VIC Effect and Phase-Dependent Optical Properties of Five-Level K-Type Atoms Interacting with Coherent Laser Fields

Commun. Theor. Phys. (Beijing China) 50 (2008) pp. 741 748 c Chinese Physical Society Vol. 50 No. 3 September 15 2008 VIC Effect and Phase-Dependent Optical Properties of Five-Level K-Type Atoms Interacting with Coherent Laser Fields JIA Wen-Zhi 1 and WANG Shun-Jin 12 1 Center of Theoretical of Physics School of Physics Sichuan University Chengdu 610064 China 2 Center of Theoretical Nuclear Physics National Laboratory of Heavy Ion Accelerator Lanzhou 730000 China (Received September 24 2007) Abstract Due to interaction with the vacuum of the radiation field a K-type atomic system with near-degenerate excited and ground levels which is driven by two strong coherent fields and two weak probe fields has additional coherence terms the vacuum-induced coherence (VIC) terms. In this paper we find that if the interference is optimized the two-photon absorption properties of this atom system can be significantly modified and electromagnetically-induced transparency (EIT) is dependent on this interference. Furthermore we find that in all the cases the coherence can suppress or enhance the partial two-photon transparency while the complete transparency window is still strictly preserved which means that it cannot be affected by the VIC. Another important result is the finding of the crucial role played by the relative phase between the probe and coupling fields: the relative height of absorption peaks can be modulated by the relative phase. The physical interpretation of the phenomena has been given. PACS numbers: 42.50.Gy 42.50.Hz 42.50.-p Key words: vacuum-induced coherence electromagnetically-induced transparency phase-dependent optical properties 1 Introduction The interference induced by a strong resonant coupling field makes an opaque atomic system transparent for a probe field. This phenomenon which is termed electromagnetically induced transparency (EIT) has been studied in all kinds of three-level atomic configurations including V Λ Y and tripod scheme. [1] Many experimental observations on the electromagnetically induced twophoton transparency were reported such as the inhibition of two-photon absorption in sodium atoms [2] two-photon transparency in Rb 85 atoms [3] one- and two-photon transparencies for the isotopes Rb 85 and Rb 87 [4] the transparency against one- and two-photon absorptions in the Λ-type four-level atom [5] etc. When the interactions of the transitions from two nearly degenerate excited levels to a single ground level with the same vacuum field occur in a V-configuration three-level atom vacuum-induced coherence (VIC) can be created due to the quantum interference between the two transitions. VIC leads to many remarkable phenomena of atomic systems such as modification of absorption properties transparency and gain with or without inversion [6 8] and spontaneous emission spectra. [9] Besides the VIC in a V-configuration the effect of VIC in a Λ-configuration three-level atom which is created by the quantum interference between the two transitions from a single excited level to two nearly degenerate levels on the coherent population trapping and absorption properties has been studied in Refs. [10] [14]. Subsequently the investigation of VIC is generalized from three-level to four-level atomic systems such as the four-level atomic systems in double- V and inverted-y configurations. [15 17] As considering the VIC effect the absorption properties [1317] and spontaneous emission spectra [16] can be related to the relative phase between the probe and coupling fields. Recently by using a control field to couple the intermediate level of the Y-type (inverted-y type) atom with another ground (or excited) level Hou et al. introduced a five-level atomic system in K-type configuration and found some new phenomena such as the split of the twophoton transparency window. [18] In the present paper we assume that the ground level and excited levels of the coupling transitions in a five-level K-type atomic system are nearly degenerate. In this case the effect of VIC which is resulted from the quantum interference between two transitions is very remarkable. In addition due to the existence of VIC we should consider the phases of the probe and coupling fields. We will investigate the effects of VIC and the relative phase on the probe absorption properties. These studies may be important in the investigation of two-photon lasing the pulse propagation [1920] and the two-photon entanglement in quantum information process. [2122] This paper is organized as follows. In Sec. 2 we describe the model and present the corresponding densitymatrix equations of motion for the system. In Sec. 3 we investigate the effect of VIC on the two-photon absorption properties. The phase dependence of the absorption The project supported in part by National Natural Science Foundation of China under Grant Nos. 90503088 and 10775100 and the Fund of Theoretical Nuclear Center of HIRFL of China E-mail: jiawenz1979@126.com Corresponding author E-mail: sjwang@home.swjtu.edu.cn

742 JIA Wen-Zhi and WANG Shun-Jin Vol. 50 properties is discussed in Sec. 4. Finally a conclusion is drawn in Sec. 5. 2 Atomic Model and Density Matrix Equations The energy scheme of the five-level K-type atom is shown in Fig. 1. The experimental system for this atomic scheme can be realized by a Rb 87 atom with 5S 1/2 F = 1 5P 3/2 F = 2 7S 1/2 F = 1 5S 1/2 F = 2 and 5D 5/2 F = 3 denoted by 1 2 3 4 and 5 respectively. The two-photon transition of the atom under consideration is induced by two weak probe lasers with frequencies (amplitudes) w 1 ( ε 1 ) and w 2 ( ε 2 ) and with Rabi frequencies G 1 = µ 21 ε 1 / driving 2 1 and G 2 = µ 32 ε 2 / driving 3 2 respectively. A control laser with frequency (amplitude) w 3 ( ε 3 ) and Rabi frequency G 3 = µ 24 ε 3 / is applied to couple the intermediate level 2 with a ground level 4 and another control laser with frequency (amplitude) w 4 ( ε 4 ) and Rabi frequency G 4 = µ 52 ε 4 / is used to couple the level 2 with an excited level 5. 2γ 1 and 2γ 3 are the spontaneous decay rates from level 2 to levels 1 and 4 and 2γ 2 and 2γ 4 correspond to the decay rates from levels 3 and 5 to 2. The detunings of the probe and coupling lasers are defined as 1 = w 21 w 1 2 = w 32 w 2 3 = w 24 w 3 and 4 = w 52 w 4 respectively. Fig. 1 The energy scheme of the five-level K-type atom. With dipole couplings under rotation-wave approximation and fixing the level 2 the interacting Hamiltonian of the atom with the coherent laser fields reads Ĥ I = [ 1 1 1 + 2 3 3 3 4 4 + 4 5 5 (G 1 2 1 + G 2 3 2 + G 3 2 4 + G 4 5 2 + G 1 1 2 + G 2 2 3 + G 3 4 2 + G 4 2 5 )]. (1) In this paper we use the density matrix approach to analyze the system. We shall pay special attention to the effect of the vacuum on the system by considering the vacuum-atom interaction. The dynamical evolution of the system including spontaneous emission terms is governed by the density-matrix equations in a Liouvillian form t ˆρ = i [ĤI ˆρ] + 2ρ 22 (γ 1 1 1 + γ 3 4 4 ) (γ 1 + γ 3 ){ˆρ 2 2 } + 2 2 2 (γ 2 ρ 33 + γ 4 ρ 55 ) γ 2 {ˆρ 3 3 } γ 4 {ˆρ 5 5 } + p γ 2 γ 4 [2(ρ 35 + ρ 53 ) 2 2 ˆρ 3 5 ˆρ 5 3 3 5 ˆρ 5 3 ˆρ] + 2q γ 1 γ 3 ρ 22 ( 1 4 + 4 1 ) (2) where the brackets [ ˆB] and { ˆB} denote the commutation and anti-commutation relations of the operators  and ˆB respectively. The terms with p γ 2 γ 4 are related to the quantum interference effect resulted from the cross-coupling between spontaneous emissions 3 2 and 5 2 and the terms with q γ 1 γ 3 are related to the quantum interference effect resulted from the crosscoupling between spontaneous emissions 2 1 and 2 4 where p = cosθ with θ being the angle between the two induced dipole matrix elements d 32 and d 52 and q = cosθ with θ being the angle between the two induced dipole matrix elements d 21 and d 24. Obviously the parameter p and q play a very important role in the creation of coherence. When d 32 and d 52 d 21 and d 24 are orthogonal to each other no such spontaneously generated interference effect exists. On the contrary when d 32 and d 52 or d 21 and d 24 are parallel we obtain a maximal spontaneously generated interference effect. The origin of the term p γ 2 γ 4 can be explained intuitively as follows: it represents the effect of spontaneous emission of one photon from the transition 3 2 and absorption of the same photon from the transition 2 5 or vice versa. Namely this term is the result of quantum interference of spontaneous emission from the two closely lying upper levels. A similar explanation is suitable for the term q γ 1 γ 3. Equation (2) can be written in terms of matrix elements of ˆρ as follows: ρ 11 = 2γ 1 ρ 22 ig 1 ρ 12 + ig 1ρ 21 ρ 33 = 2γ 2 ρ 33 + ig 2 ρ 23 ig 2ρ 32 p γ 2 γ 4 (ρ 35 + ρ 53 ) ρ 44 = 2γ 3 ρ 22 ig 3 ρ 42 + ig 3ρ 24 ρ 55 = 2γ 4 ρ 55 + ig 4 ρ 25 ig 4ρ 52 p γ 2 γ 4 (ρ 35 + ρ 53 ) ρ 12 = (i 1 γ 1 γ 3 )ρ 12 + ig 1(ρ 22 ρ 11 ) ig 2 ρ 13 ig 3ρ 14 ig 4 ρ 15 ρ 13 = (i 1 + i 2 γ 2 )ρ 13 + ig 1ρ 23 ig 2ρ 12 p γ 2 γ 4 ρ 15 ρ 14 = (i 1 i 3 )ρ 14 + ig 1ρ 24 ig 3 ρ 12 + 2q γ 1 γ 3 ρ 22 ρ 15 = (i 1 + i 4 γ 4 )ρ 15 + ig 1ρ 25 ig 4ρ 12 p γ 2 γ 4 ρ 13

No. 3 VIC Effect and Phase-Dependent Optical Properties of Five-Level K-Type Atoms Interacting with Coherent Laser Fields 743 ρ 23 = (i 2 γ 1 γ 2 γ 3 )ρ 23 + ig 1 ρ 13 + ig 2(ρ 33 ρ 22 ) + ig 3 ρ 43 + ig 4ρ 53 p γ 2 γ 4 ρ 25 ρ 24 = ( i 3 γ 1 γ 3 )ρ 24 + ig 1 ρ 14 + ig 2ρ 34 + ig ( 3 ρ 44 ρ 22 ) + ig 4ρ 54 ρ 25 = (i 4 γ 1 γ 3 γ 4 )ρ 25 + ig 1 ρ 15 + ig 2ρ 35 + ig 3 ρ 45 + ig 4(ρ 55 ρ 22 ) p γ 2 γ 4 ρ 23 ρ 34 = ( i 2 i 3 γ 2 )ρ 34 + ig 2 ρ 24 ig 3 ρ 32 p γ 2 γ 4 ρ 54 ρ 35 = ( i 2 + i 4 γ 2 γ 4 )ρ 35 + ig 2 ρ 25 ig 4ρ 32 p γ 2 γ 4 (ρ 33 + ρ 55 ) ρ 45 = (i 3 + i 4 γ 4 )ρ 45 + ig 3ρ 25 ig 4ρ 42 p γ 2 γ 4 ρ 43. (3) The closure of this atomic system requires that ρ 11 +ρ 22 + ρ 33 + ρ 44 + ρ 55 = 1 and ρ ij = ρ ji. In this paper we just consider linearly polarized electric fields with the restriction of ε 1 d 24 = 0 ε 3 d 21 = 0 ε 2 d 52 = 0 and ε 4 d 32 = 0. Thus Rabi frequencies G 2 and G 4 are connected to the angle θ by the relation of G 2 = g 20 sin θ and G 4 = g 40 sin θ while G 1 and G 3 are connected to the angle θ by the relation of G 1 = g 10 sin θ and G 3 = g 30 sin θ with g 10 = µ 21 ε 1 / and g 20 = µ 32 ε 2 / g 30 = µ 24 ε 3 / and g 40 = µ 52 ε 4 /. To gain the remarkable VIC effect on optical properties of the system we assume that the excited levels and ground levels are nearly degenerate i.e. w 35 0 w 14 0 because in the case that the energy difference of the excited levels or ground levels is large the terms with p γ 2 γ 4 and q γ 1 γ 3 accompanied by exponential factors e ±iw35t and e ±iw14t respectively [which are not shown in Eqs. (2) and (3) since the nearly degenerate case is considered in this paper] will average out and no effect from spontaneous emission interference can be obtained. In the conventional treatment where the behavior of the system only depends on the amplitudes and detunings of the external coherent fields but not on their phases the Rabi frequencies are treated as real parameters. However in our case the system is sensitive to the phases of the probe and coupling fields due to the existence of VIC. Taking phases of the probe G 1 and G 2 as well as the coherent field G 3 and G 4 into account we rewrite the Rabi frequencies as G 1 = g 1 e iφ 1 G 2 = g 2 e iφ 2 G 3 = g 3 e iφ 3 and G 4 = g 4 e iφ 4 where g 1 g 2 g 3 and g 4 are assumed to be real. By redefining the atomic variables as σ ii = ρ ii σ 12 = ρ 12 e iφ 1 σ 13 = ρ 13 e i(φ 1+φ 2 ) σ 14 = ρ 14 e i(φ 1 φ 3 ) σ 15 = ρ 15 e i(φ 1+φ 4 ) σ 23 = ρ 23 e iφ 2 σ 24 = ρ 24 e iφ 3 σ 25 = ρ 25 e iφ 4 σ 34 = ρ 34 e i(φ 2+φ 3 ) σ 35 = ρ 35 e i(φ 2 φ 4 ) and σ 45 = ρ 45 e i(φ 3+φ 4 ) we obtain the equations of motion for the redefined density matrix elements σ ij as follows: σ 11 = 2γ 1 σ 22 ig 1 (σ 12 σ 21 ) σ 33 = 2γ 2 σ 33 + ig 2 (σ 23 σ 32 ) p γ 2 γ 4 (σ 35 e iφ 24 + σ 53 e iφ 24 ) σ 44 = 2γ 3 σ 22 ig 3 (σ 42 σ 24 ) σ 55 = 2γ 4 σ 55 + ig 4 (σ 25 σ 52 ) p γ 2 γ 4 (σ 35 e iφ 24 + σ 53 e iφ 24 ) σ 12 = (i 1 γ 1 γ 3 )σ 12 + ig 1 (σ 22 σ 11 ) ig 2 σ 13 ig 3 σ 14 ig 4 σ 15 e iφ 24 σ 13 = (i 1 + i 2 γ 2 )σ 13 + ig 1 σ 23 ig 2 σ 12 p γ 2 γ 4 σ 15 e iφ 24 σ 14 = (i 1 i 3 )σ 14 + ig 1 σ 24 ig 3 σ 12 + 2q γ 1 γ 3 σ 22 e iφ 13 σ 15 = (i 1 + i 4 γ 4 )σ 15 + ig 1 σ 25 ig 4 σ 12 p γ 2 γ 4 σ 13 e iφ 24 σ 23 = (i 2 γ 1 γ 2 γ 3 )σ 23 + ig 1 σ 13 + ig 2 (σ 33 σ 22 ) + ig 3 σ 43 + ig 4 σ 53 p γ 2 γ 4 σ 25 e iφ 24 σ 24 = ( i 3 γ 1 γ 3 )σ 24 + ig 1 σ 14 + ig 2 σ 34 + ig 3 (σ 44 σ 22 ) + ig 4 σ 54 σ 25 = (i 4 γ 1 γ 3 γ 4 )σ 25 + ig 1 σ 15 + ig 2 σ 35 + ig 3 σ 45 + ig 4 (σ 55 σ 22 ) p γ 2 γ 4 σ 23 e iφ 24 σ 34 = ( i 2 i 3 γ 2 )σ 34 + ig 2 σ 24 ig 3 σ 32 p γ 2 γ 4 σ 54 e iφ 24 σ 35 = ( i 2 + i 4 γ 2 γ 4 )σ 35 + ig 2 σ 25 ig 4 σ 32 p γ 2 γ 4 (σ 33 + σ 55 ) e iφ 24 σ 45 = (i 3 + i 4 γ 4 )σ 45 + ig 3 σ 25 ig 4 σ 42 p γ 2 γ 4 σ 43 e iφ 24 (4) where φ 24 = φ 2 φ 4 and φ 13 = φ 1 φ 3 are the relative phases of the probe field G 2 (G 1 ) and the control fields G 4 (G 3 ). In the following we consider the two-photon absorption behaviors in the five-level K-type atomic system including the simultaneous applications of the two control fields G 3 and G 4. For simplicity and without loss of generality we assume that the two control fields with an identical value of their Rabi frequencies detune the same magnitudes from their corresponding transitions. In this paper we restrict our discussion on the two-photon resonant condition 1 + 2 = 0 and set the parameters to the values γ 1 = γ 2 = γ 3 = γ 4 = 10. The Rabi frequencies of the weak probe fields and control fields are given by the values g 10 = g 20 = 1 and g 30 = g 40 = 50 respectively throughout the paper. 3 VIC-Dependent Absorption Properties At first we shall investigate the two-photon absorption behaviors described by the σ 33 numerically under the steady-state condition. Now we only consider the quantum interference effect resulted from the cross-coupling between spontaneous emissions 3 2 and 5 2 namely assuming p 0 q = 0. In Fig. 2 we plot the two-photon absorption curve versus 1 with different p

744 JIA Wen-Zhi and WANG Shun-Jin Vol. 50 under the two-photon resonant condition 1 + 2 = 0 and set the detunings of the control fields by the values 3 = 4 = 30. To show the spectra with different p clearly logarithmic coordinate is adopted in Fig. 2(b). In the solid curve of Fig. 2(b) we can see that when the two control fields G 3 and G 4 simultaneously drive the atom there appears an absorption peak at 1 = 0 which splits the transparency windows into two parts indicating the phenomenon of double two-photon transparency explained in terms of dressed states. [14] We can see from the solid curve that when p = 0 there are two transparency windows located at 1 = 30 and 1 = 30 respectively the left window is partially transparent and the right one is completely transparent. Clearly the VIC can suppress the partially transparency as compared with the solid curve (p = 0 without VIC). For example it is seen from the dash-dot-dotted curve that when the parameter p = 0.99 the VIC enhances the two-photon absorption up to σ 33 = 0.023 while the absolute two-photon transparency for probe detuning at 1 = 30 is still strictly preserves as shown in Fig. 2(b). Fig. 2 The two-photon absorption σ 33 as a function of the detuning 1 plotted in ordinary coordinate (a) and logarithmic coordinate (b) with 3 = 4 = 30 γ 1 = γ 2 = γ 3 = γ 4 = 10 g 10 = g 20 = 1 g 30 = g 40 = 50 φ 24 = φ 13 = 0 q = 0 and different p. p = 0 (no VIC) (solid curve) p = 0.5 (dashed curve) p = 0.8 (dotted curve) p = 0.9 (dash-dotted curve) and p = 0.99 (dash-dot-dotted curve). Then we turn to the quantum interference effect resulted from the cross-coupling between spontaneous emissions 2 1 and 2 4 namely assuming that p = 0 q 0 and other parameters are the same as those in Fig. 2. Comparing with the solid curve in Fig. 3(a) we can see that the VIC enhances the partial transparency while the absolute transparency at 1 = 30 is still strictly preserved. In addition the VIC can widen the left transparency window and narrow the right one. To show it clearly we replot the spectrum diagram by using logarithmic coordinate in Fig. 3(b). Fig. 3 The two-photon absorption σ 33 as a function of the detuning 1 plotted in ordinary coordinate (a) and logarithmic coordinate (b) with p = 0 and different q. Other parameters are the same as those in Fig. 2 and q = 0 (no VIC) (solid curve) q = 0.5 (dashed curve) q = 0.8 (dotted curve) q = 0.9 (dash-dotted curve) q = 0.99 (dash-dot-dotted curve). Subsequently we consider another interesting case where the detunings of the control fields obey the relation 3 + 4 = 0. Again we first consider the case of p 0 and q = 0. From the solid curve in Fig. 4(a) with 3 = 4 = 30 it is shown that the in-between peaks near 1 = 30 displayed in Figs. 2 and 3 disappear

No. 3 VIC Effect and Phase-Dependent Optical Properties of Five-Level K-Type Atoms Interacting with Coherent Laser Fields 745 indicating that the double two-photon transparency windows are merged into a single one. The minimum of twophoton absorption only occurs at 1 = 30. In this case with increase of p the two absorption peaks shift closer than those in the absence of VIC (p = 0) the absorption peaks are enhanced by VIC in a wide range and the height of absorption peak decreases with increase of p as it is close to 1. It is seen from the dotted curve that the two-photon absorption is up to 0.000 841 (left peak) and 0.000 313 (right peak) as p = 0.8 while as p = 0 (no VIC) the corresponding values are 0.000 017 5 and 0.000 023 5 respectively much weaker compared with the first case. In addition the shape of transparency window with p 0 is different from that with p = 0. For the diagram of absorption spectrum against 1 the shape of transparency window with p = 0 is concave while in the case of p 0 its shape looks like a goblet. Figure 4(b) displays details of the spectra with different p near the completely transparent point 1 = 30. Fig. 4 The two-photon absorption σ 33 as a function of the detuning 1 (a) and the detail of the spectra near the completely transparent point from 1 = 40 to 1 = 20 (b) with 3 = 4 = 30 q = 0 and different p. Other parameters are the same as those in Fig. 2 and p = 0 (no VIC) (solid curve) p = 0.5 (dashed curve) p = 0.8 (dotted curve) p = 0.9 (dash-dotted curve) and p = 0.99 (dash-dot-dotted curve). Now we consider the case of p = 0 and q 0 where the transparent points with different q are all at 1 = 30 and the magnitude of the absorption peak decreases drastically with q. In a wide range of q the width and shape of the transparency windows are almost the same on the whole except that the local structure of the area near the transparency point 1 = 30 is quite different as shown in Fig. 5. Fig. 5 The two-photon absorption σ 33 as a function of the detuning 1 (a) and the detail of the spectra near the completely transparent point from 1 = 40 to 1 = 20 (b) with p = 0 and different q. Other parameters are the same as those in Fig. 4 and q = 0 (no VIC) (solid curve) q = 0.5 (dashed curve) q = 0.8 (dotted curve) q = 0.9 (dash-dotted curve) and q = 0.99 (dash-dot-dotted curve). As setting 3 = 4 = 30 we can see that the atom displays a similar absorption behavior except that the minimum in two-photon absorption is located at 1 = 30 and the curves in this case (not shown here) are just the mirror reflections of the ones with 3 = 4 = 30. At the end of this section we consider the special case where the detunings of the control fields are set to be 3 = 4 = 0. Figures 6 and 7 display how the asymmetric two-photon transparency profiles in Figs 4 and 5 are changed into symmetric ones.

746 JIA Wen-Zhi and WANG Shun-Jin Vol. 50 Fig. 6 The two-photon absorption σ 33 as a function of the detuning 1 (a) and the detail of the spectra near the completely transparent point from 1 = 10 to 1 = 10 (b) with 3 = 4 = 0q = 0 and different p. Other parameters are the same as those in Fig. 2 and p = 0 (no VIC) (solid curve) p = 0.5 (dashed curve) p = 0.8 (dotted curve) p = 0.9 (dash-dotted curve) and p = 0.99 (dash-dot-dotted curve). Fig. 7 The two-photon absorption σ 33 as a function of the detuning 1 (a) and the detail of the spectra near the completely transparent point from 1 = 10 to 1 = 10 (b) with p = 0 and different q. Other parameters are the same as those in Fig. 6 and q = 0 (no VIC) (solid curve) q = 0.5 (dashed curve) q = 0.8 (dotted curve) q = 0.9 (dash-dotted curve) and q = 0.99 (dash-dot-dotted curve). 4 Phase-Dependent Optical Properties In the conventional treatment the behavior of the system only depends on the amplitudes and detunings of the external coherent fields but not on their phases. However our numerical results show that the behavior of the system studied in this paper is sensitive to the phase φ 24 while the influence of φ 13 is not so remarkable. Thus in the following we only investigate effect of the relative phase φ 24 on two-phonon absorption. Fig. 8 The two-photon absorption σ 33 as a function of the detuning 1 with p = 4/5 q = 0 3 = 4 = 30 and different φ 24. Other parameters are the same as those in Fig. 2. (a) φ 24 = 0 (solid curve) φ 24 = π/6 (dashed curve) φ 24 = π/3 (dotted curve) and φ 24 = π/2 (dash-dotted curve). (b) φ 24 = 0 (solid curve) φ 24 = π/2 (dashed curve) φ 24 = π (dotted curve) and φ 24 = 3π/2 (dash-dotted curve).

No. 3 VIC Effect and Phase-Dependent Optical Properties of Five-Level K-Type Atoms Interacting with Coherent Laser Fields 747 φ 24 on the two-photon absorption at 1 = 56.123 (solid line) 1 = 12.731 (dashed line) and 1 = 69.956 (dotted line) is shown in Fig. 9. It is seen that the absorption profiles are periodic with a period of 2π. Fig. 9 The two-photon absorption σ 33 as a function of the relative phase φ 24 with p = 4/5 q = 0 and different 1 : 1 = 56.123 (solid line) 1 = 12.731 (dashed line) and 1 = 69.956 (dotted line). Other parameters are the same as those in Fig. 2. Fig. 11 The two-photon absorption σ 33 as a function of the relative phase φ 24 with p = 2/3 q = 0 and different 1 : 1 = 77.712 (solid line) and 1 = 47.129 (dashed line). Other parameters are the same as those in Fig. 4. Fig. 10 The two-photon absorption σ 33 as a function of the detuning 1 with p = 2/3 q = 0 and different φ 24. Other parameters are the same as those in Fig. 4. (a) φ 24 = 0 (solid curve) φ 24 = π/6 (dashed curve) φ 24 = π/3 (dotted curve) and φ 24 = π/2 (dash-dotted curve). (b) φ 24 = 0 (solid curve) φ 24 = π/2 (dashed curve) φ 24 = π (dotted curve) and φ 24 = 3π/2 (dashdotted curve). We plot the two-phonon absorption versus probe detuning with p = 4/5 q = 0 and 3 = 4 = 30 given values of relative phase φ 24 in Fig. 8. We can see from the figure that in this case the relative phase affects the minimum of the two-photon absorption very slightly. With the same parameters the modulation of the relative phase Fig. 12 The two-photon absorption σ 33 as a function of the detuning 1 with p = 2/3 q = 0 and different φ 24. Other parameters are the same as those in Fig. 6. (a) φ 24 = 0 (solid curve) φ 24 = π/6 (dashed curve) φ 24 = π/3 (dotted curve) and φ 24 = π/2 (dash-dotted curve). (b) φ 24 = 0 (solid curve) φ 24 = π/2 (dashed curve) φ 24 = π (dotted curve) and φ 24 = 3π/2 (dashdotted curve). Then we consider the case with p = 2/3 q = 0 and 3 = 4 = 30. The spectra of the two-photon absorption versus probe detuning with given values of relative

748 JIA Wen-Zhi and WANG Shun-Jin Vol. 50 phase φ 24 are shown in Fig. 10 where the other parameters are the same as those in Fig. 4. We can see that the relative phase does not affect the minimum of the twophoton absorption (transparency point) at 1 = 30 while the absorption peaks are oscillating with varying of φ 24. The modulation of the relative phase φ 24 on the two-photon absorption at 1 = 77.712 (solid line) and 1 = 47.129 (dashed line) is shown in Fig. 11. It is seen that the absorption profiles are periodic with a period of 2π. Fig. 13 The two-photon absorption σ 33 as a function of the relative phase φ 24 with p = 2/3 q = 0 and different 1 : 1 = 60.586 (solid line) and 1 = 60.586 (dashed line). Other parameters are the same as those in Fig. 6. Finally we consider the special case where the detunings of the control fields are set to be 3 = 4 = 0. We see that the relative phase does not affect the minimum of the two-photon absorption (transparency point) at 1 = 0. The two-photon absorption profile with φ 24 = π shown by the dotted curve in Fig. 12(b) is like that in absence of relative phase shown by the solid curve in Fig. 12(b) but the height of the former two symmetric absorption peaks is higher than that of the latter. The two-photon absorption profile for φ 24 = π/2 [dashed line see Fig. 12(b)] is symmetric with respect to that in the case of φ 24 = 3π/2 (dash-dotted line). The modulation of the relative phase φ 24 on the two-photon absorption at 1 = 60.586 (solid line) and 1 = 60.586 (dashed line) shown in Fig. 13 also results in oscillation with a period of 2π. It can be seen from Fig. 13 that the maximum absorption depends not only on the different detunings but also on different relative phases in this case. 5 Conclusions In summary we have investigated the electromagnetically induced two-photon transparencies without or with VIC in the five level K-type atom. In this atomic system with closely-lying excited levels or ground levels the absorption spectra can be remarkably changed due to the existence of the quantum interference effect among spontaneous decay channels. It is very interesting that the VIC cannot affect the complete transparency point in all the cases we have investigated. Additionally due to the existence of VIC the absorption spectra are quite sensitive to the relative phases between the probe and coupling lasers. References [1] K.J. Boller A. Imamolu and S.E. Harris Phys. Rev. Lett. 66 (1991) 2593; S.E. Harris Phys. Today 50 (1997) 36. [2] L.V. Hau S.E. Harris Z. Dutton and C.H. Behroozi Nature (London) 397 (1999) 594. [3] C. Liu Z. Dutton C.H. Behroozi and L.V. Hau Nature (London) 409 (2001) 490. [4] M.M. Kash V.A. Sautenkov A.S. Zibrov L. Hollberg G.R. Welch M. D Lukin Y. Rostovtsev E.S. Fry and M.O. Scully Phys. Rev. Lett. 82 (1999) 5229. [5] D.F. Phillips A. Fleischhauer A. Mair R.L. Walsworth and M.D. Lukin Phys. Rev. Lett. 86 (2001) 783. [6] P. Zhou and S. Swain Phys. Rev. Lett. 78 (1997) 832. [7] E. Paspalakis S.Q. Gong and P.L. Knight Opt. Commun. 152 (1998) 293. [8] J. Wu Z. Yu and J. Gao Opt. Commun. 211 (2002) 257; W. Xu J. Wu and J. Gao Opt. Commun. 215 (2003) 345. [9] P. Zhou and S. Swain Phys. Rev. Lett. 77 (1996) 3995; S.Y. Zhu H. Chen and H. Huang Phys. Rev. Lett. 79 (1997) 205; S.Y. Zhu Phys. Rev. A 52 (1995) 710; P. Zhou and S. Swain Phys. Rev. A 56 (1997) 3011. [10] J. Javanainen Europhys. Lett. 17 (1992) 407. [11] S. Menon and G.S. Agarwal Phys. Rev. A 57 (1998) 4014. [12] J. Evers D. Bullock and C.H. Keitel Opt. Commun. 209 (2002) 173. [13] J.H. Wu and J.Y. Gao Phys. Rev. A 65 (2002) 063807; W.H. Xu J.H. Wu and J.Y. Gao Phys. Rev. A 66 (2002) 063812. [14] J. Wu W. Xu H. Zhang and J. Gao Opt. Commun. 206 (2002) 135. [15] S.Y. Zhu and M.O. Scully Phys. Rev. Lett. 76 (1996) 388; F.L. Li and S.Y. Zhu Phys. Rev. A 59 (1999) 2330. [16] E. Paspalakis and P.L. Knight Phys. Rev. Lett. 81 (1998) 293. [17] B.P. Hou S.J. Wang W.L. Yu and W.L. Sun Phys. Rev. A 69 (2004) 053805. [18] B.P. Hou S.J. Wang W.L. Yu and W.L. Sun Phys. Lett. 352 (2006) 462. [19] G.S. Agarwal and W. Harshawardhan Phys. Rev. Lett. 77 (1996) 1039. [20] J.Y. Gao S.H. Yang D. Wang X.Z. Guo K.X. Chen Y. Jiang and B. Zhao Phys. Rev. A 61 (2000) 023401. [21] D. Petrosyan and G. Kurizki Phys. Rev. A 64 (2001) 023810. [22] M. Paternostro and M.S. Kim Phys. Rev. A 67 (2003) 023811.