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Elimination of the transient absorption in a microwave-driven Λ-type atomic system
Physics Letters A 314 (2003) 23–28 www.elsevier.com/locate/pla
Elimination of the transient absorption in a microwave-driven -type atomic system Wei-Hua Xu a,b , Jin-Hui Wu a,b , Jin-Yue Gao a,b,c,∗ a College of Physics, Jilin University, Changchun 130023, PR China b Key Laboratory of Coherent Light, Atomic and Molecular Spectroscopy, Educational Ministry of China, PR China c CCAST (World Laboratory), P.O. Box 8370, Beijing 100080, PR China
Received 15 April 2003; accepted 30 May 2003 Communicated by P.R. Holland
Abstract We investigate the absorption properties of the transient process in a three-level -type atomic system driven by a coherent field and a microwave field. It is shown that, by modulating the relative phase of the applied fields, the absorption properties of the transient process can be dramatically changed. The transient absorption can be eliminated just by tuning the relative phase of the applied fields into proper regions. 2003 Elsevier B.V. All rights reserved. PACS: 42.50.Gy; 42.50.Hz
1. Introduction Over the past few years much attention has been devoted to the effects of quantum interference between multiple atomic transitions pathways. It has been demonstrated that interference due to spontaneous emission can arise when spontaneous emission from one level can strongly affect a neighboring transition. This decay-induced coherence happens in two cases. One occurs when a single excited state decays to a closely spaced lower doublet (the -configuration). And the other appears when a closely spaced excited doublet decays to a single ground state (the * Corresponding author.
E-mail address: [email protected] (J.-Y. Gao).
V-configuration). It has been shown that the quantum interference among spontaneous decay channels can change the steady-state response of the medium, and can modify significantly the absorption or spontaneous emission spectra of a near-degenerate system [1–14]. Moreover, it has been shown that atomic systems with decay-induced interference are usually sensitive to the relative phase of the applied fields [2,8,14]. The existence of the decay-induced quantum interference effect depends on the nonorthogonality of the two dipole matrix elements, which can be obtained from the mixing of the levels arising from internal fields [15] or external fields [16–18]. Decay-induced coherence also can modify the transient properties of the medium. Recently, we studied the effect of decay-induced coherence on the transient process in a three-level -type system with near-
0375-9601/03/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0375-9601(03)00869-7
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W.-H. Xu et al. / Physics Letters A 314 (2003) 23–28
degenerate levels in the case of a weak probe [19]. We found that due to the decay-induced coherence, the transient property of the -type system is quite different from that of a conventional -system. Especially, the transient gain can be greatly enhanced, and the transient absorption can be completely eliminated just by choosing the proper relative phase between the coupling and probe fields, which cannot be realized in a conventional -system [20]. Noting that a microwave field also can generate quantum coherence on a dipole-forbidden transition with two closely lying levels [21–24], in the present paper, we consider another scheme for three-level -type atoms to achieve the phase control of the elimination of the transient absorption, where a microwave field is used to drive the two ground levels. We show that in such a microwavedriven atomic system, the absorption properties in the transient process also can be controlled by the relative phase of the applied fields. By modulating the relative phase of the applied fields, the absorption properties of the transient process can be dramatically changed. The transient absorption can be completely eliminated just by choosing the proper values of the relative phase of the applied fields, just as in the scheme with decayinduced coherence [19]. In such a microwave-driven scheme, the two involved dipole moments can be either parallel or orthogonal, and the frequency difference between the two ground levels can be much larger than spontaneous decay rates.
2. The system and the numerical analysis The closed, -type, three-level atomic system with two closely lying lower levels is shown in Fig. 1. A coupling field of frequency ωc with a Rabi frequency Ωc drives transition |2 ↔ |3. A resonant microwave field of frequency ωd with a Rabi frequency Ωd couples level |1 into level |2 through an allowed magnetic transition. A weak probe field of frequency ωp with Rabi frequency Ωp is applied to the transition |1 ↔ |3. 2γ31 and 2γ32 denote the spontaneous decay rates from level |3 to levels |1 and |2, respectively. ∆c = ω32 − ωc , ∆p = ω31 − ωp are atomfield detunings of the coupling and probe fields. In this scheme, the quantum coherence is created by coupling the two lower states of the -system by means of a microwave field instead of the sharing of the vac-
Fig. 1. A three-level -type atomic system driven by a microwave field, a strong coherent field, and a weak probe.
uum modes by the two transitions. We assume that the ground levels |1 and |2 are closely spaced, but the corresponding frequency difference ω21 is large compared to the decay rates 2γ31 and 2γ32 . This approximation allows us neglect the decay-induced coherence √ terms (proportional to γ31 γ32 ) in the following density matrix equations. The proposed scheme is therefore independent of the alignment of the dipole moments, it requires three driving fields but is more convenient in its experimental realization. With the electric-dipole approximation and the rotating-wave approximation, the density-matrix equations of motion in the interaction picture can be written as: σ˙ 22 = 2γ32σ33 − iΩc σ23 + iΩc∗ σ32 + iΩd σ12 − iΩd∗ σ21 , σ˙ 11 = 2γ31σ33 − iΩp σ13 + iΩp∗ σ31 + iΩd∗ σ21 − iΩd σ12 , σ˙ 23 = (−γ31 − γ32 + i∆c )σ23 + iΩc∗ (σ33 − σ22 ) − iΩp∗ σ21 + iΩd σ13 , σ˙ 13 = (−γ31 − γ32 + i∆p )σ13 − iΩc∗ σ12 + iΩp∗ (σ33 − σ11 ) + iΩd∗ σ23 , σ˙ 12 = i(∆p − ∆c )σ12 − iΩc σ13 + iΩp∗ σ32 + iΩd∗ (σ22 − σ11 ).
(1)
The above equations are constrained by σij = σj∗i and σ11 + σ22 + σ33 = 1.
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We treat the Rabi frequencies of the three fields as complex parameters, and take the canonical transformation of Eqs. (1) as follows: we use φp , φc , and φd denote phases of the probe, the coupling, and the microwave fields, respectively; we rewrite Rabi frequencies as Ωp = Gp exp(−iφp ), Ωc = Gc exp(−iφc ), and Ωd = Gd exp(−iφd ), where Gp , Gc , and Gd are chosen to be real; then we redefine atomic variables as ρii = σii , ρ13 = σ13 exp(−iφp ), ρ23 = σ23 exp(−iφc ), and ρ12 = σ12 exp[−i(φp − φc )]; finally, we obtain the equations of motion for the redefined density matrix elements ρij from Eqs. (1) as follows: ρ˙22 = 2γ32ρ33 + iGc (ρ32 − ρ23 ) + iGd eiΦ ρ12 − e−iΦ ρ21 ,
Fig. 2. Time evolution of the gain–absorption coefficient Im(ρ13 ) in the absence of the microwave field. Solid curve: there is no incoherent pump applied on transition |1 ↔ |3. Dashed curve: there is an incoherent pump with the rate Λ = 3γ31 applied on transition |1 ↔ |3. The parameters are γ32 = 2γ31 , ∆c = ∆p = 0, Gc = 20γ31 , Gp = 0.1γ31 .
ρ˙11 = 2γ31ρ33 + iGp (ρ31 − ρ13 ) + iGd e−iΦ ρ21 − eiΦ ρ12 , ρ˙23 = (−γ31 − γ32 + i∆c )ρ23 + iGc (ρ33 − ρ22 ) − iGp ρ21 + iGd ρ13 , ρ˙13 = (−γ31 − γ32 + i∆p )ρ13 − iGc ρ12 + iGp (ρ33 − ρ11 ) + iGd e−iΦ ρ23 , ρ˙12 = i(∆p − ∆c )ρ12 − iGc ρ13 + iGp ρ32 + iGd (ρ22 − ρ11 ),
(2)
where Φ = φd + φc − φp is the relative phase of the three fields. From Eqs. (2), it is obvious that, due to the existence of the microwave field, this atomic system becomes sensitive to the relative phase Φ, for we cannot eliminate it after the canonical transformation. That is to say, we can change the absorption properties just by tuning one of the three phases, for example φd , and keeping the other two constant. The leading role of the microwave field is to induce the quantum coherence between levels |1 and |2, which is necessary for phase-dependent effects in this atomic system. As is known, in the limit of a weak probe, the gainabsorption coefficient for the probe laser on transition |3 ↔ |1 is proportional to the imaginary part of ρ13 , which can be obtained from Eqs. (2). In our notation, if Im(ρ13 ) > 0, the probe laser will be amplified. Considering the case where both coupling laser and the probe laser are at resonance (∆c = ∆p = 0), with the initial condition ρ11 (0) = 1 and the other ρij = 0 (i, j = 0–3), we derive the time-dependent numerical solutions of Eqs. (2) in the following. Note
that in this Letter, parameters Gc , Gp , Gd , ∆p , ∆c , and γ32 are scaled by γ31 . First, we consider the case that without the microwave field. In Fig. 2, with γ32 = 2γ31 , Gc = 20γ31, Gp = 0.1γ31 , we plot the time evolution of the gain– absorption coefficient Im(ρ13 ) for two cases: one is that there is no incoherent pump applied on transition |1 ↔ |3 (solid curve); another is that there is an incoherent pump applied on transition |1 ↔ |3 (dashed curve, in the presence of the incoherent pump, steadystate gain can be obtained). It shows that, in the absence of the microwave field, the probe field shows oscillatory behavior versus time, it exhibits periodic amplification and absorption before reaching the steadystate, regardless of whether there exists the incoherent pump or not. Such phase-independent property has been discussed in Ref. [20]. Then we consider the case that in the presence of the microwave field. We suppose the microwave field is weak, i.e., Gd ≺ γ31 , γ32 , here we take Gd = 0.5γ31. In this case, the quantum coherence between level |1 and level |2 generated by the microwave field is of the order of the decay-induced coherence, which will have to be considered if level |1 and level |2 lie closely enough and the two dipole moments are nonorthogonal. In Fig. 3(a), we plot the time evolution of the gain–absorption coefficient Im(ρ13 ) in the presence of the microwave field, with a set of different
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Fig. 3. Time evolution of the gain–absorption coefficient Im(ρ13 ) in the presence of a weak microwave field Ωd = 0.5γ31 , for different values of Φ, (a) Φ = 0, π (dotted curve), Φ = π/2 (solid curve), Φ = 3π/2 (dashed curve). (b) Φ = π/6, π + π/6 (dotted curves), Φ = π/4, π + π/4 (dashed curves) and Φ = π/3, π/3 + π (solid curves). Other parameters are the same as in Fig. 2.
values of the relative phase Φ = 0, π (dotted curve), Φ = π/2 (solid curve), Φ = 3π/2 (dashed curve). It is shown that the transient property is greatly affected by the relative phase Φ, it exhibits different features with different values of the relative phase Φ. With Φ = 0, the transient behavior of the probe field is similar to the case that without the microwave field, it exhibits periodic amplification and absorption versus time. When Φ = π , the curve is completely superposed with that of Φ = 0, which indicates that the transient behavior of the probe field is the same as that of Φ = 0. While with Φ = π/2, the transient behavior of the probe field is dramatically altered, the transient
absorption disappears, only leaves the transient gain oscillating above the zero-absorption line, around the steady-state value, and eventually reaches the positive steady-state value. On the contrary, with Φ = 3π/2, the transient gain disappears, only leaves the transient absorption oscillating bellow the zero-absorption line, around the steady-state value, and at last reaches the negative steady-state value. It can be seen that the transient behavior of the probe field with Φ = π/2 and that with Φ = 3π/2 are exactly symmetric about Im(ρ13 ) = 0. Under such two cases the probe laser no longer exhibits periodic amplification and absorption as in the case of Φ = 0 (π). Besides, the amplitude of the oscillatory is much larger than that of Φ = 0 (π). From Fig. 3(a) we can see that due to the microwave-induced coherence, transient absorption can be eliminated just by choosing proper values of the relative phase Φ, e.g., Φ = π/2. We have investigated the dependence of the elimination of the transient absorption on the relative phase Φ, and found that there is a wide range for the relative phase Φ that can eliminate the transient absorption. The transient absorption can be eliminated as long as we tune the relative phase Φ into regions of 2kπ ≺ Φ ≺ (2k + 1)π (where k = 0, 1, 2, . . .); and on the contrary, the transient gain can be eliminated as long as we tune the relative phase Φ into regions of 2kπ + π ≺ Φ ≺ 2kπ + 2π . As an example, in Fig. 3(b), we plot the time evolution of the gain–absorption coefficient Im(ρ13 ) with another set of different values of the relative phase Φ = π/6, π + π/6 (dotted curves), Φ = π/4, π + π/4 (dashed curves) and Φ = π/3, π +π/3 (solid curves). It can be found that, when Φ = π/6, π/4, π/3 (i.e., 2kπ ≺ Φ ≺ (2k + 1)π), the transient absorption is eliminated, only leaves transient gain; when Φ = π + π/6, π + π/4, π + π/3 (i.e., 2kπ + π ≺ Φ ≺ 2kπ + 2π ), the transient gain is eliminated, only leaves transient absorption; and the transient behavior of the probe field with relative phase Φ and that with relative phase Φ + π are exactly symmetric about Im(ρ13 ) = 0. It also should be pointed out that the transient behavior of the probe field with relative phase Φ and that with relative phase π − Φ are exactly the same. From Fig. 3(b), it also can be found that we can change the magnitude of the transient gain (absorption) just by changing the values of the relative phase Φ, while the oscillatory frequency of the transient gain (absorption) remains the same.
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while the vacuum induced coherence is proportional √ to γ31 γ32 . Above discussions are for the case of a weak microwave field. Now, we consider a special case of a strong microwave field, i.e., the quantum coherence between levels |1 and |2 generated by the microwave field is much more stronger than decay-induced coherence. In Fig. 4, with Gd = 5γ31 , we plot time evolution of the gain–absorption coefficient Im(ρ13 ) with different values of the relative phase Φ. Analogous to Fig. 3(a), in Fig. 4(a), Φ = 0 (π) (dotted curve), Φ = π/2 (solid curve), Φ = 3π/2 (dashed curve); analogous to Fig. 3(b), in Fig. 4(b), Φ = π/6, π + π/6 (dotted curves), Φ = π/4, π + π/4 (dashed curves) and Φ = π/3, π/3 + π (solid curves). Fig. 4 shows the similar results as in Fig. 3, except that the magnitude of the transient gain (absorption) becomes much more larger than that in Fig. 3. That is to say, the quantum interference between levels |1 and |2 generated by the microwave field, either weak or strong, leads to similar phase-dependent behavior in the transient process for the probe field.
3. Conclusions
Fig. 4. Time evolution of the gain–absorption coefficient Im(ρ13 ) in the presence of a strong microwave field Ωd = 5γ31 , for different values of Φ, (a) Φ = 0, π (dotted curve), Φ = π/2 (solid curve), Φ = 3π/2 (dashed curve). (b) Φ = π/6, π + π/6 (dotted curves), Φ = π/4, π + π/4 (dashed curves) and Φ = π/3, π/3 + π (solid curves). Other parameters are the same as in Fig. 2.
The transient behavior of the probe in such a microwave-driven system is similar to that obtained in Ref. [19] except a phase difference of π/2, for example, the transient behavior with Φ = π/2 here is similar to that with Φ = π in Ref. [19]. In order to eliminate the transient absorption, we have to choose the relative phase Φ into the region of 2kπ ≺ Φ ≺ (2k + 1)π in a microwave-driven -system; while in a -system with decay-induced coherence it should be chosen as in the region of 2kπ + π/2 ≺ Φ ≺ 2kπ + 3π/2. The phase difference of π/2 is originated from the fact that the microwave-induced coherence between |1 and |2 is proportional to iGd
In summary, we have investigated the absorption properties of the transient process in a three-level type atomic system, where the two closely lying lower levels are coupled by a microwave field. We find that due to the quantum coherence between the two lower levels induced by the applied microwave field, the transient behavior of the probe field can be related to the relative phase of the applied fields. By modulating the relative phase of the applied fields, the absorption properties of the transient process can be dramatically changed, the transient absorption can be eliminated just by tuning the relative phase into the proper regions. The transient behavior of the probe in such a microwave-driven system is similar to that in a system with spontaneously generated coherence [19], where the interference between the two lower levels arises from the vacuum of the electromagnetic field, which depends on the nonorthogonality of the two dipole matrix elements. The microwave-driven scheme is independent of the alignment of the dipole moments, and the frequency difference between the two ground levels can be much larger than spontaneous de-
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cay rates, such a scheme is more convenient in its experimental realization.
Acknowledgements The authors would like to thank the support from the National Natural Science Foundation of China and the support from the Doctoral Program Foundation of Institution of Higher Education of China.
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Elimination of the transient absorption in a microwave-driven -type atomic system Wei-Hua Xu a,b , Jin-Hui Wu a,b , Jin-Yue Gao a,b,c,∗ a College of Physics, Jilin University, Changchun 130023, PR China b Key Laboratory of Coherent Light, Atomic and Molecular Spectroscopy, Educational Ministry of China, PR China c CCAST (World Laboratory), P.O. Box 8370, Beijing 100080, PR China
Received 15 April 2003; accepted 30 May 2003 Communicated by P.R. Holland
Abstract We investigate the absorption properties of the transient process in a three-level -type atomic system driven by a coherent field and a microwave field. It is shown that, by modulating the relative phase of the applied fields, the absorption properties of the transient process can be dramatically changed. The transient absorption can be eliminated just by tuning the relative phase of the applied fields into proper regions. 2003 Elsevier B.V. All rights reserved. PACS: 42.50.Gy; 42.50.Hz
1. Introduction Over the past few years much attention has been devoted to the effects of quantum interference between multiple atomic transitions pathways. It has been demonstrated that interference due to spontaneous emission can arise when spontaneous emission from one level can strongly affect a neighboring transition. This decay-induced coherence happens in two cases. One occurs when a single excited state decays to a closely spaced lower doublet (the -configuration). And the other appears when a closely spaced excited doublet decays to a single ground state (the * Corresponding author.
E-mail address: [email protected] (J.-Y. Gao).
V-configuration). It has been shown that the quantum interference among spontaneous decay channels can change the steady-state response of the medium, and can modify significantly the absorption or spontaneous emission spectra of a near-degenerate system [1–14]. Moreover, it has been shown that atomic systems with decay-induced interference are usually sensitive to the relative phase of the applied fields [2,8,14]. The existence of the decay-induced quantum interference effect depends on the nonorthogonality of the two dipole matrix elements, which can be obtained from the mixing of the levels arising from internal fields [15] or external fields [16–18]. Decay-induced coherence also can modify the transient properties of the medium. Recently, we studied the effect of decay-induced coherence on the transient process in a three-level -type system with near-
0375-9601/03/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0375-9601(03)00869-7
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W.-H. Xu et al. / Physics Letters A 314 (2003) 23–28
degenerate levels in the case of a weak probe [19]. We found that due to the decay-induced coherence, the transient property of the -type system is quite different from that of a conventional -system. Especially, the transient gain can be greatly enhanced, and the transient absorption can be completely eliminated just by choosing the proper relative phase between the coupling and probe fields, which cannot be realized in a conventional -system [20]. Noting that a microwave field also can generate quantum coherence on a dipole-forbidden transition with two closely lying levels [21–24], in the present paper, we consider another scheme for three-level -type atoms to achieve the phase control of the elimination of the transient absorption, where a microwave field is used to drive the two ground levels. We show that in such a microwavedriven atomic system, the absorption properties in the transient process also can be controlled by the relative phase of the applied fields. By modulating the relative phase of the applied fields, the absorption properties of the transient process can be dramatically changed. The transient absorption can be completely eliminated just by choosing the proper values of the relative phase of the applied fields, just as in the scheme with decayinduced coherence [19]. In such a microwave-driven scheme, the two involved dipole moments can be either parallel or orthogonal, and the frequency difference between the two ground levels can be much larger than spontaneous decay rates.
2. The system and the numerical analysis The closed, -type, three-level atomic system with two closely lying lower levels is shown in Fig. 1. A coupling field of frequency ωc with a Rabi frequency Ωc drives transition |2 ↔ |3. A resonant microwave field of frequency ωd with a Rabi frequency Ωd couples level |1 into level |2 through an allowed magnetic transition. A weak probe field of frequency ωp with Rabi frequency Ωp is applied to the transition |1 ↔ |3. 2γ31 and 2γ32 denote the spontaneous decay rates from level |3 to levels |1 and |2, respectively. ∆c = ω32 − ωc , ∆p = ω31 − ωp are atomfield detunings of the coupling and probe fields. In this scheme, the quantum coherence is created by coupling the two lower states of the -system by means of a microwave field instead of the sharing of the vac-
Fig. 1. A three-level -type atomic system driven by a microwave field, a strong coherent field, and a weak probe.
uum modes by the two transitions. We assume that the ground levels |1 and |2 are closely spaced, but the corresponding frequency difference ω21 is large compared to the decay rates 2γ31 and 2γ32 . This approximation allows us neglect the decay-induced coherence √ terms (proportional to γ31 γ32 ) in the following density matrix equations. The proposed scheme is therefore independent of the alignment of the dipole moments, it requires three driving fields but is more convenient in its experimental realization. With the electric-dipole approximation and the rotating-wave approximation, the density-matrix equations of motion in the interaction picture can be written as: σ˙ 22 = 2γ32σ33 − iΩc σ23 + iΩc∗ σ32 + iΩd σ12 − iΩd∗ σ21 , σ˙ 11 = 2γ31σ33 − iΩp σ13 + iΩp∗ σ31 + iΩd∗ σ21 − iΩd σ12 , σ˙ 23 = (−γ31 − γ32 + i∆c )σ23 + iΩc∗ (σ33 − σ22 ) − iΩp∗ σ21 + iΩd σ13 , σ˙ 13 = (−γ31 − γ32 + i∆p )σ13 − iΩc∗ σ12 + iΩp∗ (σ33 − σ11 ) + iΩd∗ σ23 , σ˙ 12 = i(∆p − ∆c )σ12 − iΩc σ13 + iΩp∗ σ32 + iΩd∗ (σ22 − σ11 ).
(1)
The above equations are constrained by σij = σj∗i and σ11 + σ22 + σ33 = 1.
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We treat the Rabi frequencies of the three fields as complex parameters, and take the canonical transformation of Eqs. (1) as follows: we use φp , φc , and φd denote phases of the probe, the coupling, and the microwave fields, respectively; we rewrite Rabi frequencies as Ωp = Gp exp(−iφp ), Ωc = Gc exp(−iφc ), and Ωd = Gd exp(−iφd ), where Gp , Gc , and Gd are chosen to be real; then we redefine atomic variables as ρii = σii , ρ13 = σ13 exp(−iφp ), ρ23 = σ23 exp(−iφc ), and ρ12 = σ12 exp[−i(φp − φc )]; finally, we obtain the equations of motion for the redefined density matrix elements ρij from Eqs. (1) as follows: ρ˙22 = 2γ32ρ33 + iGc (ρ32 − ρ23 ) + iGd eiΦ ρ12 − e−iΦ ρ21 ,
Fig. 2. Time evolution of the gain–absorption coefficient Im(ρ13 ) in the absence of the microwave field. Solid curve: there is no incoherent pump applied on transition |1 ↔ |3. Dashed curve: there is an incoherent pump with the rate Λ = 3γ31 applied on transition |1 ↔ |3. The parameters are γ32 = 2γ31 , ∆c = ∆p = 0, Gc = 20γ31 , Gp = 0.1γ31 .
ρ˙11 = 2γ31ρ33 + iGp (ρ31 − ρ13 ) + iGd e−iΦ ρ21 − eiΦ ρ12 , ρ˙23 = (−γ31 − γ32 + i∆c )ρ23 + iGc (ρ33 − ρ22 ) − iGp ρ21 + iGd ρ13 , ρ˙13 = (−γ31 − γ32 + i∆p )ρ13 − iGc ρ12 + iGp (ρ33 − ρ11 ) + iGd e−iΦ ρ23 , ρ˙12 = i(∆p − ∆c )ρ12 − iGc ρ13 + iGp ρ32 + iGd (ρ22 − ρ11 ),
(2)
where Φ = φd + φc − φp is the relative phase of the three fields. From Eqs. (2), it is obvious that, due to the existence of the microwave field, this atomic system becomes sensitive to the relative phase Φ, for we cannot eliminate it after the canonical transformation. That is to say, we can change the absorption properties just by tuning one of the three phases, for example φd , and keeping the other two constant. The leading role of the microwave field is to induce the quantum coherence between levels |1 and |2, which is necessary for phase-dependent effects in this atomic system. As is known, in the limit of a weak probe, the gainabsorption coefficient for the probe laser on transition |3 ↔ |1 is proportional to the imaginary part of ρ13 , which can be obtained from Eqs. (2). In our notation, if Im(ρ13 ) > 0, the probe laser will be amplified. Considering the case where both coupling laser and the probe laser are at resonance (∆c = ∆p = 0), with the initial condition ρ11 (0) = 1 and the other ρij = 0 (i, j = 0–3), we derive the time-dependent numerical solutions of Eqs. (2) in the following. Note
that in this Letter, parameters Gc , Gp , Gd , ∆p , ∆c , and γ32 are scaled by γ31 . First, we consider the case that without the microwave field. In Fig. 2, with γ32 = 2γ31 , Gc = 20γ31, Gp = 0.1γ31 , we plot the time evolution of the gain– absorption coefficient Im(ρ13 ) for two cases: one is that there is no incoherent pump applied on transition |1 ↔ |3 (solid curve); another is that there is an incoherent pump applied on transition |1 ↔ |3 (dashed curve, in the presence of the incoherent pump, steadystate gain can be obtained). It shows that, in the absence of the microwave field, the probe field shows oscillatory behavior versus time, it exhibits periodic amplification and absorption before reaching the steadystate, regardless of whether there exists the incoherent pump or not. Such phase-independent property has been discussed in Ref. [20]. Then we consider the case that in the presence of the microwave field. We suppose the microwave field is weak, i.e., Gd ≺ γ31 , γ32 , here we take Gd = 0.5γ31. In this case, the quantum coherence between level |1 and level |2 generated by the microwave field is of the order of the decay-induced coherence, which will have to be considered if level |1 and level |2 lie closely enough and the two dipole moments are nonorthogonal. In Fig. 3(a), we plot the time evolution of the gain–absorption coefficient Im(ρ13 ) in the presence of the microwave field, with a set of different
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Fig. 3. Time evolution of the gain–absorption coefficient Im(ρ13 ) in the presence of a weak microwave field Ωd = 0.5γ31 , for different values of Φ, (a) Φ = 0, π (dotted curve), Φ = π/2 (solid curve), Φ = 3π/2 (dashed curve). (b) Φ = π/6, π + π/6 (dotted curves), Φ = π/4, π + π/4 (dashed curves) and Φ = π/3, π/3 + π (solid curves). Other parameters are the same as in Fig. 2.
values of the relative phase Φ = 0, π (dotted curve), Φ = π/2 (solid curve), Φ = 3π/2 (dashed curve). It is shown that the transient property is greatly affected by the relative phase Φ, it exhibits different features with different values of the relative phase Φ. With Φ = 0, the transient behavior of the probe field is similar to the case that without the microwave field, it exhibits periodic amplification and absorption versus time. When Φ = π , the curve is completely superposed with that of Φ = 0, which indicates that the transient behavior of the probe field is the same as that of Φ = 0. While with Φ = π/2, the transient behavior of the probe field is dramatically altered, the transient
absorption disappears, only leaves the transient gain oscillating above the zero-absorption line, around the steady-state value, and eventually reaches the positive steady-state value. On the contrary, with Φ = 3π/2, the transient gain disappears, only leaves the transient absorption oscillating bellow the zero-absorption line, around the steady-state value, and at last reaches the negative steady-state value. It can be seen that the transient behavior of the probe field with Φ = π/2 and that with Φ = 3π/2 are exactly symmetric about Im(ρ13 ) = 0. Under such two cases the probe laser no longer exhibits periodic amplification and absorption as in the case of Φ = 0 (π). Besides, the amplitude of the oscillatory is much larger than that of Φ = 0 (π). From Fig. 3(a) we can see that due to the microwave-induced coherence, transient absorption can be eliminated just by choosing proper values of the relative phase Φ, e.g., Φ = π/2. We have investigated the dependence of the elimination of the transient absorption on the relative phase Φ, and found that there is a wide range for the relative phase Φ that can eliminate the transient absorption. The transient absorption can be eliminated as long as we tune the relative phase Φ into regions of 2kπ ≺ Φ ≺ (2k + 1)π (where k = 0, 1, 2, . . .); and on the contrary, the transient gain can be eliminated as long as we tune the relative phase Φ into regions of 2kπ + π ≺ Φ ≺ 2kπ + 2π . As an example, in Fig. 3(b), we plot the time evolution of the gain–absorption coefficient Im(ρ13 ) with another set of different values of the relative phase Φ = π/6, π + π/6 (dotted curves), Φ = π/4, π + π/4 (dashed curves) and Φ = π/3, π +π/3 (solid curves). It can be found that, when Φ = π/6, π/4, π/3 (i.e., 2kπ ≺ Φ ≺ (2k + 1)π), the transient absorption is eliminated, only leaves transient gain; when Φ = π + π/6, π + π/4, π + π/3 (i.e., 2kπ + π ≺ Φ ≺ 2kπ + 2π ), the transient gain is eliminated, only leaves transient absorption; and the transient behavior of the probe field with relative phase Φ and that with relative phase Φ + π are exactly symmetric about Im(ρ13 ) = 0. It also should be pointed out that the transient behavior of the probe field with relative phase Φ and that with relative phase π − Φ are exactly the same. From Fig. 3(b), it also can be found that we can change the magnitude of the transient gain (absorption) just by changing the values of the relative phase Φ, while the oscillatory frequency of the transient gain (absorption) remains the same.
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while the vacuum induced coherence is proportional √ to γ31 γ32 . Above discussions are for the case of a weak microwave field. Now, we consider a special case of a strong microwave field, i.e., the quantum coherence between levels |1 and |2 generated by the microwave field is much more stronger than decay-induced coherence. In Fig. 4, with Gd = 5γ31 , we plot time evolution of the gain–absorption coefficient Im(ρ13 ) with different values of the relative phase Φ. Analogous to Fig. 3(a), in Fig. 4(a), Φ = 0 (π) (dotted curve), Φ = π/2 (solid curve), Φ = 3π/2 (dashed curve); analogous to Fig. 3(b), in Fig. 4(b), Φ = π/6, π + π/6 (dotted curves), Φ = π/4, π + π/4 (dashed curves) and Φ = π/3, π/3 + π (solid curves). Fig. 4 shows the similar results as in Fig. 3, except that the magnitude of the transient gain (absorption) becomes much more larger than that in Fig. 3. That is to say, the quantum interference between levels |1 and |2 generated by the microwave field, either weak or strong, leads to similar phase-dependent behavior in the transient process for the probe field.
3. Conclusions
Fig. 4. Time evolution of the gain–absorption coefficient Im(ρ13 ) in the presence of a strong microwave field Ωd = 5γ31 , for different values of Φ, (a) Φ = 0, π (dotted curve), Φ = π/2 (solid curve), Φ = 3π/2 (dashed curve). (b) Φ = π/6, π + π/6 (dotted curves), Φ = π/4, π + π/4 (dashed curves) and Φ = π/3, π/3 + π (solid curves). Other parameters are the same as in Fig. 2.
The transient behavior of the probe in such a microwave-driven system is similar to that obtained in Ref. [19] except a phase difference of π/2, for example, the transient behavior with Φ = π/2 here is similar to that with Φ = π in Ref. [19]. In order to eliminate the transient absorption, we have to choose the relative phase Φ into the region of 2kπ ≺ Φ ≺ (2k + 1)π in a microwave-driven -system; while in a -system with decay-induced coherence it should be chosen as in the region of 2kπ + π/2 ≺ Φ ≺ 2kπ + 3π/2. The phase difference of π/2 is originated from the fact that the microwave-induced coherence between |1 and |2 is proportional to iGd
In summary, we have investigated the absorption properties of the transient process in a three-level type atomic system, where the two closely lying lower levels are coupled by a microwave field. We find that due to the quantum coherence between the two lower levels induced by the applied microwave field, the transient behavior of the probe field can be related to the relative phase of the applied fields. By modulating the relative phase of the applied fields, the absorption properties of the transient process can be dramatically changed, the transient absorption can be eliminated just by tuning the relative phase into the proper regions. The transient behavior of the probe in such a microwave-driven system is similar to that in a system with spontaneously generated coherence [19], where the interference between the two lower levels arises from the vacuum of the electromagnetic field, which depends on the nonorthogonality of the two dipole matrix elements. The microwave-driven scheme is independent of the alignment of the dipole moments, and the frequency difference between the two ground levels can be much larger than spontaneous de-
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cay rates, such a scheme is more convenient in its experimental realization.
Acknowledgements The authors would like to thank the support from the National Natural Science Foundation of China and the support from the Doctoral Program Foundation of Institution of Higher Education of China.
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