Optics Communications 426 (2018) 553–557 Contents lists available at ScienceDirect Optics Communications journal homepage: www.elsevier.com/locate/optcom Pulse propagation in an atomic medium under spontaneously generated coherence, incoherent pumping, and relative laser phase Dong Hoang Minh a, b , Doai Le Van a , Bang Nguyen Huy a, * a Vinh University, 182 Le Duan, Vinh City, Viet Nam b The Central College of Transport No. 4, Vinh City, Viet Nam ARTICLE INFO Keywords: Quantum interferences Electromagnetically induced transparency Pulse propagation ABSTRACT Influences of spontaneously generated coherence (SGC) and relative phase of laser fields on a probe laser pulse propagating in a three-level cascade atomic medium under incoherent pumping and electromagnetically induced transparency (EIT) are studied theoretically. It is shown that the leading edge of the pulse oscillates during its propagation. Furthermore, the oscillating magnitude is enhanced by growing SGC, however, it can be depressed by choosing a suitable relative phase between the laser fields. Indeed, the probe pulse depends sensitively on the relative phase with a period of 2  in which it oscillates strongest at  =0,  and 2 ; whereas it is unchanged at  = ∕2 and 3/2. On the other hand, the influences of SGC and the relative phase on the pulse envelope are more effective as growing the incoherent pumping rate. 1. Introduction Electromagnetically induced transparency (EIT) is a quantum inter- ference effect 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. The effect was observed by Harris and coworkers in 1991 [1]. Since then, the EIT has attracted tremendous interest [2– 5] due to its unusual optical properties and promising applications in nonlinear and quantum optics [6–13]. Several dynamical processes of light pulses propagate in an EIT medium that permits a remained pulse shape at low intensity, were also studied [14–24] because of their potential applications in the fields of quantum information [25], all- optical switching [26], and storage and retrieval of light pulses [27]. Among various sources resulting quantum interferences, sponta- neous emission interference in the atomic systems with nonorthogonal- ity of electric dipole moments induced by coherent fields is a special case. Such interference creates an additional atomic coherence which is called as spontaneously generated coherence (SGC) [28]. The first experiment of SGC in the sodium molecules was carried out by Xia et al. [29]. So far, the influences of SGC on lasing without population inversion [30], absorption and dispersion [31–33], slow light [34–36], enhancement of Kerr nonlinearity [37,38], and optical bistability [39], are investigated in the steady-state regime. It is shown that atomic response under the SGC is sensitive to the relative phase of the applied fields [40]. Fan et al., [41] investigated the effects of SGC and relative * Corresponding author. E-mail address: bangnh@vinhuni.edu.vn (B. Nguyen Huy). phase on the absorption and dispersion in a three-level cascade atomic system with an incoherent pumping at a steady-state regime. Their results show that the atomic medium is switched from absorption to amplification and vice versa by changing the incoherent pumping rate. Up to date, influences of SGC and relative phase on optical properties of EIT media for the case of steady-state regime are studied by numerous works [28–41], but those for the case of dynamical regime are quite moderate [42–45]. This situation is contradictive with many potential applications of laser pulse propagation in EIT media. Recently, pulse propagation in a three-level cascade EIT medium in which the probe pulse can be maintained as a soliton by choosing proper parameters of laser fields has been studied [45,46]. However, these works neglect the influence of the SGC and the relative phase on laser pulse under EIT condition. In this work, using semiclassical theory and density matrix formalism, we study propagation dynamics of a probe laser pulse in three-level cascade EIT medium under various conditions of the SGC, the relative phase between the probe and coupling fields, and incoherent pumping rate. 2. Theoretical model We consider a three-level cascade-type system with nearly equis- paced levels interacting with two laser pulses as in Fig. 1. A weak probe field with frequency   drives the transition |1⟩ ↔ |2⟩, while the transition |2⟩ ↔ |3⟩ is driven by a strong coupling field with frequency https://doi.org/10.1016/j.optcom.2018.06.008 Received 17 March 2018; Received in revised form 1 June 2018; Accepted 3 June 2018 0030-4018/© 2018 Elsevier B.V. All rights reserved.