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By solving the density matrix equations in the steady state, we have derived analytical expressions for the absorption, dispersion, and group index of a five-level cascade-type atomic system as functions of laser intensity and frequency, spontaneously generated coherence (SGC), and relative phase of applied fields. The influences of SGC and relative phase on absorption, dispersion, and group index of this system under electromagnetically induced transparency (EIT) are studied. It is shown that the three EIT windows of system become deeper and narrower as the strength of SGC increases. These lead to an increase in the slope and the amplitude of the dispersion curves at three EIT windows. As a result, the amplitude of the group index at these three EIT windows also becomes larger when the strength of SGC increases. In particular, the group index can easily be switched between negative and positive values i.e., the light propagation can easily be converted between superluminal to subluminal modes by adjusting the strength of SGC, relative phase or the coupling laser intensity.

We have achieved the negative refractive index for the probe light in a degenerated three-level $$\Lambda $$ -type atomic medium under electromagnetically induced transparency. The width of the frequency band of the negative refractive index can be changed by adjusting the coupling light intensity, while the position of the frequency band of the negative refractive index can be shifted to low or high frequency region by varying the coupling light frequency or the external magnetic field. Furthermore, the positive refractive index for a given probe frequency can be converted to the negative refractive index and vice versa by adjusting the strength or the sign of the magnetic field. This means that we can use the external magnetic field as a “knob” to control the sign of the refractive index of the material which can obtain the desired material with positive or negative index.

In this paper, we analyze the role of incoherent pumping field on the control of optical bistability (OB) via the spontaneously generated coherence (SGC) and relative phase between laser fields in a closed three-level ladder-type atomic system. It is shown that in the presence of incoherent pumping field, the influences of the SGC and the relative phase of the OB behavior become more effective. Simultaneously, the threshold intensity and width of the OB reduce significantly when the incoherent pumping rate increases.

We have achieved a negative refractive index with significantly reduced absorption in a three-level Λ-type atomic gas medium under Doppler broadening. It shows that the conditions for obtaining negative refractive index in the presence of Doppler broadening are very different from those of Doppler broadening absent. In particular, in order to obtain negative refractive index in the case of Doppler broadening the coupling laser intensity must be approximately ten times greater than that when the Doppler broadening is ignored. Meanwhile, the frequency band of negative refractive index with Doppler broadening is significantly expanded (about a hundred times) compared to that without Doppler broadening, however, the amplitude of negative refractive index decreases with increasing temperature (or Doppler width). Even in some cases as temperature (Doppler width) increases, the left-handedness of the material can disappear. In addition, we also show that the amplitude and the frequency band of negative refractive index can be changed by adjusting the intensity and the frequency of coupling laser. Our theoretical investigation can be useful for selection of laser parameters under different temperature conditions to achieve negative refractive index in experimental implementation.

We present an analytical model for cross-Kerr nonlinear coefficient in a four-level N-type atomic medium under Doppler broadening. The model is applied to 87Rb atoms to analyze the dependence of the cross-Kerr nonlinear coefficient on the external light field and the temperature of atomic vapor. The analysis shows that in the absence of electromagnetically induced transparency (EIT) the cross-Kerr nonlinear coefficient is zero, but it is significantly enhanced when the EIT is established. It means that the cross-Kerr effect can be turned on/off when the external light field is on or off. Simultaneously, the amplitude and the sign of the cross-Kerr nonlinear coefficient are easily changed according to the intensity and frequency of the external light field. The amplitude of the cross-Kerr nonlinear coefficient remarkably decreases when the temperature of atomic medium increases. The analytical model can be convenient to fit experimental observations and applied to photonic devices.

We study a pair of probe and coupling laser pulses propagating in a Doppler broadened three-level cascade atomic medium. Influence of the coupling pulse area on the probe pulse shape is studied in a wide region of pulse duration, from micro to pico second. It is found that the needed value of coupling pulse area to establish an electromagnetically induced transparency (EIT) for the probe light, namely the probe pulse shape is unchanged, is smaller at shorter pulse duration. On the other hand, influence of Doppler broadening is more efficient in the long duration side. These results can find interesting applications in all optical switching, quantum information processing, and transmission.

The magnitude and sign of self-Kerr nonlinear coefficient in a degenerate two-level inhomogeneously broadened medium are simply controlled by an external magnetic field. By changing the external magnetic field we can switch a negative peak of Kerr nonlinear coefficient into a positive peak and vice versa, and transform a zero value of Kerr nonlinear coefficient into a positive or negative peak in resonant region. Such a controllable Kerr nonlinear coefficient can be used to manipulate the working characteristics of photonic devices such as optical switches and bistable elements.

Electromagnetically induced transparency (EIT) in the five-level cascade scheme of 85Rb atoms is investigated by analytical method. In the weak field limit of the probe light, the linear susceptibility, absorption, and dispersion coefficients have been derived in analytical forms. Unusual EIT signatures of the system are studied in details.

In this paper, we propose an analytical model to study the effect of Doppler broadening on self-Kerr nonlinearity in a five-level ladder-type atomic system. First- and third-order susceptibilities and the self-Kerr nonlinear coefficient are found as the function of temperature and parameters of laser fields. The analytical model is applied to hot $^{85}{\rm Rb}$85Rb and $^{87}{\rm Rb}$87Rb atoms, and it is shown that under the electromagnetically induced transparency (EIT) effect, the self-Kerr nonlinear coefficient is enhanced around three transparent spectral regions. When the temperature of atomic vapor increases (i.e., Doppler width increases), the depth and width of the EIT windows decrease accordingly, and therefore the amplitude of the Kerr nonlinear coefficient decreases significantly. In addition, because the frequency gaps between hyperfine levels of upper excited state ${{5\rm D}_{5/2}}$5D5/2 of $^{85}{\rm Rb}$85Rb atoms are much smaller than those of $^{87}{\rm Rb}$87Rb atoms, the EIT windows as well as the nonlinear dispersion curves for $^{85}{\rm Rb}$85Rb atoms are closer than those for $^{87}{\rm Rb}$87Rb atoms as the Doppler effect presents. The analytical results agree well with the experimental observation when reducing to a three-level atomic system. The analytical model can be used to easily fit the experimental observations of self-Kerr nonlinearity in a five-level atomic system under different temperature conditions and apply to a variety of applications relating to all-optical-switching techniques.

We proposed a model for manipulating giant cross-Kerr nonlinearity in an atomic gaseous medium consisting of six-level inverted-Y systems. The absorption, dispersion, and cross-Kerr nonlinear coefficients of the medium are derived as analytical functions of the parameters of probe, coupling, and signal fields. It is shown that the cross-Kerr nonlinearity is enhanced significantly in three transparent windows under electromagnetically induced transparency (EIT). Furthermore, the cross-Kerr nonlinearity can be manipulated between positive and negative values by controlling intensity and/or frequency of the coupling laser. Such controllable giant cross-Kerr nonlinearity with the analytical interpretation is convenient to find experimental parameters and is useful for studying applications of controllable multi-channel quantum phase gates.

In this work, we study the influence of Doppler broadening on cross-Kerr nonlinearity in a four-level inverted-Y atomic system under electromagnetically induced transparency (EIT) condition. The first- and third-susceptibilities in the presence of Doppler effect are derived as a function of probe, signal and coupling beams and temperature of medium. Under EIT condition, cross-Kerr nonlinearity is enhanced several orders of magnitude compared to that without EIT. The Doppler effect leads to a reduction in the transparent efficiency and thus reduces the amplitude of cross-Kerr nonlinear coefficient. For hot atomic gaseous medium, such consideration of the Doppler effect may be useful for experimental observations and apply to photonic devices operating at different temperature conditions.

We present a comparative study of the control of group velocity in three-level atomic systems including Λ-, Ladder-, and V-types via spontaneously generated coherence (SGC) and relative phase of laser fields. In the steady regime, expressions of group index are derived as an analytical function of controllable parameters related to SGC and relative phase. It is shown that the group velocity can be further slowed down or speeded up under SGC in which modifications of group velocity due to the SGC in Λ- and V-systems are stronger than that in Ladder-system. In particular, when interference strength of SGC p increases from 0.7 to 1.0, the light propagation is switched from superluminal to subluminal modes (for Λ- and Ladder-systems) or from subluminal to superluminal modes (for V-system). In the presence of relative phase, the group velocity changes between subluminal and superluminal modes with a period of 2π. In resonant region, the largest positive value of group index reaches at the relative phase φ = π while the largest negative value of group index reaches at φ = 0 and 2π for the Λ- and Ladder-systems, otherwise, for the V-system the largest positive group index occurs at φ = 0 and 2π, and the largest negative group index occurs at φ = π.

We have measured the dispersive profile of a multiwindow electromagnetically induced transparency (EIT) spectrum in a medium consisting of Rb atoms in the presence of Doppler broadening. The atomic medium is excited by a strong coupling light and a weak probe laser light via the V-type transitions within the D2 manifold. Under the EIT effect, an anomalous dispersive region of the medium is basically modified into multinormal and anomalous dispersive regions. Furthermore, the slope and position of the dispersion can be controlled with the intensity and frequency of the coupling light. An analytic model is proposed to simulate the observed spectrum with a good agreement. Such controllable dispersive properties with their analytic description would be useful for finding applications related to multiwindow EIT phenomena.

Optical bistability (OB) of an electromagnetically induced transparency (EIT) medium placed in a conventional unidirectional ring cavity is investigated numerically. The medium is excited by a coupling and probe laser lights via a five-level cascade scheme. It is shown that optical bistabe states with controllable switching threshold intensities and width are established simultaneously in three spectral regions corresponding to EIT windows. The current numerical result is compared to a previous analytical result showing influence of coherence terms neglected in the analytical model.

We propose an optical bistability (OB) model consisting of a five-level cascade electromagnetically induced transparency (EIT) medium placed in a unidirectional ring cavity. An input-output intensity relationship for a weak probe light field is derived as a function of parameters of a strong coupling light and cooperation C. The OB behaviors of the system are investigated with variation of the parameters. It has been shown that the OB behaviors occur simultaneously in three EIT windows. Furthermore, switching thresholds and the hysteresis loop can be controlled by the parameters of the coupling light and/or a cooperation of atomic medium. Such controllable OB behaviors can find interesting applications in bistable photonic devices working at low light intensity and multiple frequencies.

We develop an analytical approach on electromagnetically induced transparency (EIT) in a Doppler broadened medium consisting of five-level cascade systems excited by a strong coupling and weak probe laser fields. In a weak field limit of the probe light, EIT spectrum is interpreted as functions of controllable parameters of the coupling light and temperature of the medium. The theoretical interpretation of EIT spectrum is applied to the case of 85Rb atoms and compared with available experimental observation. Such an analytical interpretation provides quantitative parameters to control properties of the Doppler broadened EIT medium, and it is useful to find related applications.

Using density-matrix theory, an analytical expression of the self-Kerr nonlinear coefficient of a three-level lambda EIT medium for a weak probe light is derived. Influences of the coupling light and Doppler broadening on the self-Kerr coefficient are investigated and compared to experimental observation with a good agreement. The self-Kerr nonlinearity of the medium is modified and greatly enhanced in the spectral region of EIT window. Furthermore, sign, slope, and magnitude of the self-Kerr coefficient can be controlled with frequency and intensity of the coupling light and temperature of the medium. In particular, for a given set of fixed values of the parameter coupling and probe lights, it is possible to choose an optimized temperature with which to obtain the largest magnitude of the self-Kerr coefficient. Such a controllable Kerr nonlinearity can find interesting applications in optoelectronic devices working with low-light intensity at various temperature conditions.

Using density-matrix theory, an analytical expression of the self-Kerr nonlinear coefficient of a three-level lambda EIT medium for a weak probe light is derived. Influences of the coupling light and Doppler broadening on the self-Kerr coefficient are investigated and compared to experimental observation with a good agreement. The self-Kerr nonlinearity is basically modified and greatly enhanced in the spectral region corresponding to EIT transparent window. Furthermore, sign, slope, and magnitude of the self-Kerr coefficient can be controlled with frequency and intensity of the coupling light and temperature. Such controllable Kerr nonlinearity can find interesting applications in optoelectronic devices working with low-light intensity.

We propose to use a five-level cascade system to enhance self-Kerr nonlinearity under an electromagnetically induced transparency (EIT) condition. Using density-matrix theory, an expression of the self-Kerr nonlinear coefficient for a weak probe light is derived. Variations of the self-Kerr coefficient with respect to the frequency and intensity of a strong coupling light are investigated. The Kerr nonlinearity is basically modified and enhanced greatly in the spectral regions corresponding to EIT transparent windows. Furthermore, the sign, slope, and magnitude of the self-Kerr coefficient can be controlled with the frequency and intensity of the coupling light. Such a controllable Kerr nonlinearity can find interesting applications in optoelectronic devices working with low light intensity at multiple frequencies.

In the framework of the semiclassical theory, we develop an analytical approach on electromagnetically induced transparency (EIT) in the medium consists of five-level cascade scheme of cold 85Rb atoms. In the weak field limit of the probe light, an analytical representation of EIT spectra is derived for the first time. Signatures of EIT spectra, including transparency efficiency, have been investigated. We see a tremendous agreement between our theoretical results and experimental observations.