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MINISTRY OF EDUCATION AND TRAINING VINH UNIVERSITY - LUONG THI YEN NGA CONTROLLABLE OF PULSE PROPAGATION AND OPTICAL SWITCHING IN A DEGENERATED TWOLEVELS MEDIUM Specialization: OPTICS Code: 9440110 ABSTRACT OF DOCTORAL THESIS IN PHYSICS NGHE AN - 2020 The work is accomplished at Vinh University Supervisors: Prof Dr Nguyen Huy Bang Reviewer 1: Reviewer 2: Reviewer 3: The thesis was defended before the doctoral admission board of Vinh University at … h… , … , … , 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 Nowadays, the strong development of information and communication field, the need for ultra-small and high-speed optical switches to meet the transmission of huge information data in microprocessors handling However, the basic studies on optical switching have not met the requirements of practical devices with low switching power, high switching speed and low signal attenuation Therefore, the search for materials with high nonlinear response, high switching speed is a topic that has always attracted the interest of scientists because of their potential applications in optical communication systems, optical data switching and processing systems, quantum computers In particular, recently, the discovery of the electromagnetic induction transparency (EIT) effect, it has opened up many interesting applications in quantum optics because of its ability to reduce the absorption of the medium The EIT medium not only reduces absorption but also enhances the medium's nonlinear properties Furthermore, because the height and slope of the dispersion curve can be controlled according to the parameters of the associated laser fields, the nonlinear coefficients are also controllable [16] Therefore, the EIT medium has been extensively studied in a variety of topics such as Lasing Without Inversion (LWI), group velocity slowing, quantum information, low threshold nonlinearity and surge Nonlinear Kerr strength, optical bistability and optical switching, propagation and formation of optical solitons On the other hand, the use of low intensity light controlled by coherence and quantum interference has also received great attention in recent years because of its outstanding advantages such as high response rates, switching power is low compared with switching photoelectricity and switching using silicon waveguide or conventional fiber optic systems Several methods of optical switching have been proposed and demonstrated both theoretically and experimentally based on these quantum interference and coherence effects Recently, some research groups have used the transition from the EIT effect to the electromagnetically induced absorption effect (EIA) and vice versa to study all-optical switching control Although there has been a lot of work in this field on multi-energy atom systems, in which all of the interacting fields need to be controlled synchronously However, the synchronous control of many laser fields when applied in practice will be difficult technically Hence, a simple stimulus scheme, for example, two levels that can respond to and resolve these problems is a useful and appropriate solution to the situation above Furthermore, studies often ignore the degeneration of Zeeman levels, so it is necessary to take into account the separation of energy levels when the atoms are placed in an external magnetic field or when there is polarization of laser fields be taken into consideration Therefore, we have proposed a simple model in which both optical and all-optical switches are integrated in a two-level degenerate atomic medium under the influence of an external magnetic field and the EIT effect Besides, propagating pulses with stable form (optical soliton) are always receiving research interest from research groups around the world because it has important applications in communication and optical data processing To date, most of the works on EIT-based slow light propagation have been studied in both three-level, four-level and five-level environments Therefore, we have proposed a simple model for soliton propagation and formation based on a degradated two-level atomic medium under the influence of static magnetic field and EIT Our proposed scheme will likely be useful in applications of all-optical and magnetic switches, optical and all-optical storage devices in communication signal processing, and logic gates… Given the urgency of the research problem and the reasons mentioned above, we chose the research topic "Control of pulse propagation and optical switching in a degenerated two-levels atomic medium" Purpose - Proposing an optical switches model that integrates both optical-magnetic switches and all optical switching in a degenerated two-levels atomic medium; - Proposing a model to control the ultraslow group velocity of light soliton and dark soliton in a degenerated two-levels atomic medium Object The object of study is a sample of Rb atom in a degenerated two-level atomic medium under the influence of EIT effects and an external magnetic field Methods: - Using density matrix theory to establish equations describing the process of the interaction between atoms and laser fields in the rotating wave approximation and the electric dipole approximation; - Using the rotational wave approximation and the slowly varying inclusion approximation to lead to the equation of laser pulse propagation in a degenerated twolevels atomic medium under the influence of an external magnetic field; - Using numerical methods to solve problems of propagating laser pulses and optical switching in a degenerated two-levels atomic medium in the presence of an external magnetic field under the influence of EIT effect Chapter OVERVIEW OF OPTICAL SWITCHES AND PULSE PROPAGATION IN ATOMIC MEDIUM 1.1 Introduce Optical switches or optical switch is a device used to "On" or "Off" an optical circuit An optical switch has one or more input gates and two or more output gates that we often call 1xN or NxN optical switch Based on different working principles, there are different types of switches and can be classified according to different characteristics, work functions and applications For example, according to switching agents, they are divided into optical-mechanical (mechanical agents), optical-magnetic (the agent is a magnetic field), optical-optical or all-optical (the agent is the optical field) ; according to space-time, the signal beam is divided into space optical switching, time optical switching The basic knowledge in this chapter is used to develop the main research in chapters and of the thesis on the time-based optical switching model and pulse propagation in a two-level degenerative medium 1.2 Space optical switching Space optical switching is the directional change of the signal beam when driven by different switching agents (such as mechanical, thermal, magnetic, optical) For quantum information networks, it is important to develop optical switches excited by a single photon However, the intensity of the nonlinear optical interactions of most materials is so small that achieving single-photon switching is extremely difficult To solve this problem, it is necessary to increase the nonlinear interaction by applying quantum interference effects In the work that Dawes AMC's team have studied, the author obtained the spatial optical switching pattern shown in Figure 1.1, which illustrates the switching where a signal light beam passes through a nonlinear material and emits a certain direction, this is called the “On” state of the switch The "Off" state of the switch is achieved when a weakly switched beam hits the nonlinear optical material changing the direction of the output beam Figure 1.1 Space switching scheme (all optical): switching beam impacting on nonlinear medium changes direction of signal beam when exiting the medium 1.3 Time optical switching Time optical switching is the conversion of the signal beam intensity between the states "On" and "Off" without changing the direction (space) when transmitted through the medium under the action of a switching agent Another desirable feature of all-optical switching is that the output beam is controlled by a weaker switching beam so that they can be used as classical computational elements or quantum computation components However, current switches tend to use an intense beam of light to drive a low intensity beam In the work of Dawes AMC’s team proposed in 2005, a highly sensitive all optical switching of the horizontal optical patterns produced is unstable with minor disturbances by means of quantum interference The team studied an optical sample using a beam with a power up to 6500 times weaker than the capacity contained in the sample itself, suggesting that the switch could operate at a single-photon level with optimization systems such as pump beam sizing or sample in steam form In this work, the authors presented the type of time-based all-optical switch illustrated in Figure 1.2, which illustrates the state of the switch that a signal beam passes through a nonlinear material and emits in a certain direction This is one of two states of switching; is called a "On" state The switch is switched to the "Off" state, when the beam light switches, is fed into the nonlinear optical material In this case, the light of both the signal beam and the switching beam is absorbed by the material and there is no output light beam Figure 1.2 Time-optical switching scheme: the switching beam acts on a nonlinear medium that absorbs the signal beam when passing through the medium [80] 1.4 Pulse propagation in an atomic medium 1.4.2 Maxwell – Bloch equations Consider an optical field that propagates along the z-axis and is written as a carrier envelope: → → E ( z, t ) = E ( z, t ) e − i ( kz −t ) + c.c , (1.1) where 𝐸⃗ (𝑧, 𝑡)is the envelope function, ω is the frequency of light, and k = ω/c is the wavenumber We are interested in the near-resonance interaction and assume that the field frequency ω is adjusted very close to the atomic shift frequency We assume a single optical field propagating in the z direction, then: 𝜕2 𝜕2 𝜕2 𝜕𝑧 𝑐 𝜕𝑡 𝜕𝑡 𝐸⃗ − 𝐸⃗ = 𝜇0 𝑃⃗, (1.2) Where ⃗⃗⃗𝑃 is polarized vector (space and time dependent) Next,we assume a slowly varying envelope optical field propagating to the carrier and using slowly varying envelope approximation: E z E t k E; 2E z k E , z (1.3) E; 2E t E t (1.4) Similarly, the rotating-wave variables are slowly-varying by definition: 12RW t 12RW (1.5) Using the SVEA and RWA and noting that k = ω/c, c = 1/ 0 , we find slowlyvarying wave equation: 𝜕 𝜕 −𝑖𝜔 𝑅𝑊 𝑁𝑑12 𝜌12 ( + ) 𝐸⃗ = 𝜕𝑧 𝑐 𝜕𝑡 (1.6) 2𝜀0 𝑐 or, in terms of the Rabi frequency 1 RW + = −2i12 , z c t (1.7) Here 𝜇𝑝 = 𝜔𝑁|𝑑12 |2 2𝜀0 𝑐ℏ , (1.8) is the atom-field coupling parameter The system of equations density matrix and (1.7) is called the system of equations Maxwell-Bloch considering a two-levels atom system 1.4.5 Nonlinear Schrödinger equation As stated before, strong laser pulses can induce a nonlinear polarization in matter In this case the induced polarization ⃗⃗⃗𝑃 can be decomposed to its linear and nonlinear parts as 𝑃⃗ = 𝑃⃗(𝐿) + 𝑃⃗(𝑁𝐿) (1.9) where ⃗⃗⃗𝑃(L) and 𝑃⃗(NL) are the linear and nonlinear parts of ⃗⃗⃗𝑃 Then the wave propagation equation can be expressed as 𝛻 𝐸⃖⃗ − 𝜕2 𝐸⃗ 𝑐 𝜕𝑡 = 𝜕2 𝑃⃗(𝐿) ∈0 𝑐 𝜕𝑡 +𝑠 𝜕2 𝑃⃗(𝑁𝐿) ∈0 𝑐 𝜕𝑡 , (1.10) Using the slowly varying envelope approximation one can cast the Nonlinear Schrödinger equation (NLSE) 𝜕𝐴 𝜕𝑧 = −𝑖𝛽 𝜕2 𝐴 𝜕𝜏2 + 𝑖ℚ|𝐴|2 𝐴, (1.11) where the term with β is due to the group velocity dispersion, while the term with represents the effect of third order Kerr nonlinearity Equation (1.11) describes intense optical pulse propagation through nonlinear dispersive media which, depending on the sign of the group velocity dispersion β, has two distinct types of solutions, bright or dark solitons 1.4.6 Slow optical solitons Formation of optical solitons with applications for optical buffers, phase shifters , switches, routers, transmission lines, wavelength converters , optical gates and others Solitons represent a specific type of stable shape preserving waves propagating through nonlinear media They can be formed due to a balance between dispersive and nonlinear effects leading to an undistorted propagation over long distance Most of optical solitons are generated with highly intense electromagnetic fields Also, far-off resonance excitation schemes are required to avoid any uncontrollable attenuation and distortion ofoptical waves propagation As a result, optical solitons formed in this way popagate with the speed close the speed of light in vacuum Fortunately, the EIT effect can result in significant reduction of the propagation velocity of an opticalfield The phenomenon of EIT leads also to the substantial enhancement of nonlinear effects such as a large enhancement of the Kerr nonlinearity in highly resonant media The question of interest is if the coherent effect of EIT can also be utilized in generation and propagation of optical solitons with slow group velocity Following a report of ultraslow optical solitons in a highly resonant atomic medium by Wu and Deng, these solitary waves have received a consider able attention It was demonstrated that the significant probe field spreading and attenuation due to group velocity dispersion can be precisely balanced by the Kerr nonlinear effect in a highly resonant four-state atomic system Hang et al showed that stable spatial optical solitons with extremely weaklight intensity can occur in a highly resonant three-state medium through the mechanism of EIT The formation and propagation of three-wave coupled vector optical solitonswithultraslowgroupvelocitiesinalifetime-broadenedseven-statetriple-Λ atomic system was also explored under Raman excitation Recently, the storage and retrieval of ultraslow optical solitons in an ultracold ladder-type three-level atomic gas was investigated by Chen and colleagues In our work, we report a new atom-light coupling scheme to realize such slow optical solitons under the effect of EIT Chapter OPTICAL SWITCHES IN A DEGENERATED TWO-LEVELS ATOMIC MEDIUM In this chapter, based on the effect of quantum interference in atoms, we have proposed a time-switching model that integrates both optical-magnetic switches and all-optical switches By using a degenerated two-levels atomic system housed in a solenoid tube fed by current of variable amperage, the optical properties of the atomic system to the detector laser field can vary with the the number of the external magnetic field and the control laser field 2.1.1 Model Fig 2.1 The degenerated two-level optical switching model (a) and (b) forming the two-levels energy diagram without an external magnetic field (c) under a static magnetic field We consider a degenerated two-level lambda atomic system under interaction of an external magnetic field, as shown in Fig 2.1 A weak probe laser field Ep with the left-circularly polarized component σ− (with carrier frequency ωp and a one-half Rabi-frequency 2 p = 21E p / drives the transition |1 |2 At the same time, as couplinglaser field Ec with the right-circularly polarized component σ+ (with carrier frequency ωc a one-half Rabi-frequency 2c = 23 Ec / ) applies the transition |3 |2 In this configuration (Fig.2.1),the probe field travels in the same the direction as the magnetic field B, which is used to eliminate the degradation between the ground states |1 and |3 (mF = ± 1) due to Zeeman with the separation is given by B = B mF g F B / ,where μB isthe Bohr magneton, gF is the Landé factor, and mF = ± is the magnetic quantum number The dynamical evolutionofthesystemcanbedescribedbytheLiouvilleequation: = −i H int , + , t (2.1) and the relevant density matrix equations obtained for this system are given as follows: 11 = 21 22 + i*p 21 − i p 12 , t (2.2a ) 22 = − ( 21 + 23 ) 22 + i p 12 − i*p 21 + ic 32 − i*c 23 , t (2.2b) 33 = 23 22 + i*c 23 − ic 32 , t (2.2c) + 21 = −(i ( p + B ) + 21 23 ) 21 − i p ( 22 − 11 ) + ic 31 , t (2.2d) 31 = −i ( p − c + 2 B ) 31 − i p 32 + i*c 21 , t (2.2e) 23 + = − i ( c − B ) + 21 23 23 − ic ( 22 − 33 ) + i p 13 , t (2.2f ) where the matrix elements obey conjugated and normalized conditions,namely, (i ≠ j) and 𝜌11 + 𝜌22 + 𝜌33 = 1, respectively 2.2 Probe Absorption Behaviors under Magnetic Fiel First, we consider the influence of the magnetic field on the absorption behaviors of the probe field in the presence of the coupling field by numerically solving the above density matrix Eqs (2.2) in the steady state with initial conditions 11 = 33 = and 22 = Fig 2.2 Probe absorption coefficients versus the probe detuning Δp for the absence B = (red dashed line) and presence (blue solid line) of the magnetic field: b) B = 2c c) B = -2c Other system parameters are chosen as Ωp = 0.0121, Ωc = 321, c = γ23 = γ21, respectively 2.3.3 The effect of probe field frequency Next, to see more clearly the influence of the different values of the control detuning on the switching signal, we consider the switching process of the different values of the control detuning p as shown in Figure 2.5 Figure 2.5 Time progression of detection pulse (solid line) and switching magnetic field (dashed line) at optical depth ξ = 50/α for different detector beam frequency deviations (a) Δp = 1γ21; (b) Δp = 2γ21; (c) Δp = 3γ21và (d) Δp = 4γ21 The other parameters are selected as shown in Figure 2.3 (a) Adjusting that the different values of the control detuning p, we find that the different values of the control detuning has a significant influence on the mode and switching efficiency of the probe pulse For different values of the control detuning increases by < p ≤ 2γ21, it can be seen that the efficiency decreases and the probe is switched asynchronously with the modulation of the magnetic field (Figure 2.5 (a) - (b) ) 2.4 All optical switching In this section, to consider the all-optical switching performed by the linked field itself as the control field modulation switching field Similar to Section 2.3, where the probe 11 field takes the form of a continuous wave and the time modulated bound field with an evolution of a nearly square pulse, is represented: (i) c ( ) = c 1 − 0.5 ( ai − ) − ( ai − 14 ) + ( ai − 24 ) − ( ai − 34 ) + ( ai − 44 ) − ( ai − 54 ) + ( ai − 64 ) − ( ai − 74 ) , (2.4) where, c0 is the peak strength of the bonding field, α1 = 0.4, α2 = 1.0, α3 = 2.0 and α4 = 4.0 corresponding to the switching period 50/γ21, 25/ γ21, 10/γ21 5/γ21 respectively and / γ21 Next we solve the number of equations of density matrix (2.2) and equation (2.4) over time by using the quadratic Runge - Kutta algorithm with the initial conditions is 11 ( = 0) = 33 ( = 0) = 1/ and 22 ( = 0) = 2.4.2 Effect of period modulation Figure 2.8 Time evolution of the probe field (solid line) and switching control field (dotted line) at the position of optical depth ξ = 50/α for different switching cycles: (a) 50/γ21; (b) 25/γ21; (c) 10/ γ21; (d) 5/γ21 Other parameters are selected f(ξ = 0, τ) = 1, Ωp = 0.01γ21, Ωc0= 3γ21, Δp = Δc = 0, ΔB = (hoặc B = 0), γ23 = γ21 In this section, Figure 2.8 shows the effect of the switching cycle of the control pulse on the switching process of the probe From Figure 2.8 we see that the probe is switched 12 synchronously with the switching control field However, the anterior rib of the probe pulses and is deviated from the square pulse when the switching cycle is reduced from 50/γ21 to 5/γ21 2.4.3 The effect of magnetic field strength Next we consider the effect of magnetic field strength on the switching process and switching efficiency by plotting the time progression of the detection field for the two cases B = 1γc and B = 3γc as shown in Figure 2.9 By comparing Figure 2.8 (a) with Figure 2.9 we can see that the magnetic field is very sensitive to the propagation of the detected light The switching efficiency in these cases is very low η ≈ 6.3% for Figure 2.9 (a) and η = 5% for Figure 2.9 (b), respectively Then, the detector pulse propagation is switched synchronously (Figure 2.9a) and switched synchronously (Figure 2.9b) according to the magnetic field strength B Figure 2.9 Time evolution of the probe the field (solid line) and switching control field (dashed line) at optical deth ξ = 50/α for different values of magnetic field: (a) B = 1γc; (b) B = 3γc Other parameters are selected as shown in Figure 2.8 (a) 2.4.4 Effect of field frequency deviation We continue to consider the effect of the probe pulse frequency deviation on the switching and pulse efficiency by plotting the time progression of the probe field at different probe deviations, as shown in Figure 2.10 The switching efficiency corresponding is η ≈ 47%, and η ≈ 1% When 221 < p 421, the absorption of the probe pulse is decreased gradually, the probe transmission is switched synchronously with the switching of the magnetic field, and the efficiency increased gradually corresponding η ≈ 5% and η ≈ 14.5%, respectively 13 Figure 2.10 Time progression of the detected detecting field (solid line) and switching magnetic field (dashed line) (b) at the position of optical depth ξ = 50/α and other detector laser frequency deviations each other: : (a) p = 121; (b) p = 221; (c) p = 321 (d) p = 421 The other parameters are selected as shown in Figure 2.8 (a) Chapter CONTROLLABLE PULSE PROPAGATION IN A DENEGERATED TWO-LEVELS ATOMIC MEDIUM In this chapter, we will study impulse propagation in a two-degenerate atomic medium Investigate the propagation kinetics in the linear domain, the propagation pulse is stable in the presence of EIT effect Studying the formation and propagation of solitons considering the effects of nonlinear parameters The results show that super slow solitons, the formation of light and dark solitons with super slow group velocities can be controlled Furthermore, we are also able to switch between the morning and evening soliton of the magnetic field 3.1 Linear pulse propagation On the basis of the model proposed in Chapter 2, we consider the propagation of a weak probe laser pulse in a degenerated two-levels atomic medium Using the 14 rotational wave approximation and the slow varying inclusion approximation, the detection pulse progression is represented by the wave equation: p ( z, t ) z + p ( z, t ) = i 2121 ( z, t ), c t (3.1) Nd Here, = p 21 is the propagation constant For convenience, we represent the Rabi 4 c 21 frequency of the probe field through the expression p ( z, t ) = p f ( z, t ) , where p is the real constant and is the maximum value of the Rabi frequency at the environmental input (i.e at z = 0) and f ( z, t ) is one-dimensional time and space pulse function In motion coordinate systems with = z and = t − z / c , the optical Bloch equations (2.2a) - (2.2f) for the density elements and the Maxwell wave equation (3.1) for the probe field p ( , ) = p f ( , ) can be rewritten: 11 = 21 22 + i p f * ( , ) 21 − i p f ( , ) 12 , (3.2a) 22 = − ( 21 + 23 ) 22 + i p f ( , ) 12 − i p f * ( , ) 21 + ic 32 − i*c 23 (3.2b) 33 = 23 22 + i*c 23 − ic 32 , (3.2c) + 21 = −(i ( p + B ) + 21 23 ) 21 − i p f ( , )( 22 − 11 ) + ic 31 , (3.2d) 31 = −i ( p − c + 2 B ) 31 − i p f ( , ) 32 + i*c 21 , (3.2e) 23 + 23 = − i ( c − B ) + 21 23 − ic ( 22 − 33 ) + i p f ( , ) 13 , f ( , ) = i 21 21 ( , ) ( ) p0 (3.2f) (3.2g) To consider the propagation of laser pulses in the medium, we solve the number of equations (3.2) by time and space by using a combination of quadratic Runge - Kutta algorithm and the right difference method term The selected probe initial condition is a Gaussian function f ( = 0, ) = exp[−(ln 2)( − 30) / 02 ] , where = / 21 is the initial pulse width 3.1.5 The probe pulse propagation dynamic To see more clearly the propagation of a probe laser pulse in the medium we represent the spatial and temporal progression of the square of the amplitudes of the normalized probe envelope |f(ξ,τ)|2 through the modulation of the external magnetic field intensity 15 when the parameters are selected Ωc = 321, c = p = As can be seen from Figure 2.9, turning on or off the magnetic field will strongly affect the absorbing pulses of the detector beam during propagation in the medium Specifically, in the absence of an external magnetic field (i.e., B = 0), the atomic medium becomes transparent to the probe, so the beam can propagate through the medium without loss Furthermore, the Gaussian impulse shape remains during propagation (Figure 3.4a) Conversely, when the external magnetic field is "On" with B = 2c, the probe pulse can be completely absorbed by the medium when propagating a very short distance (Figure 3.4b) Figure 3.4 The spacetime progression of the laser pulse intensity detects when the magnetic fields are "On" and "Off": ”: B = (a) B = 2c (b) Other parameters are selected : Ω0p = 0.0121, Ωc = 321, p = c = γ23 = γ21, respectively 3.2 Nonlinear pulse propagation 3.2.1 Theoretical model Figure 3.5 Transformation from a degenerated two-level (a) to a three-level lambda (b) configuration under a static magnetic field and two coupling and probe laser fields 16 We consider a degenerated two-level atomic system consists of an upper nondegenerated level as described in Chapter (corresponds to hyperfine state F = with magnetic quantum number mF = 0) and a lower degenerated level (correspond to hyperfine state F = with mF = ± 1), as shown in Fig 3.5a The atomic medium is placed in a longitudinal magnetic field B that removes the degeneracy of the lower states with the Zeeman shifts ±ΔB = µBmFgFB/ ћ , where μB is the Bohr magneton, gF is the Landé factor (Fig 3.5b) However, here the levels are excited according to the Raman scheme as shown in Figure 3.5 For simplicity in the calculation process here we use the probability amplitude approach to solve the solutions that describe the variation of the probe pulse in terms of higher order nonlinear terms Using the rotating-wave and the electric dipole approximations, the interaction Hamiltonian of system in the interaction picture can be written as (in the units of ћ=1): Hint = 2 B 3 + ( B − ) 2 − ( p + c + H c) (3.3) In the interaction picture, by using the time-dependent Schrödinger equations, the probability amplitudes equations for the relevant states are given by A2 = −i 2 B A2 + i*c A2 , t (3.4a) A3 = i ( − B + i ) A3 + i p A1 + i c A2 , t (3.4b) A1 + A2 + A3 = 1, 2 (3.4c) where An (n = 1, 3) represents amplitude of atomic wave function for each state, γ is decaying rate of the states |3〉 Under the slowly varying envelope and rotating-wave approximations, evolution of the probe field is represented by the following wave: p z + p = i 12 A2 A1* , c t (3.5) here 12 = N p 21 / c is the propagation constant, with N, µ12, c, and ε0, are the atomic density, dipole moment between levels |1〉 and |2〉, vacuum speed of light, and vacuum dielectric constant, respectively To the first-order of the probe field Ωp, we assume that the atomic is initially in the ground states |1〉 and |3〉 with A1(0) A3(0) 1/ and A2(0) = By performing the time Fourier transform of Eqs (2) and (3) and keeping up to the first order of Ω p, we obtained ( − 2 B )a2 (1) + c a3(1) = 0, (3.6a) 17 ( + − B + i )a3(1) + c a2(1) = − P , p z − (3.6b) i p = i12 a3(1) , c (3.6c) here an( n) (n= 1,2 ,3) and p are the Fourier transforms of An(1) p respectively, and ω is the Fourier variable 3.2.2 Formation conditions bright soliton and dark soliton To study formation of optical solitons, there should balance the interplay between group velocity dispersion and nonlinear effects We consider the nonlinear polarization on the right-hand sides of Eq (3.6c) and take a trial function p ( z , t ) = p ( z, t ) exp[i (0) z ], for Eq (3.5), we obtain the nonlinear wave equations for the slowly varying envelope p ( z , t ) −i[ 2 p + 1 (0) ] p + (0) = NLT , z t t where NLT is a nonlinear term given by 𝑁𝐿𝑇 = (3.7) (1) −𝜅12 𝐴3 𝑒𝑥𝑝[ − (1) 𝑖𝛽0 (0)𝑧](|𝐴2 | + (1) |𝐴3 | ) with parameters: A2(1) = −2 B p , 2D (3.8a) A3(1) = −*c p 2D (3.8b) It is convenient to transform Eq (3.8) it is convenient to convert equation (3.7) into a moving frame by changing ξ = z and τ = t − z/Vg , we obtain the following equation for Ωp with ϕ = Re[β0(0)] 0: i 2 p p − 2 ( 0) = W exp ( − ) p p , (3.9) where absorption coefficient = Im[0 (0)] and 12 B ( c + 4 2B ) W= (3.10) 4D D 18 Figure 3.6 (a) The absorption coefficient α; (b) the ratios of the imaginary and real parts of the coefficients β2i/β2r (solid) and W2i/W2r (dashed) versus the dimensionless Rabi frequency Ωc/γ This case corresponds to condition of bright solitons (β2r.Wr > 0) Furthermore, as we can see below, for the practical parameters one may find conditions so that the imaginary part of the complex coefficient in Eq (3.8) much smaller than their corresponding real part, i.e., 2 (0) = 2r (0) + i2i (0) = 2r (0), , and W = Wr + iWi = Wr Under the regime of these parameters, we can neglect the imaginary parts and make the Eq (3.10) to be integrable, then Eq (3.9) can be reduced to the standard nonlinear Schrödinger equation: 2 p i p − r (0) = W p, r p (3.11) which admits the solutions describing various types of solitons, such as the right (β2rWr > 0) and dark (β2rWr < 0) solitons, depending on choosing the parameters Figure 3.7 (a) The absorption coefficient α; (b) the ratios β2i/β2r (solid) and W2i/W2r (dashed) versus Ωc/γ with the parameters as same as those in Fig except for ΔB = 0.33γ or B = 0.33 γc This case corresponds to the dark solitons (β2r.Wr