matrix formalism and semi-classical theory, we have derived accurate analytical expressions of the absorption, dispersion, group refractive index and group delay for the -, - and V-type configurations According to the parameters of the control laser field: intensity, frequency, polarization and relative phase Applied to gas atomic medium 85Rb obtained the following results: Absorption coefficient changes to polarization, intensity, frequency and relative phase between laser fields, absorption can be reduced (EIT effect) or enhanced (EIA effect) Leads to a change in absorption rate in -type configuration larger than in - and V-type configurations The amplitude and slope of the dispersion (hence the group refractive index / group velocity) change simultaneously by the parameters of the laser fields In particular, the dispersion curve in V-shaped configuration reverses the sign compared to configurations  and  At the same time, the dispersion change rate according to the set of control parameters between the three configurations is also different In the presence of SGC, the group index changes significantly in the interference parameter p from 0.7 to In this parameter, the group refractive index changes rapidly with intensity, frequency and relative phase In particular, we can switch between superluminar to subluminar by changing the angle between the shifting dipole moments In weak field, we can select the optimal parameter set for light to propagate with the smallest group velocity of 2.63×10-4 m/s in the subluminal corresponding to the delay time maximum of about 1060 s by simultaneously optimizing the control parameters, maximum delay reaches the maximum value in  type In the presence of SGC, orientation of the atom non-orthogonal electric dipole moments, so the group refractive index changes with the relative phase for the period of 2 The group refractive index has a negative value with maximum amplitude at  = and 2π and a positive value with maximum amplitude at  = -π và  in both - and -type configurations However, in the V-type configuration, the group refractive index has a negative value with the largest amplitude at  = -π và , a positive value with the largest amplitude at  = and 2π So we can switch between superluminar and subluminar by changing the relative phase between two laser fields 22 The model of studying optical properties of an atom according to all parameters of the laser field (intensity, frequency, polarization, phase) proposed in this thesis is a comprehensive solution for control laser optical properties Moreover, with the results obtained by analysis, in addition to accurately describing the phenomenon, it also describes the picture of the continuously changing optical properties according to the parameters At the same time, the research results suggest the optimal excitation configuration and parameter selection in relevant applications The main research results of the thesis have been published in 02 articles in international journals in the ISI list and 03 internal articles 23 LIST OF PAPERS RELATED TO THE THESIS Nguyen Huy Bang, Le Nguyen Mai Anh, Nguyen Tien Dung, and Le Van Doai, “Comparative Study of Light Manipulation in Three-level Systems via Spontaneously Generated Coherence and Relative Phase of Laser Fields”, Communications in Theoretical Physics, 71 (2019) 947-954 Dinh Xuan Khoa, Le Van Doai, Le Nguyen Mai Anh, Le Canh Trung, Phan Van Thuan, Nguyen Tien Dung, and Nguyen Huy Bang: “Optical bistability in a five-level cascade EIT medium: An analytical approach”, Journal of the Optical Society of America B, 33, 04 (2016) 735-740 Le Nguyen Mai Anh, Le Van Doai, and Nguyen Huy Bang, “Slower light with amplification via spontaneously generated coherence in a three-level cascade-type system under incoherent pumping” 2020 (Submitted in Physics Laser) Le Nguyen Mai Anh, Nguyen Huy Bang, Nguyen Van Phu and Le Van Doai, “Effect of spontaneously generated coherence on absorption, dispersion and group velocity in a five - level cascade system”, 2020 (Submitted in The European Physical Journal D) Le Nguyen Mai Anh, Nguyen Huy Bang, Le Thi Hong Hieu Le Van Đoai, “Ảnh hưởng phân cực pha tương đối trường laser lên tính chất quang mơi trường nguyên tử ba mức Bậc thang”, Tạp chí Khoa học Trường Đại học Vinh, tập 48, số 1A (2019), trang 5-15 Le Nguyen Mai Anh, Le Van Doai, Dinh Xuan Khoa, and Nguyen Huy Bang, “Influences of spontaneously generated coherence and relative phase on group velocity in a three-level atomic medium: analytical approach”, The 5th Academic Conference on Natural Science for Young Scientists, Masters, and Ph.D Students from ASEAN Countries, (2018), Da Lat, Vietnam, ISBN: 978-604-913-088-5, pp118-123 Nguyen Thi Minh Hue, Le Van Doai, Le Nguyen Mai Anh, Vu Ngoc Sau, Đinh Xuan Khoa Nguyen Huy Bang, “Nghiên cứu ảnh hưởng định hướng giữa mômen lưỡng cực điện dịch chuyển lên đảo lộn cư trú nguyên tử ba mức cấu hình Bậc thang”, Tạp chí Khoa học Trường Đại học Vinh, tập 43, số 3A (2014), trang 35-42 24 ... propagation through the material medium Chapter Influences of polarization and relative phase between laser fields on absorption and dispersion 2.1 Atomic stimulation according to a -type configuration... of the research field and the results achieved by the group, we chose the topic "Influence of spontaneously generated coherence and relative phase between laser fields on optical properties of. .. effects of EIT and SGC on optical properties of atoms In particular, it shows the influence of SGC and relative phase on the group refractive index / group velocity and group delay in the strong and