⇦Chinese Physics Letters, 2018, Vol. 35, No. 2, Article code 024204 Phase Control of Transient Optical Properties of Double Coupled Quantum-Dot Nanostructure via Gaussian Laser Beams * J. Shiri1**, F. Shahi2, M. R. Mehmannavaz3, L. Shahrassai2 Affiliations 1Young Researchers and Elite Club, Tabriz Branch, Islamic Azad University, Tabriz, Iran 2Department of Atomic and Molecular Physics, University of Tabriz, Tabriz, Iran 3Young Researchers and Elite Club, Mashhad Branch, Islamic Azad University, Mashhad, Iran Received 31 October 2017 *
**Corresponding author. Email: jalil.shiri@chmail.ir Citation Text: Shiri J, Shahi F, Mehmannavaz M R and Shahrassai L 2018 Chin. Phys. Lett. 35 024204 Abstract We theoretically analyze the transient properties of a probe field absorption and dispersion in a coupled semiconductor double-quantum-dot nanostructure. We show that in the presence of the Gaussian laser beams, absorption and dispersion of the probe field can be dramatically influenced by the relative phase between applied fields and intensity of the Gaussian laser beams. Transient and steady-state behaviors of the probe field absorption and dispersion are discussed to estimate the required switching time. The estimated range is between 5–8 ps for subluminal to superluminal light propagation. DOI:10.1088/0256-307X/35/2/024204 PACS:42.50.-p, 42.65.-k, 42.65.Pc © 2018 Chinese Physics Society Article Text In recent years, there has been an increasing interest in the optical properties of nanostructures, especially quantum dots (QDs), due to their important role in optoelectronic devices. Recently, researchers have examined the effects of the external field on the optical properties of QDs and quantum wells.[1-7] Flexible design, controllable interference strength, similar properties to atomic vapors, long dephasing times,[1,2] and large bandwidth because of fast carrier dynamics make a potential application of quantum dot nanostructures. Coherent control[8-14] over the optical properties of the gaseous systems as well as semiconductor QDs and quantum wells has recently attracted tremendous attention.[15-20] The theory of the quantum coherence phenomenon in a semiconductor quantum dot system is presented by Chow et al.[21] Villas-Bôas et al. demonstrated that quantum coherence in a QD structure can be induced by electron tunneling or by applying a laser field.[22] Coherence induced by laser field and inter-dot tunnel coupling to the QDs system plays an important role in light–matter interaction and has found numerous implementations in semiconductor optics.[23-26] Controlling light via light or by electron tunneling has important applications to develop next-generation all-optical communication and optoelectronic devices.[27-30] An interesting application of QDs is the modification of light pulse propagation in a semiconductor system, which depends on the dispersive properties of the medium.[31] This is important due to its novel application in fast optical switching, which is an important technique in quantum computing and quantum information.[32,33] Having control over the group velocity of propagating waves, in addition to tailoring the nonlinear effects, the coupling coefficients between transmission channels are also controllable. Packets of information will be effectively managed if their transmission speed is changeable. A potential application is data storage carried by the light pulses, leading to a potential all-optical computing system. The ability to control light pulses on an integrated chip involving QDs may propose new schemes for the realization of all-optical quantum communication networks, with potentially vast improvements in ultra-low-power performance. In this Letter, we introduce a QD molecule composed of two QDs, which is fabricated as a four-level double coupled QD system. The effect of intensity of Gaussian laser field and relative phase between applied fields on absorption, dispersion, and the group velocity of a weak probe field is investigated. The dynamical response of the system is also discussed to inspect the required switching time. This may be used as a fast optical switch, which is an important technique in quantum computing and quantum information. An important application of such as a switch is the storage of large carrying information by the light pulses, which leads to a potential application in all-optical computing systems.
cpl-35-2-024204-fig1.png
Fig. 1. Schematic diagram of the epitaxial structure of the coupled semiconductor double quantum-dot nanostructure under study. Energy-band diagram of the coupled quantum-dot consist of levels $|3\rangle$ and $|4\rangle$ are the excitation levels and $|1\rangle$, $|2\rangle$ are ground electron states of the coupled system.
We consider a coupled semiconductor double-quantum-dot nanostructure interacting with weak probe fields and few-cycle pulse laser fields, as depicted in Fig. 1. The scheme consists of two metastable lower states at level $|1\rangle$ and level $|2\rangle$ and two excited levels $|3\rangle$, $|4\rangle$. Level $|1\rangle$ couples to levels $|3\rangle$ and $|4\rangle$ using a weak tunable probe terahertz-infrared field with angular frequency $\omega _{\rm p}$, while level $|2\rangle$ couples to levels $|3\rangle$ and $|4\rangle$ using strong femtosecond Gaussian coupling laser beams with angular frequency $\omega _{\rm c}$. The electric field of the laser beams in interaction between $|i\rangle$ and $|j\rangle$ is defined as ${\boldsymbol E}_{ij} (r,t)={\boldsymbol A}_{ij} (r,t)\cos (\omega _{ij} t+\omega _{Dij} +\varphi_{ij})$, where ${\boldsymbol A}_{ij} (r,t)$ is the space- and time-dependent field amplitude vector. Also, $\omega _{ij}$ and $\varphi_{ij}$ refer to the carrier frequency and the absolute phase of the pulse. Moreover, $\omega _{D_{ij}} ={\boldsymbol k}_{ij}{\boldsymbol \nu}$ is the frequency detuning. We assume that the field amplitude has a Gaussian profile, i.e., ${\boldsymbol A}_{ij}(r,t)=E_{0ij}\hat {e}_{ij}\exp(-\frac{r^2}{w^2}-\frac{t^2}{\tau^2})$, where $E_{0ij}$, $\hat {e}_{ij}$, $w$ and $\tau$ denote the peak amplitude of the field, unit polarization vector, beam waist and the temporal width of the pulse, respectively. In general, the Rabi frequency can be written in the form ${\it \Omega}_{ij}={\it \Omega}_{0ij} \exp (-\frac{r^2}{w^2}-\frac{t^2}{\tau ^2})$, where ${\it \Omega} _{0ij} =E_{0ij} \hat {e}_{ij}{\boldsymbol d}_{ij}/\hbar$ with ${\boldsymbol d}_{ij}$ being the dipole moment of the corresponding transition. Under the dipole and rotating wave approximations, the Hamiltonian of the system can be written as $$\begin{align} H_{\rm int} =\,&({\it \Delta} _{\rm p} -{\it \Delta} _{\rm c})|2\rangle \langle 2|-{\it \Delta} _{\rm c}|3 \rangle \langle 3 |-{\it \Delta} _{\rm c} |4 \rangle \langle 4|\\ &-({\it \Omega} _{\rm p_1}|4 \rangle \langle 1|+{\it \Omega} _{\rm p_2}|3 \rangle \langle 1|+{\it \Omega} _{\rm c_2}|3 \rangle \langle 2|\\ &+{\it \Omega} _{\rm c_1} e^{-i\Phi}|4 \rangle \langle 2|+{\rm H.c.}). \end{align} $$ The density matrix equations of motion are obtained as $$\begin{align} \dot {\rho}_{12} =\,&[-{\it \Gamma}_{12}/2+i({\it \Delta} _{\rm p} -{\it \Delta} _{\rm c})]\,\rho _{12} +i{\it \Omega} _{\rm p_2} \rho _{32}\\ &-i{\it \Omega} _{\rm c_2} \rho _{13} +i{\it \Omega} _{\rm p_1} \rho _{42} -i{\it \Omega} _{\rm c_1} \rho _{14} e^{-i\varphi},\\ \dot {\rho}_{13} =\,&[-{{\it \Gamma}_{13}}/2+i{\it \Delta} _{\rm p} ]\,\rho _{13} +i{\it \Omega} _{\rm p_2} (\rho _{33} -\rho _{11})\\ &-i{\it \Omega} _{\rm c_2} \rho _{12} +i{\it \Omega} _{\rm p_1} \rho _{43},\\ \dot {\rho}_{14} =\,&[-{{\it \Gamma}_{14}}/2+i{\it \Delta} _{\rm p} ]\,\rho _{14} +i{\it \Omega} _{\rm p_1} (\rho _{44} -\rho _{11})\\ &+i{\it \Omega} _{\rm p_2} \rho _{34} -i{\it \Omega} _{\rm c_1} \rho _{12} e^{i\varphi},\\ \dot {\rho}_{23} =\,&[-{{\it \Gamma}_{23}}/2+i{\it \Delta} _{\rm c} ]\,\rho _{23} +i{\it \Omega} _{\rm c_1} (\rho _{33} -\rho _{22})\\ &+i{\it \Omega} _{\rm c_1} \rho _{43} e^{i\varphi}-i{\it \Omega} _{\rm c_1} \rho _{21},\\ \dot {\rho}_{24} =\,&[-{{\it \Gamma}_{24}}/2+i{\it \Delta} _{\rm c} ]\rho _{24} +i{\it \Omega} _{\rm c_1} (\rho _{44} -\rho _{22})e^{i\varphi}\\ &+i{\it \Omega} _{\rm c_2} \rho _{34} -i{\it \Omega} _{\rm p_1} \rho _{21},\\ \dot {\rho}_{34} =\,&-({{\it \Gamma}_{34}}/2)\rho _{34} +i{\it \Omega} _{\rm p_2} \rho _{14} +i{\it \Omega} _{\rm c_2} \rho _{24}\\ &-i{\it \Omega} _{\rm c_1} \rho _{32} e^{i\varphi}-i{\it \Omega} _{\rm p_1} \rho _{31},\\ \dot {\rho}_{22} =\,&-\gamma_2 \rho _{22} +\gamma_{32} \rho _{33} +\gamma_{42} \rho _{44} +i{\it \Omega} _{\rm c_2} \rho _{32}\\ &-i{\it \Omega} _{\rm c2} \rho _{23} +i{\it \Omega} _{\rm c_1} \rho _{42} e^{i\varphi}-i{\it \Omega} _{\rm c_1} \rho _{24} e^{-i\varphi},\\ \dot {\rho}_{33} =\,&-\gamma_3 \rho _{33} -i{\it \Omega} _{\rm p_2} \rho _{31} +i{\it \Omega} _{\rm p_2} \rho _{13}\\ &+i{\it \Omega} _{\rm c_2} \rho _{23} -i{\it \Omega} _{\rm c_2} \rho _{32},\\ \dot {\rho}_{44} =\,&-\gamma_4 \rho _{44} -i{\it \Omega} _{\rm p_1} \rho _{41} +i{\it \Omega} _{\rm p_1} \rho _{14}\\ &-i{\it \Omega} _{\rm c_1} \rho _{42} e^{i\varphi}+i{\it \Omega} _{\rm c_1} \rho _{24} e^{-i\varphi},\\ &\rho _{11} +\rho _{22} +\rho _{33} +\rho _{44} =1.~~ \tag {1} \end{align} $$ The detuning parameters of coupling laser, and probe laser fields with respect to the corresponding transitions are ${\it \Delta} _{\rm c} =\omega _0 -\omega _{\rm c}$, ${\it \Delta} _{\rm p} =\omega _0 -\omega _{\rm p}$, where $\omega _0 =(\omega _4 +\omega _3)/2$. The total decay rates ${\it \Gamma}_{ij} (i\ne j)$ are given by ${\it \Gamma}_{12} =\gamma_2 +\gamma_{12}^{\rm dph}$, ${\it \Gamma}_{13} =\gamma_{31} +\gamma_{32} +\gamma_{13}^{\rm dph}$, ${\it \Gamma}_{14} =\gamma_{41} +\gamma_{42} +\gamma_{14}^{\rm dph}$, ${\it \Gamma}_{23} =\gamma_2 +\gamma_{31} +\gamma_{32} +\gamma_{23}^{\rm dph}$, ${\it \Gamma}_{24} =\gamma_2 +\gamma_{41} +\gamma_{42} +\gamma_{24}^{\rm dph}$, and ${\it \Gamma}_{34} =\gamma_{31} +\gamma_{32} +\gamma_{41} +\gamma_{42} +\gamma_{34}^{\rm dph}$. Here $\gamma_{mn}^{\rm dph}$ are the dephasing rates of the quantum coherence between level $|i\rangle$ and level $|j \rangle$, determined by electron–electron, interface roughness and phonon scattering processes. Population decay rate of a sub band $|i\rangle$ is also denoted by $\gamma_i$, and $\varphi =\varphi_1 +\varphi_4 -\varphi _2 -\varphi _3$ is the collective phase of the four applied fields. The perturbation theory is employed to discuss the linear susceptibility, thus we represent the density matrix elements as $\rho _{ij} =\rho _{ij}^{(0)} +\lambda \rho _{ij}^{(1)} +\lambda ^2\rho _{ij}^{(2)} +\lambda ^3\rho _{ij}^{(3)} +\ldots$ Given the fact that the probe field is much weaker than the coupling field, the zeroth order solution will be $\rho _{11}^{(0)} =1$, and the other elements are set to be zero. The linear susceptibility is then given by[34] $$ \chi=\frac{2N}{\varepsilon _0 \hbar {\it \Omega} _{\rm p}}(d_{31}^2 \rho _{31}^{(1)} +d_{41}^2 \rho _{41}^{(1)}),~~ \tag {2} $$ where we set $d_{31} =d_{41} =d$, and ${\it \Omega} _{\rm p_1} ={\it \Omega} _{\rm p_2} ={\it \Omega} _{\rm p}$. The linear dispersion and absorption are proportional to the real ${\chi}'$ and the imaginary ${\chi}''$ parts of $\chi$, respectively. The slope of dispersion with respect to the probe field detuning denotes the group velocity of the probe light. The group velocity of a weak probe beam $v_{\rm g}$ is given by[35,36] $$\begin{align} v_{\rm g} =\frac{c}{1+2\pi {\chi}'(\omega _{\rm p})+2\pi \nu _{\rm p} (\partial {\chi}'(\omega _{\rm p})/\partial \omega _{\rm p})},~~ \tag {3} \end{align} $$ where $c$ is the speed of light in vacuum. The dispersion slope is determined by $\partial {\chi}'(\omega _{\rm p})/\partial \omega _{\rm p}$, where ${\chi}'$ is a real part of the linear susceptibility. When the dispersion slope is negative, the group velocity becomes larger than the speed of light in vacuum. By changing the slope of dispersion from negative towards positive, the group velocity changes from superluminal to subluminal light propagation. This makes the possibility of designing an ultra-fast all-optical switch, which can be used in optical telecommunication. Now, we examine the effect on the system parameters, on steady state and transient evolution of probe absorption and dispersion. We inquire of the time evolution of the dispersion slope and estimate the required time for switching the group velocity from subluminal to superluminal (or vice versa). We assume that the system is initially in the ground state $|1\rangle$, i.e., $\rho _{11} ^{(0)}=1$, and other elements are set to be zero $\rho _{ij} ^{(0)}=0$ ($i,j=1,2,3,4$). In addition, it is assumed that ${\it \Omega} _{\rm c_1} ={\it \Omega} _{\rm c_2} ={\it \Omega} _{\rm c}$ and the reasonable value of the spontaneous decay rate is $\gamma_2 =\gamma =2\pi \times 0.24$ THz.[36] Other system parameters are scaled by the factor of $\gamma$. Probe field absorption and dispersion as a function of probe field detuning are shown in Fig. 2 for two different values of ${\it \Omega} _{\rm c}$. For ${\it \Omega} _{\rm c} =0.1\gamma$ under the resonance condition, a large probe absorption with a negative slope of dispersion is achieved. When the Rabi-frequency of the coupling field changes to ${\it \Omega} _{\rm c} =4\gamma$, a large transparency window with a positive slope of dispersion appears around zero probe field detuning. In fact, the large intensity of the coupling field in transition $|2\rangle \to|4\rangle (|3\rangle)$ makes the system almost transparent to the probe field in transition $|1\rangle \to|4\rangle (|3\rangle)$. Thus for the large intensity on coupling field the electromagnetically induced transparency (EIT) condition is fulfilled, and the superluminal light propagation changes to subluminal light propagation.
cpl-35-2-024204-fig2.png
Fig. 2. Real (dashed) and imaginary (solid) parts of susceptibility versus probe detuning. The selected parameters are (a) $\gamma=1$ THz, ${\it \Omega} _{\rm c_2}={\it \Omega} _{\rm c_1}=0.1\gamma$, and (b) ${\it \Omega} _{\rm c_2}={\it \Omega} _{\rm c_1}=4\gamma$. Other parameters are ${\it \Delta} _{\rm c}=0$, ${\it \Omega} _{\rm p}= 0.001\gamma$, $\gamma_{31}=\gamma_{41}=3\gamma$, $\gamma_{32}=\gamma_{42}= 0.5\gamma$, $\gamma_2=1\gamma$, $\gamma_{12} ^{\rm dph}= \gamma_{13} ^{\rm dph}= \gamma_{14} ^{\rm dph}= \gamma_{23} ^{\rm dph}= \gamma_{24} ^{\rm dph}= \gamma_{34} ^{\rm dph}= 0.1\gamma$.
cpl-35-2-024204-fig3.png
Fig. 3. Imaginary parts of susceptibility versus probe detuning. The selected parameters are $\varphi =(\pi,\pi /2,0)$, ${\it \Omega} _{\rm c_1}={\it \Omega} _{\rm c_2}=6\gamma$. The other parameters are the same as those in Fig. 2.
cpl-35-2-024204-fig4.png
Fig. 4. (a) The steady state of the real parts of susceptibility versus probe detuning, and (b) transient behavior of dispersion slope. The selected parameters are $\varphi =(\pi,\pi /2,0)$, ${\it \Omega} _{\rm c_1}={\it \Omega} _{\rm c_2}=6\gamma$. The other parameters are the same as those in Fig. 2.
In Fig. 3 we demonstrate that the probe field absorption is sensitive to the relative phase of coupling and probe fields. For $\varphi =\pi$ and $3\pi$, the absorption of the probe field reaches a large amount, and for $\varphi =0$ and $\pi$ the media become transparent around zero probe field detuning. We show that for $\varphi =\pi/2$ and $3\pi/2$ the probe field absorption reduces from a large amount to three low absorption peaks. Figure 4(a) shows that changing the relative phase from $\varphi =0, 2\pi$ to $\varphi =\pi/2, 3\pi/2$ alters the dispersion slope from positive to negative. Therefore, the probe field propagation changes from subluminal to superluminal as well as the relative phase continuing toward the dispersion slope will be steeper, which leads to a larger superluminal. The transient behavior of the dispersion slope for different values of relative phase is shown in Fig. 4(b). As can be seen, the amounts of dispersion slopes after a short time fluctuation have a stable trend and constant values, which make the system suitable for an optical switch. We now demonstrate that the semiconductor double-quantum-dot (SDQD) system can be employed as an optical switch between large absorption and nearly transparency (negligible absorption) through modulation of the intensity of coupling field. We are looking for the required switching time for changing the light absorption case to the nearly transparency or vice versa by properly manipulating $\varphi$ and ${\it \Omega} _{\rm c}$.
cpl-35-2-024204-fig5.png
Fig. 5. Dynamical behavior for the absorption of the probe field. The selected parameters are (a) $\varphi =(\pi,\pi /2,0)$, ${\it \Omega} _{\rm c_1}={\it \Omega} _{\rm c_2} ={\rm 0.1}\gamma$, and (b) $\varphi =(\pi,\pi /2,0)$, ${\it \Omega} _{\rm c_1}={\it \Omega} _{\rm c_2}=6\gamma$. The other parameters are the same as those in Fig. 2(a).
cpl-35-2-024204-fig6.png
Fig. 6. Dynamical behavior (a) and switching process for the absorption of the probe field, and (b) ${\it \Omega} _{\rm c_1}={\it \Omega} _{\rm c_2} =({0.1\gamma,6\gamma})$, $\varphi =0$. The other parameters are the same as those in Fig. 2(a).
cpl-35-2-024204-fig7.png
Fig. 7. Dynamical behavior (a) and switching process for the slope of dispersion of the probe field, and (b) ${\it \Omega} _{\rm c_1}={\it \Omega} _{\rm c_2} =(0.1\gamma,6\gamma)$, $\varphi =0$. The other parameters are the same as those in Fig. 2(a).
In Fig. 5, we investigate the transient behavior of the absorption for different values of the relative phase in the presence of ${\it \Omega} _{\rm c1}={\it \Omega} _{\rm c2} =0.1\gamma$ (Fig. 5(a)) and ${\it \Omega} _{\rm c1}={\it \Omega} _{\rm c2} ={\it \Omega} _{\rm c}=6\gamma$ (Fig. 5(b)). As depicted in Fig. 5(a), the transient behavior of the absorption is not susceptible to changes of the relative phase in a low intensity of coherent control fields, while increasing intensity of coherent control fields from $1\gamma$ to $6\gamma$, dynamic absorption system goes down to zero when the relative phase changes from $\varphi =\pi$ to $\varphi =0$, Fig. 5(b). This makes the system dynamically stable in the EIT state, which is suitable for optical devices and switching. We now demonstrate that the QDs system can be employed as an optical switch between large absorption and nearly transparency (negligible absorption) through modulating the intensity of coupling field. We are looking for the required switching time for changing the light absorption case to the nearly transparency or vice versa by properly manipulating the ${\it \Omega} _{\rm c}$. The transient behavior of the probe absorption is displayed for two various values of ${\it \Omega} _{\rm c}$ in Fig. 6. Increasing ${\it \Omega} _{\rm c}$ leads to a significant decrease of steady-state probe absorption (Fig. 6(a)). Figure 6(b) shows that the required switching time from large absorption to nearly transparent case or vice versa is about $8\gamma$. Note that an optical switching, in which the propagation of a light pulse can be controlled with another pulse, has potential applications in optical information processing and transmission. A high-speed optical switch is an important technique for quantum information network and communication. We also discuss the switching time as the group velocity when it changes from subluminal to superluminal light propagation and vice versa. Accordingly, the slope of the dispersion versus the normalized time is displayed in Figs. 7(a) and 7(b). It is clearly found that the slope of the dispersion changes from positive to negative just by adjusting the Rabi-frequency of coupling field. For ${\it \Omega} _{\rm c} =0.1\gamma$ the slope of the dispersion is negative corresponding to superluminal light propagation, while it changes to be positive corresponding to subluminal light propagation. The required switching time on this case is about $7\gamma$ (Fig. 7(b)).
cpl-35-2-024204-fig8.png
Fig. 8. Dynamical behavior (a) and switching process for the absorption of the probe field, and (b) ${\it \Omega} _{\rm c_1}={\it \Omega} _{\rm c_2} =6\gamma$, $\varphi =(0,\pi)$. The other parameters are the same as those in Fig. 2.
cpl-35-2-024204-fig9.png
Fig. 9. Dynamical behavior (a) and switching process for the slope of dispersion of the probe field, and (b) ${\it \Omega} _{\rm c_1}={\it \Omega} _{\rm c_2} =6\gamma$, $\varphi =(0,\pi)$. The other parameters are the same as those in Fig. 2.
cpl-35-2-024204-fig10.png
Fig. 10. Dynamical behavior (a) and switching process for the slope of dispersion of the probe field, and (b) ${\it \Omega} _{\rm c_1}={\it \Omega} _{\rm c_2} =6\gamma$, $\varphi =(0,\pi /2)$. The other parameters are the same as those in Fig. 2.
In Figs. 8(a), 8(b), 9(a), 9(b), 10(a), and 10(b), the effect of the relative phase of the applied fields on transient behavior of the probe beam in fixing the Rabi frequency at ${\it \Omega} _{\rm c_1} ={\it \Omega} _{\rm c_2} =6\gamma$ is shown. It can be seen from Fig. 8(a) that the probe absorption increases gradually for $\varphi=\pi$ and finally reaches a large absorption at the steady state. However, it substantially decreases only by adjusting the relative phase from $\pi$ to zero. The required switching time in this case is about $5\gamma$. Note that by the successive change of the relative phase, one can design an optical switch for quantum information networks and communications. Figure 9(a) shows the transient behavior of the dispersion slope for the parameters ${\it \Omega} _{\rm c_1} ={\it \Omega} _{\rm c_2} =6\gamma$, $\varphi =(0,\pi)$. We find that the slope of steady-state dispersion is positive for $\varphi =0$, while it is too negative for $\varphi =\pi$. In Fig. 9(b), we plot the switching diagram of the dispersion slope for two different values of $\varphi$. We observe that the switching time from subluminal to superluminal light propagation is about 6 ps, and from superluminal to subluminal light propagation is about 8 ps. The transient behavior of the dispersion slope is displayed in Fig. 10 for two different relative $\varphi =0$ and $\frac{\pi}{2}$. For this case, the switching time for switching the group velocity from subluminal to superluminal light propagation is about $6\gamma$. In conclusion, the transient and the steady-state behaviors of the probe field absorption and the dispersion in the SDQD have been investigated. We show that the absorption and the dispersion can be controlled in quantum dot structure just by the Rabi-frequency of the coupling field and relative phase of applied fields. It has also been shown that the medium can be used as a fast optical switch that can be controlled by the intensity of coupling field and the relative phase of applied field. The switching time is estimated to be 5–8 ps. Thus it may provide some new possibilities for technological applications in optoelectronics, solid-state quantum information science. 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