Electromagnetically induced transparency in an inverted Y-type four-level system
aa r X i v : . [ qu a n t - ph ] S e p Electromagnetically induced transparency in aninverted Y-type four-level system
Jianbing Qi
Department of Physics and Astronomy, Penn State University, Berks Campus,Tulpehocken Road, P.O. Box 7009, Reading, PA 19610E-mail: [email protected]
Abstract.
The interaction of a weak probe laser with an inverted-Y type four-levelatomic system driven by two additional coherent fields is investigated theoretically.Under the influence of the coherent coupling fields, the steady-state linear susceptibilityof the probe laser shows that the system can have single or double electromagneticallyinduced transparency windows depending on the amplitude and the detuning of thecoupling lasers. The corresponding index of refraction associated with the groupvelocity of the probe laser can be controlled at both transparency windows by thecoupling fields. The propagation of the probe field can be switched from superluminalnear the resonance to subluminal on resonance within the single transparency windowwhen two coupling lasers are on resonance. This provides a potential application inquantum information processing. We propose an atomic Rb system for experimentalobservation.PACS numbers: 42.50.Gy, 42.25.Kb lectromagnetically induced transparency in an inverted Y-type four-level system
1. Introduction
In recent years, substantial attention has been paid to the study of coherence effectsin atomic and molecular systems [1, 2, 3, 4, 5, 6, 7, 8, 9]. The interaction of thecoherent light with multilevel atomic ensembles results in many striking quantumphenomena. Three-level atomic systems, such as Lambda, Vee, and cascade schemesare the most widely used level schemes for study [10, 11, 12, 13]. Among them, theelectromagnetically induced transparency(EIT), which is based on the phenomenonof coherent population trapping [14], has attracted considerable attention [15, 16,17, 18, 19, 20, 21, 22]. Some extraordinary effects associated with EIT havebeen studied theoretically, and observed experimentally, including ultraslow pulsepropagation of light [1, 23], light storage in an atomic vapor [23, 24], superluminal lightpropagation [11, 25, 26, 27, 28], coherent control of the optical information processingwith BEC [29]. An EIT system that results from a quantum interference effect candramatically reduce the group velocity of a propagating probe laser with greatly reducedor even vanishing absorption of the probe laser. The essential physical mechanismis that the internal structure of atoms can be modified by the interaction with boththe coupling(or ”control”) field and the probe field. Therefore the interaction of theprobe field with the atoms can be manipulated by the coupling field [30]. A properlychosen and prepared atomic system is essential for a successful experimental observation.Alkali atoms have served as test species for these effects due to the availability of laserwavelength and spectroscopy data. Multilevel rubidium atomic systems provide anexcellent test ground and a starting point for extending the control dimensions withinexpensive available laser frequencies for its atomic energy structures.In this paper, we investigate the response of a probe laser in an inverted Y-typefour-level system driven by two additional coherent fields. This scheme has been usedin the study of the Autler-Townes effect in a sodium dimer [31], and of two-photonfluorescence suppression in an ultracold rubidium atom [32], respectively. Here, westudy the absorption and dispersion of a weak probe laser using probability amplitudeand equivalent density matrix methods to obtain the linear susceptibility of the probelaser. We show that the system exhibits two electromagnetically induced transparencywindows for the probe field. The transparency windows can be controlled by theamplitudes and frequency detunings of the coupling fields. The index of refractionassociated with the group velocity of the probe laser can be very different at the twotransparency windows and can be controlled by the two coupling fields. We propose anatomic Rb system for experimental observation of this phenomenon.
2. Equations of Motion
We consider an inverted-Y type four-level system interacts with three lasers, L1, L2and L3 as shown in figure 1(a). Two ground states, | i and | i , are coupled by laserL1 and L2 to a common excited state | i , and the excited state | i is coupled by laser lectromagnetically induced transparency in an inverted Y-type four-level system | i , respectively. laser L1 is a weak probe laser, L2 andL3 are two coupling (or ”control”) lasers. The corresponding dressed-state( | + i , | i ,and |−i )diagram of two coupling lasers interacting with level | i , | i and | i is shownin figure 1(b). All the transitions are electric dipole allowed. The Hamiltonian of thesystem is given by H = H + H I , (1)where H = X i =1 ~ ω i | i ih i | (2)is the atomic Hamiltonian, ~ ω i is the energy of the isolated atom in state | i i , and H I isthe dipole interaction Hamiltonian, which is given by H I = X i = j h i | ( − ~µ · ~E ) | j i = − X i = j µ ij E ij , (3)where µ ij is the electric dipole moment for | i i ↔ | j i transition, and E ij is thecorresponding coupling laser field. In the rotating-wave-approximation, the interactionHamiltonian can be written as: H I = − ~ e − iν t | ih | + Ω e − iν t | ih | + Ω e − iν t | ih | ) + h.c., (4)where ν i is the laser frequency and Ω i = µ ij E ij / ~ is the corresponding Rabi frequencywhich is assumed positive in our calculation. The Hamiltonian of the system in theinteraction representation can be written as H int = − ~ e − iδ t | ih | + Ω e − iδ t | ih | + Ω e − iδ t | ih | ) + h.c., (5)where δ = ν − ω , δ = ν − ω , and δ = ν − ω are the frequency detunings of theprobe laser L1, coupling lasers L2, and L3, respectively. ω ij = ω i − ω j is the | i i ↔ | j i resonance transition frequency. We assume ω = 0 for simplicity and the energy of allother states are measured relative to state | i . The atomic wave-function of the system in the interaction picture at any time t can beexpanded in terms of bare-state eigenvectors as | Ψ int ( t ) i = a ( t ) | i + a ( t ) | i + a ( t ) | i + a ( t ) | i , (6)where a i ( t ) is the time-dependent probability amplitude of the atomic state | i i . TheSchr¨odinger equation in the interaction picture reads as ∂ | Ψ int ( t ) i ∂t = − i ~ H int | Ψ int ( t ) i (7) lectromagnetically induced transparency in an inverted Y-type four-level system |2>|4>|1> L2 |3> L1 L3 (a) L1 |1> |+>|0>|-> (b) Figure 1. (Color online)(a) Energy level scheme for an inverted Y-type four-levelatom. (b) Corresponding dressed-state diagram of laser L2 and L3 interacting with | i , | i , and | i . By introducing the wave-function Ψ int , and the interaction Hamiltonian H int of equation(5) into the Schr¨odinger equation, and after making some rotating transformations, weobtain the equations for the evolution of probability amplitudes of the wave function asfollows: ˙ a ( t ) = i Ω a ( t ) (8 a )˙ a ( t ) = i ( δ − δ ) a ( t ) + i Ω a ( t ) (8 b )˙ a ( t ) = iδ a ( t ) + i Ω a ( t ) + i Ω a ( t ) + i Ω a ( t ) (8 c )˙ a ( t ) = i ( δ + δ ) a ( t ) + i Ω a ( t ) (8 d )We assume that the probe laser is weak and the population is initially in level | i , a = 1. We solve the above equations to the first order in terms of the Rabi frequencyΩ of the probe laser , and to all orders in Ω and Ω of the coupling lasers under thesteady-state condition. From equation (8b)-(8d) we obtain˙ a (1)2 ( t ) − i ( δ − δ ) a (1)2 ( t ) = i Ω a (1)3 ( t ) (9 a )˙ a (1)3 ( t ) − iδ a (1)3 ( t ) = i Ω i Ω a (1)2 ( t ) + i Ω a (1)4 ( t ) (9 b )˙ a (1)4 ( t ) − i ( δ + δ ) a (1)4 ( t ) = i Ω a (1)3 ( t ) (9 c ) lectromagnetically induced transparency in an inverted Y-type four-level system a (1)3 is given by a (1)3 = Ω − δ + Ω δ − δ + Ω δ + δ ! − . (10)The susceptibility at the probe frequency is given by χ = 2 N µ a ∗ a ǫ E = N | µ | ǫ ~ − δ + Ω δ − δ + Ω δ + δ ! − , (11)where N is the atomic number density. Now we include the effects of damping using aphenomenological description. Let γ / γ / | i and | i , respectively. By inspecting equation (9b) and (9c) we cansee that the effects of damping can be included by replacing δ by δ + iγ / δ + δ by δ + δ + iγ / χ = N | µ | ǫ ~ (cid:18) − δ − iγ / δ − δ + Ω / δ + δ + iγ / (cid:19) − (12) We can also model this system by density matrix equations. The master equation ofmotion for the density operator in the interaction representation is given by ∂̺∂t = − i ~ [ H int , ̺ ] + ( ∂̺∂t ) inc , (13)where the second term on the left hand-side represents the damping due to spontaneousemission and other irreversible processes. In the rotating-wave-approximation it isstraightforward to obtain the density matrix equations as follows:˙ ρ = i Ω ρ − ρ ) − γ ρ (14)˙ ρ = i Ω ρ − ρ ) + i Ω ρ − ρ ) + i Ω ρ − ρ )+ γ ρ − γ ρ (15)˙ ρ = i Ω ρ − ρ ) + W ρ − γ ρ (16) ρ + ρ + ρ + ρ = 1 (17)˙ ρ = ( iδ − γ ) ρ + i Ω ρ − ρ ) + i Ω ρ + i Ω ρ (18)˙ ρ = [ i ( δ + δ ) − γ ] ρ + i Ω ρ − i Ω ρ (19)˙ ρ = [ i ( δ − δ ) − γ ] ρ + i Ω ρ − i Ω ρ (20) lectromagnetically induced transparency in an inverted Y-type four-level system ρ = ( iδ − γ ) ρ + i Ω ρ + i Ω ρ + i Ω ρ − ρ ) (21)˙ ρ = [ i ( δ + δ ) − γ ] ρ + i Ω ρ − i Ω ρ (22)˙ ρ = ( iδ − γ ) ρ − i Ω ρ + i Ω ρ − ρ ) − i Ω ρ (23)where γ i is the population decay rate of level | i i , W ij is the branch decay ratefrom level | i i to | j i , and γ ij represents the coherence decay rate which is given by γ ij = γ ji = ( γ i + γ j ) / γ cij . γ cij the collision dephasing rate.We assume that the population is initially in its ground state level | i , and theprobe laser is weak so that ρ (0) ≈ ρ associated with the susceptibility of the probe field. By setting the derivatives to zeroand keeping the first order term in Ω in equations (18)-(20) we obtain the steady-statesolution of ρ to the first order of Ω ρ (1)31 = Ω ρ (0)2 (cid:18) − δ − iγ + Ω / δ − δ + iγ + Ω / δ + δ + iγ (cid:19) − (24)Then the susceptibility reads χ = N µ ρ ǫ E = N | µ | ǫ ~ (cid:18) − δ − iγ + Ω / δ − δ + iγ + Ω / δ + δ + iγ (cid:19) − . (25)The similarity of equation (25) to equation (12) is clear. If the decay rates of the groundstates( | i and | i ) and the collision dephasing rate are small compared to γ , and canbe neglected, by setting γ = γ = γ cij = 0, equation (25) is identical to equation (12).
3. Discussion and Numerical Results
It is well-known that the imaginary part of the susceptibility gives the absorption andthe real part gives the dispersion of the probe field. The susceptibility χ in equation(12) can be separated into the real( χ ′ ) and imaginary( χ ′′ ) parts as χ = χ ′ + iχ ′′ . Theexplicit expressions for χ ′ and χ ′′ are χ ′ = N | µ | ǫ ~ ( δ − δ )[ A ( δ + δ ) + Bγ / A + B (26 a ) χ ′′ = N | µ | ǫ ~ ( δ − δ )[ Aγ / − B ( δ + δ )] A + B , (26 b )with A = ( δ + δ ) Ω − ( δ − δ ) Ω
4+ ( δ − δ )[ δ ( δ + δ ) − γ γ lectromagnetically induced transparency in an inverted Y-type four-level system - - ∆ (cid:144) Γ Χ H ´ - L H a L - - ∆ (cid:144) Γ Χ H ´ - L H b L - - ∆ (cid:144) Γ Χ H ´ - L H c L - - ∆ (cid:144) Γ Χ H ´ - L H d L Figure 2. (Color online) The imaginary part of the linear susceptibility of the probelaser as a function of the probe frequency detuning. There are two transparencywindows at δ = δ , and δ = − δ , respectively. (a) δ = 0 . γ , Ω = 0 . γ , δ = 0,and Ω = γ . (b) When γ = 0 the probe laser at δ = − δ is also completelytransparent. Other parameters are the same as in (a). (c) δ = 0, Ω = 0 . γ , δ = − . γ , and Ω = 2 γ . (d) Solid line is the same as in (c); dashed lines are forthe cascade scheme, | i − | i − | i : Ω = 0 and δ = − . γ ; dotdashed lines are forthe Λ scheme, | i − | i − | i : Ω = 0 and δ = 0. B = ( δ − δ )[ δ γ δ + δ ) γ γ . In our following numerical calculations we assume the atoms are at ultracoldtemperatures and the Doppler effect can be neglected. An ultracold atomic samplecan be obtained in a magnetic-optical-trap(MOT), such as an ultracold Rb atom trap.For example, 10 ∼ atoms can be trapped in a ”dark-spot” MOT within a sphericalcloud of a size of 1.0 mm in diameter,and the atomic number density N can be around10 ∼ /cm [33, 34]. A transition dipole moment of 2 . × − Cm is used for µ in our calculations which is corresponding to the value of Rb D transition [35].Combining these parameters, we obtain the coefficient N | µ | / ( ǫ ~ ) ≈ . × /s .For demonstration we assume in our model that the upper state decays much slowerthan the intermediate excited state | i , such as γ = 0 . γ in our calculation. We willdiscuss this with more details later in this paper.By inspecting equation (26b), we see that the absorption spectrum of the probelaser can have two electromagnetically induced transparency windows as long as the twocoupling lasers are neither on resonance simultaneously nor at a Raman detuning( δ = lectromagnetically induced transparency in an inverted Y-type four-level system − δ ), that is, when the two coupling lasers have different frequency detunings, theabsorption of the probe laser displays two minima at δ = δ and δ = − δ , respectively,as demonstrated in figure 2(a)-(c). The absorption of the probe laser becomes zero(orthe atomic system becomes completely transparent to the probe laser) as the probe laserfrequency is detuned at δ = δ . The second minima is at δ = − δ but the absorption isnot completely zero at this detuning due to the decay of the upper excited state | i . Ifthe upper state | i does not decay, the absorption will also be zero at δ = − δ as shownin figure 2(b). However, the absorption is significantly reduced at δ = − δ even for adecaying upper excited state provided that the coupling laser L3 is strong with respectto the decay rate. If we choose an atomic system with a small decay rate γ of theupper state, such as a metastable state, then the absorption of the probe laser can bereduced greatly by a relatively strong coupling laser L3 as illustrated in figure 2(c). Letus compare this scheme with the widely used three-level Λ and cascade systems in theEIT study. Equation (26) can be used for a Λ scheme | i − | i − | i by setting Ω = 0,and for a cascade scheme | i − | i − | i by setting Ω = 0, respectively. In both casesthere is only one transparency window as illustrated in figure 2(d) by the dotdashedline and dashed line. The combination of both systems brings another dimension ofcontrol of EIT. As one can see from figure 2(d), an absorption line emerges withinthe transparency window of the Λ scheme by introducing laser L3, and the absorptionlinewidth can be subnatural. It is clear that the absorption spectrum of the probe laser depends on both the detuningand the Rabi frequency of the coupling lasers. We plot the imaginary part(absorption)and the real part(dispersion) of the linear susceptibility of the probe laser in the sameframe as a function of the frequency detuning of the probe laser in figure 3(a). Thedispersion curves(see dashed line) display a very high positive slope at the center of thetransparency windows; therefore the group velocity of the probe laser can be sloweddown without absorption. The group velocity of the probe field can be calculated by v g = c/n g , where c is the speed of light in vacuum and the group velocity index is givenby [19] n g = 1 + 12 χ ′ + ν ∂χ ′ ∂ν , (27)which is evaluated at the carrier frequency of the probe laser.The group index n g based on equation (27) is simultaneously plotted in figure 3(b)as a function of the frequency detuning of the probe laser corresponding to the sameparameters as in figure 3(a). One can see that the group velocity of the probe laser canbe reduced by as much as a factor of 10 for the chosen parameters without absorption.When the Rabi frequency of laser L3 increases as shown in figure 3(c), the absorptioncorresponding to the transparency window at δ = − δ decreases and the width of thetransparency window increases, while the slope of the dispersion curves decreases at lectromagnetically induced transparency in an inverted Y-type four-level system - - ∆ (cid:144) Γ - - Χ H ´ - L H a L - - - ∆ (cid:144) Γ - - - - n g H ´ L H b L - - ∆ (cid:144) Γ - - Χ H ´ - L H c L - - - ∆ (cid:144) Γ - - - - - n g H ´ L H d L - - ∆ (cid:144) Γ - - Χ H ´ - L H e L - - - ∆ (cid:144) Γ - - - - - n g H ´ L H f L - - ∆ (cid:144) Γ - - Χ H ´ - L H g L - - - ∆ (cid:144) Γ - - - n g H ´ L H h L Figure 3. (Color online) Left panel: Absorption (solid line) and dispersion (dashedline) of the probe laser (L1); Right panel: the corresponding group index. Theparameters for the calculations are: (a)-(b) Ω = 1 . γ , Ω = 0 . γ . (c)-(d)Ω = 1 . γ , Ω = 0 . γ . (e)-(f) Ω = 1 . γ , Ω = 1 . γ . (g)-(h) Ω = 2 . γ ,Ω = 1 . γ . Other parameters are: δ = 0, δ = − . γ , γ = 0 . γ . lectromagnetically induced transparency in an inverted Y-type four-level system δ = − δ , and therefore the group index at δ = − δ decreases as shown in figure 3(d).When the Rabi frequency of the laser L2 increases the width of the EIT window at δ = δ increases and the central absorption peak is pushed toward to the second EITwindow at δ = − δ as shown in figure 3(e), while the group index decreases at δ = δ ,and increases at δ = − δ as shown in figure 3(f). When we further increase Ω thecentral absorption component is pushed back toward the EIT window at δ = δ and thesecond transparency window becomes wider and deeper as shown in figure 3(g). Theslopes of the dispersion curves as well as the group index at the EIT windows decreasesaccordingly as shown in figure 3(h). Clearly, the absorption at the two EIT windows, andtherefore the corresponding group index can be very different depending on the couplinglasers. Consequently, the EIT can be controlled by the detunings and Rabi frequenciesof two coupling lasers as well as the group velocities at these two windows. This canbe very useful for quantum information processing and transfer. One can control thepropagation of probe signals at two adjacent frequencies with the two coupling fields.To uncover the responsible physical parameters for the group index within each EITwindow, we write the group index equation (27) explicitly.(1) For the δ = δ transparency window: n g ( δ = δ ) = 1 + 2 κ × (cid:18) δ + ω Ω (cid:19) , (28)with κ = N | µ | ǫ ~ , which is a function of δ and Ω of laser L2, but independent of laserL3. For a given detuning δ , the group index at the δ = δ transparency window isinversely proportional to the intensity of laser L2 as illustrated in figure 4(a).(2) For the δ = − δ transparency window: n g ( δ = − δ ) = 1 − κ ( δ + δ ) βγ α + β )+ κ ( ω − δ ) (cid:16) β γ − ( δ + δ )( α + ∂β∂δ γ ) (cid:17) α + β + κγ ( ω − δ )( δ + δ ) β ( α ∂α∂δ + β ∂β∂δ )( α + β ) (29)with α = ( δ + δ )( ω + γ γ / β = γ (cid:0) Ω / − δ ( δ + δ ) (cid:1) ∂α∂δ = Ω + Ω + γ γ / − δ ( δ + 3 δ ) ∂β∂δ = − δ ( γ γ / − δ γ + γ δ = − δ , the group index depends on bothlasers and it is a function of the three parameters δ , Ω and Ω for a given δ . In figure4(b)-(d) we show the group index as a function of Ω , δ , or Ω , respectively, when the lectromagnetically induced transparency in an inverted Y-type four-level system a( ) n g1 = )( x10 ) / b( ) n g1 = - )( x ) / c( ) n g1 = - )( x10 ) / d( ) n g1 = - )( x10 ) / Figure 4. (Color online) (a) The group index at the center of the EIT window, δ = δ , as a function of Ω . Other parameters are δ = 0 . γ , δ = 0, Ω = 1 . γ and γ = 0 . γ . The group index at the center of the EIT window δ = − δ = − . γ : (b)as a function of Ω for δ = 0 and Ω = 0 . γ ; (c) as a function of δ for Ω = 1 . γ and Ω = 0 . γ ; (d) as a function of Ω , for δ = 0 and Ω = 1 . γ . two other parameters are given at the transparency window δ = − δ . This clearlyshows that the group index can be manipulated by the parameters of both couplinglasers.When both coupling lasers are on resonance or in Raman detuning with eachother( δ = − δ ), the two transparency windows merge into one as shown in figure 5(a)and 5(b), respectively. The width of the transparency window increases with the Rabifrequencies, Ω and Ω , as shown in figure 5(c). If we inspect the spectrum carefullywe notice that there is a small feature at the center of the transparency window. Weplot it in an expanded scale in figure 5(d). We find that the dispersion changes betweennormal and abnormal within a very narrow frequency region. Consequently, the groupindex changes from negative to positive; in other words, the probe laser can be switchedfrom a anomalous dispersion associated with a negative group index(or superluminal)to a positive group index(or subluminal) without being absorbed within this window. lectromagnetically induced transparency in an inverted Y-type four-level system - - ∆ (cid:144) Γ - - Χ H ´ - L H a L - - ∆ (cid:144) Γ - - Χ H ´ - L H b L - - ∆ (cid:144) Γ - - Χ H ´ - L H c L - - - ∆ (cid:144) Γ - - n g H ´ L , Χ H ´ - L H d L Figure 5. (Color online) The susceptibility and the group index as a functionof the probe detuning δ when two coupling lasers are on resonance or on Ramandetuning( δ = − δ ). (a) δ = δ = 0, Ω = Ω = 0 . γ . (b) For laser L2 and L3are on Raman detuning: δ = − δ = 0 . γ , Ω = Ω = 0 . γ . (c) δ = δ = 0,Ω = 2 . γ , and Ω = 0 . γ . (d) we plot (c) on an expanded scale to show the detailswithin the EIT window(solid lines: imaginary part of χ , dashed lines: the real part of χ ; dotdashed lines: the group index). Although the transparency is not one hundred percent for the negative group indexregion the absorption is small.Based on the above analysis, the scheme can be realized in a four-level rubidium Rb atom. Two hyperfine ground state levels, | i = | S / , F ′′ = 1 i , and | i = | S / , F ′′ = 2 i are coupled by laser L1 and laser L2 to a common intermediate excited hyperfinelevel, | i = | P / , F ′ = 1 i , respectively. The third laser L3 couples the intermediatelevel to a higher excited 5 D / hyperfine level | i = | D / , F = 2 i . In order to observethe phenomena experimentally, an ultracold Rb atomic ensemble formed in a MOTwith a typical temperature of 100 µK would satisfy all the conditions correspondingto our above calculations. The lifetime of 5 D / state is very long compared to thatof the 5 P / state( γ ≃ . γ ) [35]. So all the conditions assumed in our analysis canbe satisfied in this system. We expect the experimental observations can be readilyrealized. lectromagnetically induced transparency in an inverted Y-type four-level system
4. Summary
In conclusion, we have shown that the response of a probe laser in an inverted Y-typefour-level system driven by two additional coherent fields exhibits double transparencywindows for the probe laser. The reliability of the calculations is established by theagreement in the susceptibility of the probe laser obtained by both wavefunction anddensity matrix methods. The transparency windows can be controlled by the amplitudeand frequency detuning of the coupling fields. The group index associated with thegroup velocity of the probe laser can be very different at the two transparency windows;hence it can be controlled by the coupling fields. The propagation of the probe field canbe switched from superluminal near the resonance to subluminal on resonance of theprobe transition within the single transparency window when the two coupling lasers aredetuned on resonance. This provides a potential application in quantum informationprocessing. This scheme may be realized in an ultracold Rb system and can be usedto investigate both superluminal and slow light.
5. Acknowledgement
This work is supported by the Research Development Grant from Penn State University.
References [1] Hau, L.V., Harris, S.E., Dutton, Z., and Behroozi, C.H., Nature 397, 594 (1999).[2] M.D. Lukin, S.F. Yelin, M. Fleischhauer, and M.O. Scully, Phys. Rev. A 60, 3225 (1999).[3] G.S. Agwarwal and S. Menon, Phys. Rev. A 63, 023818 (2001).[4] J. P. Marangos, J. of Modern Optics, vol. 43, No.3, 471 (1998).[5] Ryan M. Camacho, Michael V. Pack, John C. Howell, Aaron Schweinsberg and Robert W. Boyd,Phys. Rev. Lett. 98, 153601 (2007).[6] Michael M. Kash, Vladimir A. Sautenkov, Alexander S. Zibrov, L. Hollberg, George R. Welch,Mikhail D. Lukin, Yuri Rostovtsev, Edward S. Fry, and Marlan O. Scully, Phys. Rev. Lett. 82,5229 (1999).[7] Ray-Yuan Chang, Wei-Chia Fang, Zong-Syun He, Bai-Cian Ke, Pei-Ning Chen, and Chin-ChunTsai, Phys. Rev. A 76, 053420 (2007).[8] James Owen Meatherall, Christopher P. Search, and Markku J¨a¨askel¨ainen, Phys. Rev. A 78, 013830(2008).[9] J. Qi, F. C. Spano, T. Kirova, A. Lazpudis, J. Magnes, L. Li, L.M. Narducci, R. W. Field, and A.M. Lyyra, Phys. Rev. Lett. 88, 173003 (2002).[10] J. Mompart and R. Corbaln, Optics Communications 156, 133 (1998).[11] Ying Wu, and Xiaoxue Yang, Phys. Rev. A 71, 053806 (2005).[12] F. Carreo, Oscar G. Caldern, M. A. Antn, and Isabel Gonzalo, Phys. Rev. A 71, 063805 (2005).[13] Kristian Rymann Hansen and Klaus Mølmer, Phys. Rev. A 75, 053802 (2007).[14] For a review of this subject, see E. Arimondo, in Progress in Optics XXXV, edited by E.Wolf(North-Holland,Amsterdam, 1996)[15] S. E. Harris, J.E. Field and A, Kasapi, Phys. Rev. A 46, R29 (1992).[16] Julio Gea-Banacloche, Yong-qing Li, Shao-zheng Jin, and Min Xiao, Phys Rev. A 51, 576(1995).[17] S. E. Harris, Phys. Today 50, 736 (1997).[18] M.D. Lukin , and A. Imamoglu, Nature 413, 273 (2001). lectromagnetically induced transparency in an inverted Y-type four-level system14