Role of strongly modulated coherence in transient evolution dynamics of probe absorption in a three-level atomic system

Optics Communications 309 (2013) 95–102

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Role of strongly modulated coherence in transient evolution dynamics of probe absorption in a three-level atomic system Pradipta Panchadhyayee n Department of Physics, P. K. College, Contai, Purba Medinipur 721 401, W. B., India

art ic l e i nf o

a b s t r a c t

Article history: Received 30 May 2013 Received in revised form 2 July 2013 Accepted 4 July 2013 Available online 18 July 2013

We investigate the dynamical behaviour of atomic response in a closed three-level V-type atomic system with the variation of different relevant parameters to exhibit transient evolution of absorption, gain and transparency in the probe response. The oscillations in probe absorption and gain can be efficiently modulated by changing the values of the Rabi frequency, detuning and the collective phase involved in the system. The interesting outcome of the work is the generation of coherence controlled loop-structure with varying amplitudes in the oscillatory probe response of the probe field at various parameter conditions. The prominence of these structures is observed when the coherence induced in a one-photon excitation path is strongly modified by two-step excitations driven by the coherent fields operating in closed interaction contour. In contrast to purely resonant case, the time interval between two successive loops gets significantly reduced with the application of non-zero detuning in the coherent fields. & 2013 Elsevier B.V. All rights reserved.

Keywords: Transient response Absorption Inversionless gain Collective phase Coherence induced loop structures

1. Introduction In the last two decades, extensive studies in the area of nonlinear optics and laser spectroscopy have led to considerable interest in the study of optical response of the atomic system interacting with a number of coherent fields. In ideal three-level systems (Λ, V and Ξ) [1], the atomic coherence can be dynamically induced by the application of a strong field driving one excitation path. In the presence of an additional field probing another excitation path, the induced coherence leads to various quantum optical effects like electromagnetically induced transparency (EIT) [2], electromagnetically induced absorption (EIA) [3], gain without inversion (GWI) or lasing without inversion (LWI) [4–9], enhancement of refractive index [10] and generation of quantum beats [11]. The traditional method of generating GWI in an ideal closed threelevel scheme producing EIT incorporates the application of an incoherent pump field to the system to populate the upper lasing level of the transition probed by a weak coherent field. The phenomenon of inversionless gain can result in the absorptive response of the probe field due to strong two-photon coherence induced in the system [6]. In a suitable level scheme, without invoking incoherent pumping, the condition of initial population in the upper lasing level is fulfilled by choosing an appropriate relaxation pathway from the excited level [12,13]. In presence of incoherent pumping, interference between two spontaneous decay channels in a V-type system with

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two closely lying upper levels leads to the formation of gain as a result of vacuum induced coherence (VIC) [14,15]. In such scheme, it is possible to obtain the control of absorption and gain by the relative phase of the fields involved in the system [16–22]. In absence of VIC, similar phase control of absorption and gain is also achieved in a Vtype [23] and Λ-type [24–26] schemes where a microwave field is employed in the low-frequency induced transitions to form the closeloop interaction model. Such model shows its robustness to produce gain when the system is much dissipative [27]. All the works [4–27] relating to GWI as we have addressed so far, are concerned with the steady state properties of probe response. A lot of investigations [28–38] has been made to study the transient evolution of gain at various situations. In closed [29,31,32,34,37,38] and open [33,35,36] folded type schemes, gain-characteristics have been studied with [31,32,35–38] and without [29,30,33,34] inclusion of VIC. By adopting the close-loop interaction in a Λ-type system [32], phase dependent elimination of absorption has been noted in transient regime. Similar scheme using an external magnetic field in the low-frequency induced transition has been analyzed [34] to obtain GWI in transient regime. Motivated by these works [28–38] performed in transient regime, a closed three-level V-type scheme (Fig. 1) with close-loop interaction has been considered in this article to represent the three-ways of control of transient response of the probe field in detail. The close-loop interaction inherent to the present model is exploited in this work to exhibit the transient evolution of absorption, gain and transparency in the probe response with the variation of model parameters. The salient features of the work are presented as follows: (a) Explicit occurrence of absorption and gain in the

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where ∂ρ=∂t stands for the inclusion of irreversible decay effects with the expression

3 ωm

γ j1 ∂ρ ¼ ∑ ðfjj〉〈jj; ρg2j1〉〈jjρjj〉〈1jÞ ∂t j ¼ 2;3 2

2 γ

31

ωp

ωc

γ

In order to simplify the component equations of motion in Eq. (2), we introduce the unitary transformations for the atomic responses ρ12 ¼ ρ12 eiϕc , ρ13 ¼ ρ13 eiϕp and ρ23 ¼ ρ23 eiϕm and finally obtain the following equations for the density matrix elements:

21

1 Fig. 1. Schematic presentation of a three-level V-type atom interacting with three coherent fields designated by their frequencies: ωp for the probe field and ωj ðj ¼ c; mÞ for two control fields. γ 21 and γ 31 denote the spontaneous decay rates of the excited levels.

oscillatory probe response is obtained in many ways in the present model. (b) We have shown the generation of multi-loop-structure in the oscillatory probe response of the probe field at various conditions. This structure appears in probe transition due to strong perturbation resulted from two successive transition pathways driven by two coherent fields. Control of such structure by varying Rabi frequency, detuning and the collective phase of the control fields is presented in detail. (c) Unlike the purely resonant case, the time interval between two successive loops is found to be reduced by setting non-resonant detuning in the lasers involved in the system.

2. Basic model and related parameters We consider a closed three-level V-type system with the ground state j1〉 and the excited states j2〉 and j3〉 as shown in Fig. 1. The transition j1〉2j2〉 of frequency ω21 is driven by a coherent coupling field of frequency ωc and amplitude Ec in optical regime, and the other transition j2〉2j3〉 of frequency ω32 is driven by another coherent coupling field (microwave) of frequency ωm and amplitude Em. A weak optical field of frequency ωp and amplitude Ep probing the transition j1〉2j3〉 of frequency ω31 is so chosen that the population transfer to the uppermost level by this field becomes negligible. All the fields are considered in the continuous wave (CW) regime. In the semiclassical formulation, fields are defined as Ei ðx; tÞ ¼ ϵi cos ðωi tki zÞ ði ¼ p; c; mÞ where ki is the propagation vector along the z-direction. The spontaneous decay rates from level j3〉 ðj2〉Þ to level j1〉 is taken to be γ 31 (γ 21 ). For the transitions j1〉2j2〉, j1〉2j3〉 and j2〉2j3〉, the coherence dephasing rates are designated by Γ 21 ð ¼ γ 21 =2Þ, Γ 31 ð ¼ γ 31 =2Þ and Γ 32 ð ¼ ðγ 21 þ γ 31 Þ=2Þ, respectively. In the context of decay-induced coherence we neglect the role of vacuum modes in the transitions j1〉2j2〉 and j1〉2j3〉 with the approximation that the Raman coherence originated by the application of the microwave field Em between two upper states of the V-type system predominates over the decay-induced coherence in the limit of spontaneous decay rates much smaller than ω32 . The Hamiltonian of the atomic system assuming the electricdipole and rotating wave approximations can be expressed in the interaction picture as H ¼ ℏ½Rc eiΔc t j1〉〈2j þ Rm eiΔm t j2〉〈3j þ Rp eiΔp t j1〉〈3j þ H:c:

ð1Þ

where Δc ¼ ω21 ωc , Δm ¼ ω32 ωm and Δp ¼ ω31 ωp are the detunings and Rc ¼μ 12 :ϵ nc =2ℏ, Rm ¼μ 23 :ϵ nm =2ℏ and Rp ¼μ 13 :ϵ np =2ℏ are the complex Rabi frequencies which can be presented as Rj ¼ rjeiϕj (j ¼ p; c; m), rj being real parameters. The relation among the frequencies of coupling fields obeying the condition, ωp ¼ ωc þ ωm obviously implies that Δp ¼ Δc þ Δm . The time evolution dynamics of the system can be represented by the following density matrix equations of motion: ∂ρ i ∂ρ ¼  ½H; ρ þ ∂t ℏ ∂t

ð3Þ

ð2Þ

ρ_ 11 ¼ γ 21 ρ22 þ γ 31 ρ33 þ ir c ðρ21 ρ12 Þ þ ir p ðρ31 ρ13 Þ

ð4Þ

ρ_ 22 ¼ γ 21 ρ22 þ ir c ðρ12 ρ21 Þ þ ir m ðρ32 ρ23 Þ

ð5Þ

ρ_ 33 ¼ γ 31 ρ33 þ ir m ðρ23 ρ32 Þ þ irp ðρ13 ρ31 Þ

ð6Þ

ρ_ 12 ¼ ðΓ 21 iΔc Þρ12 þ ir c ðρ22 ρ11 Þir m ρ13 eiϕ þ irp ρ32 eiϕ

ð7Þ

ρ_ 13 ¼ ðΓ 31 iΔp Þρ13 þ ir p ðρ33 ρ11 Þir m ρ12 eiϕ þ irc ρ23 eiϕ

ð8Þ

ρ_ 23 ¼ ðΓ 32 iðΔp Δc ÞÞρ23 þ ir m ðρ33 ρ22 Þ þ ir c ρ13 eiϕ ir p ρ21 eiϕ

ð9Þ

where ρjk ¼ ρnkj . The closure of the system requires, ρ11 + ρ22 + ρ33 ¼ 1. The phase term ϕ ¼ ϕc þ ϕm ϕp is named as collective phase which arises due to the interaction of the coherent fields operating in close-loop configuration (Fig. 1). It controls the coherence property of the system dynamics for a definite set of Rabi frequencies and detuning parameters. The Fourier component of polarization Pðωp Þ induced by the probe field can be determined by performing the quantum average of the dipole moment over an ensemble of homogeneously broadened atoms. As is well known, the imaginary part of the polarization represents the absorptive properties in probe response. Here, the absorption coefficient for the probe field operating on transition j1〉2j3〉 is directly proportional to the imaginary part of ρ13 . 3. Numerical results In order to comprehend the absorptive response of the probe field at various parameter conditions, we investigate the timedependent numerical solutions of Eqs. (4)–(9) and present the temporal evolution of probe absorption by showing the time dependence of Imðρ13 Þ in resonant and non-resonant conditions of the applied fields. For our model, the condition Imðρ13 Þ 4 0 indicates that the probe laser will be amplified. In other words, the system exhibits gain for the probe field. Other conditions like Imðρ13 Þ o 0 and Imðρ13 Þ ¼ 0 are for probe absorption and transparency respectively. 3.1. Resonant case This subsection deals with the coupling lasers and the probe laser producing zero detuning (Δc ¼ Δm ¼ Δp ¼ 0) for the fields Ec, Em and Ep. We present the temporal dynamics of absorption with the variation of distinctive controlling parameters as follows: 3.1.1. Rabi frequency-induced modulation To comprehend the Rabi-frequency-induced modulation of probe absorption in the given model, we show the time evolution of atomic response Imðρ13 Þ in Figs. 2 and 3 for two sets of resonant coupling fields (r c ¼ r m ¼ 40γ and r c ¼ 10γ, r m ¼ 40γ) respectively for a fixed resonant probe field r p ¼ 0:8γ in zero-phase (ϕ ¼ 0) condition. All the rate-parameters used in computation are scaled by the decay rate γ 31 denoted as γ. To have an estimate of the population inversion (ρ33 ρ11 ) contributed in probe gain and absorption we present the atomic population distribution of the

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gain-absorption envelope diminishes with the inclusion of γ 32 . With the application of two driving fields of unequal Rabi frequencies, we show the transient evolution of atomic population and probe response in Fig. 3 with a (r c ¼ 10γ, r m ¼ 40γ; γ 32 ¼ γ), b (r c ¼ 10γ, r m ¼ 40γ; γ 32 ¼ 0). It is seen from atomic population evolution in Fig. 3 that the probability for the atoms being excited to states j2〉 and j3〉 is very small because of the Rabi frequency of the optical driving field small compared to that of the microwave field. With regard to the transient evolution of Imðρ13 Þ in Fig. 3 the probe response shows transparency in steady state after the transient periodic variation accompanied by the feature of loop structures as shown in Fig. 3b. This is to note here that the variation in the amplitudes of the transient gain-absorption envelopes in Figs. 2 and 3 may be attributed to satisfying the Raman inversion condition, ρ33 4ρ22 and mainly the competition between the real parts of the coherence terms ρ12 and ρ23 . In Figs. 2 (r c ¼ r m ¼ 40γ) and 3 (r c ¼ 10γ, r m ¼ 40γ) the Raman inversion condition is obeyed. If the values of the Rabi frequencies are interchanged between two coupling fields (r c ¼ 40γ, r m ¼ 10γ), the qualitative nature of transient gain-absorption response remains almost same as shown in Fig. 3. But the Raman coherence ρ23 in probe response plays the most significant role in the condition (ρ33 o ρ22 and ρ33 o ρ11 ) [6] of inversionless gain in bare atomic state basis. To present and discuss results more conveniently on the nature of transient variation of atomic response with other relevant parameters in following, we prefer to define the fixed Rabi-frequency-combinations (r c ¼ r m ¼ 40γ), (r c ¼ 10γ, r m ¼ 40γ),

V system for each set of parameter conditions in Figs. 2 and 3. In each part (b) of Figs. 2 and 3 we show how the gain and absorption in probe response are affected by the inclusion of the decay rate γ 32 from level j3〉 to level j2〉. The spontaneous decay rate γ 21 is kept fixed to γ. In Fig. 2a where the Rabi frequencies of the coupling field and the microwave field are equal (r c ¼ r m ¼ 40γ; γ 32 ¼ γ), it is prominent that the atomic population oscillates back and forth among the states j1〉, j2〉 and j3〉 and eventually reaches the steady-state values ρ11 ≈ρ22 ≈ρ33 ≈0:33, which shows that the populations are almost trapped. In case of plot of Imðρ13 Þ versus normalized time, numerical results show the signature of loop structures with varying amplitudes in its oscillatory behaviour versus time. We have focused on the onset and prominent emergence of the multiloop structures under different parametric conditions in the following Sections 3.1.2 and 3.1.3 in details. The probe response exhibits periodic amplification and absorption around zeroabsorption line within the domain of each loop and, in course of time, probe response attains the zero absorption in the steady state. Such vanishing of absorption is due to the destructive interference between the transition pathways: j1〉2j3〉 and j1〉2j2〉2j3〉. In Fig. 2b for the case (r c ¼ r m ¼ 40γ; γ 32 ¼ 0), the similar feature occurs for the atomic population and population inversion but with exception in steady-state values ρ11 ≈ρ33 ≈0:4; ρ22 ≈0:2. It is worth noting that the additional dissipation due to γ 32 causes a faster decay of the atomic coherence, and the atomic system reaches the steady state faster with γ 32 ¼ γ (Fig. 2a) than with γ 32 ¼ 0 (Fig. 2b). The oscillatory amplitude of the transient

0.250 1.0

ρ11 0.125

0.0

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1.0 0.5

Im (ρ13)

Population Distribution

0.5

0.000

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γt

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γt(sec) 0.250

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Population Distribution

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ρ33

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γt

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0

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γt

Fig. 2. Rabi frequency-induced transient evolution of population (left panel) and atomic response, Imðρ13 Þ (right panel) – (a) r c ¼ r m ¼ 40γ, γ 32 ¼ γ and (b) r c ¼ r m ¼ 40γ, γ 32 ¼ 0. Other parameters are set as r p ¼ 0:8γ, γ 21 ¼ γ, γ 31 ¼ γ, ϕ ¼ 0, Δc ¼ Δm ¼ Δp ¼ 0.

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0.06

1.0

ρ11 0.03

0.8 0.090

ρ22

0.045 0.000 0.250

Im (ρ13)

Population Distribution

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ρ33

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-0.03

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Im (ρ13)

Population Distribution

0.9

0.00

0.000 0.250

ρ33

-0.04

0.125 0.000

-0.08 0

2

4

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0

2

4

γt

Fig. 3. Rabi frequency-induced transient evolution of population (left panel) and atomic response, Imðρ13 Þ (right panel) – (a) r c ¼ 10γ, r m ¼ 40γ, γ 32 ¼ γ and (b) r c ¼ 10γ, r m ¼ 40γ, γ 32 ¼ 0. Other parameters are same as in Fig. 2.

and (r c ¼ 40γ, r m ¼ 10γ) with γ 32 ¼ 0 as the first, second and third combinations, respectively.

also be observed for Imðs13 Þ whose temporal change is governed by the following equation: Imðs_ 13 Þ ¼ eΓ31 t ðA sin ðΔp tÞ þ B cos ðΔp tÞÞ

3.1.2. Phase-induced modulation We analyze the effects of the collective phase, ϕ, on the transient evolution of the atomic response in the three combinations and show the relevant plots in Fig. 4 in connection with the plots for ϕ ¼ 0 in Figs. 2b and 3b. In each case of Fig. 4a–c we present the plots of Imðρ13 Þ for ϕ ¼ π=2 (left panel) and π (right panel) for three combinations respectively. Fig. 4c is added to represent the case for r c ¼ 40γ, r m ¼ 10γ. Other parameters (r p ¼ 0:8γ, γ 21 ¼ γ, γ 31 ¼ γ, γ 32 ¼ 0) are set same as those for the previous Figs. 2b and 3b. We note that the phase ϕ modulates the transient dynamics regarding transient and stationary values of gain, absorption without having any impact on the transient evolution of populations. Further, it is worth mentioning that the transient absorption/gain spectra in right panels (ϕ ¼ π) of the cases in Fig. 4 is identical with those in the figures mentioned earlier for ϕ ¼ 0. This indicates that the transient behavior of the probe field for ϕ ¼ π is the same as that for ϕ ¼ 0. As far as the loop patterns are concerned with ϕ ¼ 0 or π, we have examined that the formation of such patterns is mainly due to the competitive nature of the real parts of two coherence terms ρ23 and ρ12 associated with the two-step excitation pathways. This fact can be qualitatively understood by substituting ρ13 in the Eq. (8) on the basis of the transformation ρ13 ¼ s13 expððΓ 31 iΔp ÞtÞ. We note that the coherent part of the transient dynamics as obtained for Imðρ13 Þ can

ð10Þ

where, A¼U I cos ϕ þ V R sin ϕ and B ¼r p ðρ33 ρ11 Þ þ V R cos ϕU I sin ϕ. UI and VR represent the resultant contributions of imaginary and real parts of the coherence terms ρ12 and ρ23 respectively, i.e., U I ¼ r c Imðρ23 Þr m Imðρ12 Þ and V R ¼ r c Reðρ23 Þ r m Reðρ12 Þ. For ϕ¼0, A¼UI and B ¼ r p ðρ33 ρ11 Þ þ V R ; while for ϕ ¼ π, A will be equal to U I and B suffers only the interchange of the signs (‘ ’ and ‘+’) in the real parts of coherence terms in VR. These two terms in VR always dominate over the first term (population exchange term in B) initiated by the weak probe field. In the right panel (ϕ ¼ π) of Fig. 4a in case of the first combination, the coherence terms Reðρ23 Þ and Reðρ12 Þ are found to contribute in atomic response with nearly equal amplitudes. For the second combination (the right panel of Fig. 4b), we have observed that Reðρ12 Þ becomes predominant over Reðρ23 Þ whereas the reverse is the case for the third combination (the right panel of Fig. 4c). To analyze the change in phase dependent transient evolution of Imðρ13 Þ in three combinations, we have found that all the plots in the left panel of Fig. 4 with ϕ ¼ π=2 for a definite combination suffer an appreciable change with respect to the plots corresponding to ϕ ¼ 0. We note that the probe field no longer exhibits the loop structure with periodic amplification and absorption as in the case of ϕ ¼ 0 or π. For the first combination at ϕ ¼ π=2 (left panel of Fig. 4a), the transient behaviour of the probe field exhibits only the transient absorption oscillating below the zero-absorption line

P. Panchadhyayee / Optics Communications 309 (2013) 95–102

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Fig. 4. Phase-induced transient evolution of atomic response, Imðρ13 Þ for ϕ ¼ π=2 (left panel) and π (right panel) – (a) r c ¼ r m ¼ 40γ (b) r c ¼ 10γ, r m ¼ 40γ; and (c) r c ¼ 40γ, r m ¼ 10γ. Other parameters are set as r p ¼ 0:8γ, γ 21 ¼ γ, γ 31 ¼ γ, γ 32 ¼ 0, Δc ¼ Δm ¼ Δp ¼ 0.

with the complete suppression of transient gain. Unlike the zerophase case, the transient absorption reaches to a negative steadystate value which is not at the middle portion of the transient spectrum envelope; rather it corresponds to the lower part of the envelope. In the second combination for ϕ ¼ π=2 (left panel of Fig. 4b), only a little deviation occurs in the steady-state negative value for transient absorption residing at nearly middle portion of the transient envelope in comparison to the features obtained in same phase condition for the first combination (Fig. 4a). To speak of the transient evolution process for the third combination with respect to those for other combinations, a drastic change can be seen, where transient gain is obtained in the left panel of Fig. 4c for ϕ ¼ π=2 with an initial but little absorption only during a very short time. In the left panel of Fig. 4c, the steady-state inversionless gain occurs as we attain the steady-state probe gain in the V-type system [29]. For each case corresponding to ϕ ¼ π=2, we have observed that the Eq. (10) representing the rate of change of Imðs13 Þ shapes with A ¼ V R ¼ r c Reðρ23 Þr m Reðρ12 Þ and B ¼ r p ðρ33 ρ11 ÞU I ¼ r p ðρ33 ρ11 Þr c Imðρ23 Þ þ r m Imðρ12 Þ. So, in exact probe resonance, the coherence effects appear in the probe absorption due to the net contribution of all the three terms (specifically

r m Imðρ12 Þ and r c Imðρ23 Þ) in B. We note that the oscillatory amplitude of atomic response is nearly equal in case of the first combination whereas it appears much larger for the other two combinations for ϕ ¼ π=2 compared to their zero-phase cases. As far as the features of gain and absorption are concerned, it is verified that the plots of transient and steady-state behaviour of the probe field show the opposite features with ϕ ¼ π=2 and with ϕ ¼ 3π=2, albeit they are exactly symmetric in reference with zeroabsorption line in the gain-absorption profiles. This is due to the change in sign in A as well as B (for signs only before the second and third terms in B) when ϕ changes from π=2 to 3π=2.

3.1.3. Decay-rate-induced modulation To study the modulation of transient response induced by spontaneous decay rates we plot the phase-dependent (ϕ ¼ 0 and π=2) transient spectra of Imðρ13 Þ keeping γ 21 ¼ γ, γ 31 ¼ 0:1γ in Fig. 5 and γ 21 ¼ 0:1γ, γ 31 ¼ γ in Fig. 6 for the two combinations (a - first and b - third) mentioned above with r p ¼ 0:8γ. In each case of Figs. 5 and 6 we present the phase-dependent transient atomic response for ϕ ¼ 0 (left panel) and ϕ ¼ π=2 (right panel).

P. Panchadhyayee / Optics Communications 309 (2013) 95–102

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Fig. 5. Decay-rate-induced transient evolution of atomic response, Imðρ13 Þ for ϕ ¼ 0 (left panel) and π=2 (right panel) – (a) r c ¼ r m ¼ 40γ and (b) r c ¼ 40γ, r m ¼ 10γ. Other parameters are set as r p ¼ 0:8γ, γ 21 ¼ γ, γ 31 ¼ 0:1γ, γ 32 ¼ 0, Δc ¼ Δm ¼ Δp ¼ 0.

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Fig. 6. Decay-rate-induced transient evolution of atomic response, Imðρ13 Þ for ϕ ¼ 0 (left panel) and π=2 (right panel) – (a) r c ¼ r m ¼ 40γ and (b) r c ¼ 40γ, r m ¼ 10γ. Other parameters are set as r p ¼ 0:8γ, γ 21 ¼ 0:1γ, γ 31 ¼ γ, γ 32 ¼ 0, Δc ¼ Δm ¼ Δp ¼ 0.

With the decrease in γ 31 and γ 21 , it can be seen from Figs. 5 and 6 that the response time for the atomic medium arriving at the steady state is prolonged. In contrast to Fig. 4, the periodic gainabsorption profiles in zero-phase cases (left panels) of Figs. 5 and 6, exhibit prominent signature of loop structures (specially in

cases for the first combination) having two or more loops with larger magnitudes. With regard to the contributions of Reðρ23 Þ and Reðρ12 Þ in Imðs13 Þ as shown in Fig. 4 for the three combinations with ϕ ¼ 0 or π, it is worth mentioning that the difference in amplitudes of the two coherence terms is reduced in the

P. Panchadhyayee / Optics Communications 309 (2013) 95–102

corresponding cases when γ 31 becomes 0:1γ instead of γ. In the zero-phase condition for the first combinations in Fig. 5 and 6 the transient population dynamics in the levels j1〉 and j3〉 with the passage of time exhibits the continuous interchange of population in between two states follows the generation of multiple loops (not shown). As compared to the transient absorption spectrum for ϕ ¼ π=2 in Fig. 4a for the first combination, the steady-state value in the corresponding spectrum in Fig. 5a indicates the increasing probe absorption. The gain-absorption profile of the third combination in Fig. 6b for ϕ ¼ π=2 exhibits the same qualitative transient and steady-state (gain) features as shown in Fig. 5b. It is verified that, in such parametric variation for decay rates, the gain-absorption characteristics of the second combination for ϕ ¼ π=2 show absorption in both the transient and steady states, which is similar to that in corresponding profile in Fig. 4b. With a view to above discussions a pertinent question naturally comes how the Rabi-frequencies of the coupling fields affect the onset and prominence of loop structures with multiple loops. We have presented the related features in Fig. 7 where we use the increasing set of equal Rabi-frequencies (r c ¼ r m ¼ γ for a; r c ¼ r m ¼ 4γ for b; r c ¼ r m ¼ 60γ for c) of the coupling fields keeping the other parameters same as mentioned for the zero-phase condition of Fig. 5a. For very small Rabi-frequency values of the coupling fields (r c ¼ r m ¼ γ) no trace of loop structure is found in Fig. 7a. With the increase both in the Rabi-frequencies of coupling fields, the feature of coherence induced multiple loops sets in, as shown in Fig. 7b for r c ¼ r m ¼ 4γ accompanied by the subsequent increase in oscillatory amplitudes of atomic response of probe laser. As the applied coupling laser fields become stronger (r c ¼ r m ¼ 60γ), we observe that the steady-state transparency gets modulated in the transient process with the notable evolution of multiple loops in atomic response. We have checked that, at that condition, the competition between Reðρ23 Þ and Reðρ12 Þ yields a

0.00

Im (ρ13)

Im (ρ13)

0.15

-0.15

3.2. Non-resonant case Next, we address the issues of using probe and optical driving fields in non-resonant condition. We have chosen the first combination (r c ¼ r m ¼ 40γ) to show the impact of probe detuning (Δp ) and coupling field detuning (Δc ) on the transient response Imðρ13 Þ in Fig. 8. Other parameters are set as r p ¼ 0:8γ, γ 21 ¼ γ, γ 31 ¼ 0:1γ, ϕ ¼ 0. In reference to the Fig. 5a for ϕ ¼ 0, it is observed in Fig. 8a that, for non-zero detuning of probe (Δp ¼ 10γ) using the resonant coupling laser, the number of loops in temporal dynamics increases within the same range of normalized time. In the reverse conditions of probe and coupling detuning (Δp ¼ 0, Δc ¼ 10γ) the same feature replicates as is seen for Fig. 8a. Further, when we set both the detunings as equal (Δp ¼ Δc ¼ 10γ), the appearance of the multi-looped structure of atomic response in Fig. 8b remains almost same as that shown in Fig. 5a for ϕ ¼ 0. But it is interesting to note that the consequent reduction of time interval (with complete zero-value in atomic response) between two successive loops is observed with the application of non-zero detuning of the probe and coupling lasers.

4. Conclusion In summary, we analyze the impact of different relevant parameters like the Rabi frequency, detuning and the relative phase of the control fields on transient evolutions of gain, absorption and populations in a closed three-level V-type atomic system for both resonant and non-resonant conditions. Different combinations of Rabi-frequencies of control fields are chosen to

0.36

0.4

0.18

0.2

0.00 -0.18

-0.30 -0.45

synchronized form of the atomic response and shapes in coherence induced multi-loop patterns.

Im (ρ13)

0.30

101

0

2

4

6

8

-0.36

10

0.0 -0.2

0

2

4

γt

6

8

-0.4

10

0

2

4

γt

6

8

10

γt

0.50

0.4

0.25

0.2

Im (ρ13)

Im (ρ13)

Fig. 7. Evolution of coherence controlled loop structure in atomic response, Imðρ13 Þ for (a) r c ¼ r m ¼ γ; (b) r c ¼ r m ¼ 4γ; and (c) r c ¼ r m ¼ 60γ. Other parameters are set as r p ¼ 0:1γ, γ 21 ¼ γ, γ 31 ¼ 0:1γ, γ 32 ¼ 0, ϕ ¼ 0, Δc ¼ Δm ¼ Δp ¼ 0.

0.00

0.0

-0.2

-0.25

-0.50

-0.4 0

2

4

6

γt

8

10

0

2

4

6

8

10

γt

Fig. 8. The transient evolution of atomic response, Imðρ13 Þ under non-resonant condition of applied fields – (a) Δp ¼ 10γ, Δc ¼ 0 and (b) Δp ¼ Δc ¼ 10γ. Other parameters are set as r c ¼ r m ¼ 40γ, r p ¼ 0:8γ, γ 21 ¼ γ, γ 31 ¼ 0:1γ, γ 32 ¼ 0, ϕ ¼ 0.

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P. Panchadhyayee / Optics Communications 309 (2013) 95–102

present the nature of transient variation of atomic response. Under resonant condition of probe and control lasers the probe response shows transparency in steady state after the transient periodic variation accompanied by the signature of loop-structure. For the third Rabi frequency-combination the role of the Raman coherence is significant in the transient gain-absorption response to achieve inversionless gain in bare atomic state basis. In every case of ϕ ¼ 0 at the condition of exact probe resonance, we mention that the loop patterns in atomic response are due to the contributions of two coherence terms (Reðρ23 Þ and Reðρ12 Þ) associated with the two-step excitation pathways, while for ϕ ¼ π=2, Imðρ23 Þ and Imðρ12 Þ play the vital role in behind. As a whole, the collective phase modulates the transient dynamics showing transient and stationary values of gain, absorption and transparency without invoking any restriction to the dynamical variation of populations. On controlling radiative properties of the atomic system it is shown that the response time for the atomic medium arriving at the steady state is prolonged with the decrease in decay rates. The impact of probe Rabi frequency on the evolution of the coherence induced multi-loop patterns at resonant condition is also examined with equal Rabi frequencies of coupling lasers. In comparison with purely resonant case, we can infer that the time interval (with complete zero-value in atomic response) between two successive loops gets reduced by the non-zero detuning applied only in any one of the lasers involved in the system. Acknowledgements

[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

The author gratefully acknowledges Prof. Hong Guo for some helpful discussion and valuable suggestions. The author also acknowledges Prof. Prasanta Kumar Mahapatra and Dr. Bibhas Kumar Dutta for their valuable suggestions.

[29] [30] [31] [32] [33]

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