Optics Communications 309 (2013) 95–102

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Optics Communications

0030-40http://d

n Tel.:E-m

journal homepage: www.elsevier.com/locate/optcom

Role of strongly modulated coherence in transient evolution dynamicsof 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

a r t i c l e i n f o

Article history:Received 30 May 2013Received in revised form2 July 2013Accepted 4 July 2013Available online 18 July 2013

Keywords:Transient responseAbsorptionInversionless gainCollective phaseCoherence induced loop structures

18/$ - see front matter & 2013 Elsevier B.V. Ax.doi.org/10.1016/j.optcom.2013.07.021

+91 3220 255020.ail address: ppcontai@gmail.com

a b s t r a c t

We investigate the dynamical behaviour of atomic response in a closed three-level V-type atomic systemwith the variation of different relevant parameters to exhibit transient evolution of absorption, gain andtransparency in the probe response. The oscillations in probe absorption and gain can be efficientlymodulated by changing the values of the Rabi frequency, detuning and the collective phase involved inthe system. The interesting outcome of the work is the generation of coherence controlled loop-structurewith varying amplitudes in the oscillatory probe response of the probe field at various parameterconditions. The prominence of these structures is observed when the coherence induced in a one-photonexcitation path is strongly modified by two-step excitations driven by the coherent fields operating inclosed interaction contour. In contrast to purely resonant case, the time interval between two successiveloops gets significantly reduced with the application of non-zero detuning in the coherent fields.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

In the last two decades, extensive studies in the area of nonlinearoptics and laser spectroscopy have led to considerable interest in thestudy of optical response of the atomic system interacting with anumber of coherent fields. In ideal three-level systems (Λ, V and Ξ)[1], the atomic coherence can be dynamically induced by theapplication of a strong field driving one excitation path. Inthe presence of an additional field probing another excitation path,the induced coherence leads to various quantum optical effects likeelectromagnetically induced transparency (EIT) [2], electromagneti-cally induced absorption (EIA) [3], gain without inversion (GWI) orlasing without inversion (LWI) [4–9], enhancement of refractiveindex [10] and generation of quantum beats [11].

The traditional method of generating GWI in an ideal closed three-level scheme producing EIT incorporates the application of anincoherent pump field to the system to populate the upper lasinglevel of the transition probed by a weak coherent field. The phenom-enon of inversionless gain can result in the absorptive response of theprobe field due to strong two-photon coherence induced in thesystem [6]. In a suitable level scheme, without invoking incoherentpumping, the condition of initial population in the upper lasing level isfulfilled by choosing an appropriate relaxation pathway from theexcited level [12,13]. In presence of incoherent pumping, interferencebetween two spontaneous decay channels in a V-type system with

ll rights reserved.

two closely lying upper levels leads to the formation of gain as a resultof vacuum induced coherence (VIC) [14,15]. In such scheme, it ispossible to obtain the control of absorption and gain by the relativephase 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 V-type [23] and Λ-type [24–26] schemes where a microwave field isemployed in the low-frequency induced transitions to form the close-loop interaction model. Such model shows its robustness to producegain 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 transientevolution of gain at various situations. In closed [29,31,32,34,37,38]and open [33,35,36] folded type schemes, gain-characteristics havebeen studied with [31,32,35–38] and without [29,30,33,34] inclusionof VIC. By adopting the close-loop interaction in a Λ-type system [32],phase dependent elimination of absorption has been noted in tran-sient regime. Similar scheme using an external magnetic field in thelow-frequency induced transition has been analyzed [34] to obtainGWI in transient regime. Motivated by these works [28–38] performedin transient regime, a closed three-level V-type scheme (Fig. 1) withclose-loop interaction has been considered in this article to representthe three-ways of control of transient response of the probe field indetail.

The close-loop interaction inherent to the present model isexploited in this work to exhibit the transient evolution of absorp-tion, gain and transparency in the probe response with the variationof model parameters. The salient features of the work are presentedas follows: (a) Explicit occurrence of absorption and gain in the

mω3

cω

2

1

31γ

21γpω

Fig. 1. Schematic presentation of a three-level V-type atom interacting with threecoherent fields designated by their frequencies: ωp for the probe field and ωj

ðj¼ c;mÞ for two control fields. γ21 and γ31 denote the spontaneous decay rates ofthe excited levels.

P. Panchadhyayee / Optics Communications 309 (2013) 95–10296

oscillatory probe response is obtained in many ways in the presentmodel. (b) We have shown the generation of multi-loop-structure inthe oscillatory probe response of the probe field at various condi-tions. This structure appears in probe transition due to strongperturbation resulted from two successive transition pathwaysdriven by two coherent fields. Control of such structure by varyingRabi frequency, detuning and the collective phase of the controlfields is presented in detail. (c) Unlike the purely resonant case, thetime interval between two successive loops is found to be reduced bysetting 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 groundstate j1⟩ and the excited states j2⟩ and j3⟩ as shown in Fig. 1. Thetransition j1⟩2j2⟩ of frequency ω21 is driven by a coherent couplingfield of frequency ωc and amplitude Ec in optical regime, and theother transition j2⟩2j3⟩ of frequency ω32 is driven by anothercoherent coupling field (microwave) of frequency ωm and amplitudeEm. A weak optical field of frequency ωp and amplitude Ep probingthe transition j1⟩2j3⟩ of frequency ω31 is so chosen that thepopulation transfer to the uppermost level by this field becomesnegligible. All the fields are considered in the continuous wave(CW) regime. In the semiclassical formulation, fields are defined asEiðx; tÞ ¼ ϵi cos ðωit�kizÞ ði¼ p; c;mÞ where ki is the propagationvector along the z-direction. The spontaneous decay rates fromlevel j3⟩ ðj2⟩Þ to level j1⟩ is taken to be γ31 (γ21). For the transitionsj1⟩2j2⟩, j1⟩2j3⟩ and j2⟩2j3⟩, the coherence dephasing rates aredesignated by Γ21ð ¼ γ21=2Þ, Γ31ð ¼ γ31=2Þ and Γ32ð ¼ ðγ21 þ γ31Þ=2Þ,respectively. In the context of decay-induced coherence we neglectthe role of vacuum modes in the transitions j1⟩2j2⟩ and j1⟩2j3⟩with the approximation that the Raman coherence originated bythe application of the microwave field Em between two upper statesof the V-type system predominates over the decay-induced coher-ence in the limit of spontaneous decay rates much smaller than ω32.

The Hamiltonian of the atomic system assuming the electric-dipole and rotating wave approximations can be expressed in theinteraction picture as

H¼�ℏ½Rce�iΔct j1⟩⟨2j þ Rme�iΔmt j2⟩⟨3j þ Rpe�iΔpt j1⟩⟨3j þ H:c:� ð1Þwhere Δc ¼ ω21�ωc , Δm ¼ω32�ωm and Δp ¼ ω31�ωp are the detun-ings and Rc¼μ12:ϵ

nc=2ℏ, Rm¼μ23:ϵ

nm=2ℏ and Rp¼μ13:ϵ

np=2ℏ are the

complex Rabi frequencies which can be presented as Rj¼rjeiϕj

(j¼ p; c;m), rj being real parameters. The relation among thefrequencies of coupling fields obeying the condition, ωp ¼ωc 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∂t

¼� iℏ½H; ρ� þ ∂ρ

∂tð2Þ

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

∂ρ∂t

¼� ∑j ¼ 2;3

γj12

ðfjj⟩⟨jj; ρg�2j1⟩⟨jjρjj⟩⟨1jÞ ð3Þ

In order to simplify the component equations of motion inEq. (2), we introduce the unitary transformations for the atomicresponses ρ12 ¼ ρ12eiϕc , ρ13 ¼ ρ13e

iϕp and ρ23 ¼ ρ23eiϕm and finallyobtain the following equations for the density matrix elements:

_ρ11 ¼ γ21ρ22 þ γ31ρ33 þ ircðρ21�ρ12Þ þ irpðρ31�ρ13Þ ð4Þ

_ρ22 ¼�γ21ρ22 þ ircðρ12�ρ21Þ þ irmðρ32�ρ23Þ ð5Þ

_ρ33 ¼�γ31ρ33 þ irmðρ23�ρ32Þ þ irpðρ13�ρ31Þ ð6Þ

_ρ12 ¼�ðΓ21�iΔcÞρ12 þ ircðρ22�ρ11Þ�irmρ13e�iϕ þ irpρ32e

�iϕ ð7Þ

_ρ13 ¼�ðΓ31�iΔpÞρ13 þ irpðρ33�ρ11Þ�irmρ12eiϕ þ ircρ23e

iϕ ð8Þ

_ρ23 ¼�ðΓ32�iðΔp�ΔcÞÞρ23 þ irmðρ33�ρ22Þ þ ircρ13e�iϕ�irpρ21e

�iϕ ð9Þwhere ρjk ¼ ρnkj. The closure of the system requires, ρ11 + ρ22 +ρ33 ¼ 1. The phase term ϕ¼ ϕc þ ϕm�ϕp is named as collectivephase which arises due to the interaction of the coherent fieldsoperating in close-loop configuration (Fig. 1). It controls thecoherence property of the system dynamics for a definite set ofRabi frequencies and detuning parameters.

The Fourier component of polarization PðωpÞ induced by theprobe field can be determined by performing the quantum averageof the dipole moment over an ensemble of homogeneouslybroadened atoms. As is well known, the imaginary part of thepolarization represents the absorptive properties in proberesponse. Here, the absorption coefficient for the probe fieldoperating on transition j1⟩2j3⟩ is directly proportional to theimaginary part of ρ13.

3. Numerical results

In order to comprehend the absorptive response of the probefield at various parameter conditions, we investigate the time-dependent numerical solutions of Eqs. (4)–(9) and present thetemporal evolution of probe absorption by showing the timedependence of Imðρ13Þ in resonant and non-resonant conditionsof the applied fields. For our model, the condition Imðρ13Þ40indicates that the probe laser will be amplified. In other words, thesystem exhibits gain for the probe field. Other conditions likeImðρ13Þo0 and Imðρ13Þ ¼ 0 are for probe absorption and transpar-ency respectively.

3.1. Resonant case

This subsection deals with the coupling lasers and the probelaser producing zero detuning (Δc ¼ Δm ¼ Δp ¼ 0) for the fields Ec,Em and Ep. We present the temporal dynamics of absorption withthe variation of distinctive controlling parameters as follows:

3.1.1. Rabi frequency-induced modulationTo comprehend the Rabi-frequency-induced modulation of

probe absorption in the given model, we show the time evolutionof atomic response Imðρ13Þ in Figs. 2 and 3 for two sets of resonantcoupling fields (rc ¼ rm ¼ 40γ and rc ¼ 10γ, rm ¼ 40γ) respectivelyfor a fixed resonant probe field rp ¼ 0:8γ in zero-phase (ϕ¼ 0)condition. All the rate-parameters used in computation are scaledby the decay rate γ31 denoted as γ. To have an estimate of thepopulation inversion (ρ33�ρ11) contributed in probe gain andabsorption we present the atomic population distribution of the

P. Panchadhyayee / Optics Communications 309 (2013) 95–102 97

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 andabsorption in probe response are affected by the inclusion of thedecay rate γ32 from level j3⟩ to level j2⟩. The spontaneous decayrate γ21 is kept fixed to γ.

In Fig. 2a where the Rabi frequencies of the coupling field andthe microwave field are equal (rc ¼ rm ¼ 40γ; γ32 ¼ γ), it is promi-nent that the atomic population oscillates back and forth amongthe states j1⟩, j2⟩ and j3⟩ and eventually reaches the steady-statevalues ρ11≈ρ22≈ρ33≈0:33, which shows that the populations arealmost trapped. In case of plot of Imðρ13Þ versus normalized time,numerical results show the signature of loop structures withvarying amplitudes in its oscillatory behaviour versus time. Wehave focused on the onset and prominent emergence of the multi-loop structures under different parametric conditions in thefollowing Sections 3.1.2 and 3.1.3 in details. The probe responseexhibits periodic amplification and absorption around zero-absorption line within the domain of each loop and, in course oftime, probe response attains the zero absorption in the steadystate. Such vanishing of absorption is due to the destructiveinterference between the transition pathways: j1⟩2j3⟩ andj1⟩2j2⟩2j3⟩. In Fig. 2b for the case (rc ¼ rm ¼ 40γ; γ32 ¼ 0), thesimilar feature occurs for the atomic population and populationinversion but with exception in steady-state values ρ11≈ρ33≈0:4;ρ22≈0:2. It is worth noting that the additional dissipation due toγ32 causes a faster decay of the atomic coherence, and the atomicsystem reaches the steady state faster with γ32 ¼ γ (Fig. 2a) thanwith γ32 ¼ 0 (Fig. 2b). The oscillatory amplitude of the transient

0.0

0.5

1.0

0.0

0.5

1.0

0.0

0.5

1.0

γt

Popu

latio

n D

istri

butio

n

-0.250

-0.125

0.000

0.125

0.250

0.0

0.5

1.0

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1.0

Popu

latio

n D

istri

butio

n

0 4 6 8 10 0

-0.250

-0.125

0.000

0.125

0.250

Im (ρ

13)

2

Im (ρ

13)

γt0 4 6 8 10 02

ρ33

ρ22

ρ11

ρ33

ρ22

ρ11

Fig. 2. Rabi frequency-induced transient evolution of population (left panel) and atomγ32 ¼ 0. Other parameters are set as rp ¼ 0:8γ, γ21 ¼ γ, γ31 ¼ γ, ϕ¼ 0, Δc ¼ Δm ¼ Δp ¼ 0.

gain-absorption envelope diminishes with the inclusion of γ32.With the application of two driving fields of unequal Rabifrequencies, we show the transient evolution of atomic populationand probe response in Fig. 3 with a (rc ¼ 10γ, rm ¼ 40γ; γ32 ¼ γ),b (rc ¼ 10γ, rm ¼ 40γ; γ32 ¼ 0). It is seen from atomic populationevolution in Fig. 3 that the probability for the atoms being excitedto states j2⟩ and j3⟩ is very small because of the Rabi frequency ofthe optical driving field small compared to that of the microwavefield. With regard to the transient evolution of Imðρ13Þ in Fig. 3 theprobe response shows transparency in steady state after thetransient periodic variation accompanied by the feature of loopstructures as shown in Fig. 3b. This is to note here that thevariation in the amplitudes of the transient gain-absorptionenvelopes in Figs. 2 and 3 may be attributed to satisfying theRaman inversion condition, ρ334ρ22 and mainly the competitionbetween the real parts of the coherence terms ρ12 and ρ23. InFigs. 2 (rc ¼ rm ¼ 40γ) and 3 (rc ¼ 10γ, rm ¼ 40γ) the Raman inver-sion condition is obeyed. If the values of the Rabi frequencies areinterchanged between two coupling fields (rc ¼ 40γ, rm ¼ 10γ), thequalitative nature of transient gain-absorption response remainsalmost same as shown in Fig. 3. But the Raman coherence ρ23 inprobe response plays the most significant role in the condition(ρ33oρ22 and ρ33oρ11) [6] of inversionless gain in bare atomicstate basis.

To present and discuss results more conveniently on thenature of transient variation of atomic response with otherrelevant parameters in following, we prefer to define the fixedRabi-frequency-combinations (rc ¼ rm ¼ 40γ), (rc ¼ 10γ, rm ¼ 40γ),

γt(sec)2 4 6 8 10

γt2 4 6 8 10

ic response, Imðρ13Þ (right panel) – (a) rc ¼ rm ¼ 40γ, γ32 ¼ γ and (b) rc ¼ rm ¼ 40γ,

0.000

0.125

0.2500.000

0.045

0.090

0.8

0.9

1.0

γt

ρ33

ρ22Po

pula

tion

Dis

tribu

tion

ρ11

-0.06

-0.03

0.00

0.03

0.06

0.000

0.125

0.2500.000

0.035

0.070

0.8

0.9

1.0

Popu

latio

n D

istri

butio

n

0 2 4 8 10 0 2 4 6 8 10

-0.08

-0.04

0.00

0.04

0.08Im

(ρ13)

Im (ρ

13)

6γt

γt0 2 4 8 10 0 2 4 6 8 106

γt

ρ33

ρ22

ρ11

Fig. 3. Rabi frequency-induced transient evolution of population (left panel) and atomic response, Imðρ13Þ (right panel) – (a) rc ¼ 10γ, rm ¼ 40γ, γ32 ¼ γ and (b) rc ¼ 10γ,rm ¼ 40γ, γ32 ¼ 0. Other parameters are same as in Fig. 2.

P. Panchadhyayee / Optics Communications 309 (2013) 95–10298

and (rc ¼ 40γ, rm ¼ 10γ) with γ32 ¼ 0 as the first, second and thirdcombinations, respectively.

3.1.2. Phase-induced modulationWe analyze the effects of the collective phase, ϕ, on the

transient evolution of the atomic response in the three combina-tions and show the relevant plots in Fig. 4 in connection with theplots for ϕ¼ 0 in Figs. 2b and 3b. In each case of Fig. 4a–c wepresent the plots of Imðρ13Þ for ϕ¼ π=2 (left panel) and π (rightpanel) for three combinations respectively. Fig. 4c is addedto represent the case for rc ¼ 40γ, rm ¼ 10γ. Other parameters(rp ¼ 0:8γ, γ21 ¼ γ, γ31 ¼ γ, γ32 ¼ 0) are set same as those for theprevious Figs. 2b and 3b. We note that the phase ϕ modulates thetransient dynamics regarding transient and stationary values ofgain, absorption without having any impact on the transientevolution of populations. Further, it is worth mentioning that thetransient absorption/gain spectra in right panels (ϕ¼ π) of thecases in Fig. 4 is identical with those in the figures mentionedearlier for ϕ¼ 0. This indicates that the transient behavior of theprobe field for ϕ¼ π is the same as that for ϕ¼ 0. As far as the looppatterns are concerned with ϕ¼ 0 or π, we have examined that theformation of such patterns is mainly due to the competitive natureof the real parts of two coherence terms ρ23 and ρ12 associatedwith the two-step excitation pathways. This fact can be qualita-tively understood by substituting ρ13 in the Eq. (8) on the basis ofthe transformation ρ13 ¼ s13 expð�ðΓ31�iΔpÞtÞ. We note that thecoherent part of the transient dynamics as obtained for Imðρ13Þ can

also be observed for Imðs13Þ whose temporal change is governedby the following equation:

Imð _s13Þ ¼ eΓ31tðA sin ðΔptÞ þ B cos ðΔptÞÞ ð10Þwhere, A¼UI cos ϕþ VR sin ϕ and B¼rpðρ33�ρ11Þ þ VR cos ϕ�UI

sin ϕ. UI and VR represent the resultant contributions of imaginaryand real parts of the coherence terms ρ12 and ρ23 respectively, i.e.,UI ¼ rc Imðρ23Þ�rm Imðρ12Þ and VR ¼ rc Reðρ23Þ� rm Reðρ12Þ. Forϕ¼0, A¼UI and B¼ rpðρ33�ρ11Þ þ VR; while for ϕ¼ π, A will beequal to �UI and B suffers only the interchange of the signs (‘� ’

and ‘+’) in the real parts of coherence terms in VR. These two termsin VR always dominate over the first term (population exchangeterm in B) initiated by the weak probe field. In the right panel(ϕ¼ π) of Fig. 4a in case of the first combination, the coherenceterms Reðρ23Þ and Reðρ12Þ are found to contribute in atomicresponse with nearly equal amplitudes. For the second combina-tion (the right panel of Fig. 4b), we have observed that Reðρ12Þbecomes predominant over Reðρ23Þ whereas the reverse is the casefor the third combination (the right panel of Fig. 4c).

To analyze the change in phase dependent transient evolutionof Imðρ13Þ in three combinations, we have found that all the plotsin the left panel of Fig. 4 with ϕ¼ π=2 for a definite combinationsuffer an appreciable change with respect to the plots correspond-ing to ϕ¼ 0. We note that the probe field no longer exhibits theloop structure with periodic amplification and absorption as in thecase of ϕ¼ 0 or π. For the first combination at ϕ¼ π=2 (left panel ofFig. 4a), the transient behaviour of the probe field exhibits only thetransient absorption oscillating below the zero-absorption line

-0.09

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0.27

0.36

0.45

Im (ρ

13)

-0.06

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0.00

γt0 4 6 8 10 0 2 6 8 10

-0.2

-0.1

0.0

0.1

0.2

γt

Im (ρ

13)

Im (ρ

13)

Im (ρ

13)

Im (ρ

13)

Im (ρ

13)

2 4

γt0 4 6 8 10 0 2 6 8 10

γt2 4

γt0 4 6 8 10 0 2 6 8 10

γt2 4

Fig. 4. Phase-induced transient evolution of atomic response, Imðρ13Þ for ϕ¼ π=2 (left panel) and π (right panel) – (a) rc ¼ rm ¼ 40γ (b) rc ¼ 10γ, rm ¼ 40γ; and (c) rc ¼ 40γ,rm ¼ 10γ. Other parameters are set as rp ¼ 0:8γ, γ21 ¼ γ, γ31 ¼ γ, γ32 ¼ 0, Δc ¼ Δm ¼ Δp ¼ 0.

P. Panchadhyayee / Optics Communications 309 (2013) 95–102 99

with the complete suppression of transient gain. Unlike the zero-phase case, the transient absorption reaches to a negative steady-state value which is not at the middle portion of the transientspectrum envelope; rather it corresponds to the lower part of theenvelope. In the second combination for ϕ¼ π=2 (left panel ofFig. 4b), only a little deviation occurs in the steady-state negativevalue for transient absorption residing at nearly middle portion ofthe transient envelope in comparison to the features obtained insame phase condition for the first combination (Fig. 4a). To speakof the transient evolution process for the third combination withrespect to those for other combinations, a drastic change can beseen, where transient gain is obtained in the left panel of Fig. 4cfor ϕ¼ π=2 with an initial but little absorption only during a veryshort time. In the left panel of Fig. 4c, the steady-state inversion-less gain occurs as we attain the steady-state probe gain in theV-type system [29]. For each case corresponding to ϕ¼ π=2, wehave observed that the Eq. (10) representing the rate of change ofImðs13Þ shapes with A¼ VR ¼ rc Reðρ23Þ�rm Reðρ12Þ and B¼ rpðρ33�ρ11Þ�UI ¼ rpðρ33�ρ11Þ�rc Imðρ23Þ þ rm Imðρ12Þ. So, in exact proberesonance, the coherence effects appear in the probe absorptiondue to the net contribution of all the three terms (specifically

rm Imðρ12Þ and rc Imðρ23Þ) in B. We note that the oscillatoryamplitude of atomic response is nearly equal in case of the firstcombination whereas it appears much larger for the other twocombinations for ϕ¼ π=2 compared to their zero-phase cases. Asfar as the features of gain and absorption are concerned, it isverified that the plots of transient and steady-state behaviour ofthe probe field show the opposite features with ϕ¼ π=2 and withϕ¼ 3π=2, albeit they are exactly symmetric in reference with zero-absorption line in the gain-absorption profiles. This is due to thechange in sign in A as well as B (for signs only before the secondand third terms in B) when ϕ changes from π=2 to 3π=2.

3.1.3. Decay-rate-induced modulationTo study the modulation of transient response induced by

spontaneous decay rates we plot the phase-dependent (ϕ¼ 0and π=2) transient spectra of Imðρ13Þ keeping γ21 ¼ γ, γ31 ¼ 0:1γ inFig. 5 and γ21 ¼ 0:1γ, γ31 ¼ γ in Fig. 6 for the two combinations(a - first and b - third) mentioned above with rp ¼ 0:8γ. In eachcase of Figs. 5 and 6 we present the phase-dependent transientatomic response for ϕ¼ 0 (left panel) and ϕ¼ π=2 (right panel).

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Im (ρ

13)

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γt0 2 6 8 100 2 4 6 8 10

-0.36

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γt

Im (ρ

13)

Im (ρ

13)

Im (ρ

13)

4

γt0 2 6 8 100 2 4 6 8 10

γt4

Fig. 5. Decay-rate-induced transient evolution of atomic response, Imðρ13Þ for ϕ¼ 0 (left panel) and π=2 (right panel) – (a) rc ¼ rm ¼ 40γ and (b) rc ¼ 40γ, rm ¼ 10γ. Otherparameters are set as rp ¼ 0:8γ, γ21 ¼ γ, γ31 ¼ 0:1γ, γ32 ¼ 0, Δc ¼ Δm ¼ Δp ¼ 0.

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Im (ρ

13)

Im (ρ

13)

Im (ρ

13)

Im (ρ

13)

6

γt γt0 2 4 6 8 10 0 2 4 8 106

Fig. 6. Decay-rate-induced transient evolution of atomic response, Imðρ13Þ for ϕ¼ 0 (left panel) and π=2 (right panel) – (a) rc ¼ rm ¼ 40γ and (b) rc ¼ 40γ, rm ¼ 10γ. Otherparameters are set as rp ¼ 0:8γ, γ21 ¼ 0:1γ, γ31 ¼ γ, γ32 ¼ 0, Δc ¼ Δm ¼ Δp ¼ 0.

P. Panchadhyayee / Optics Communications 309 (2013) 95–102100

With the decrease in γ31 and γ21, it can be seen from Figs. 5 and 6that the response time for the atomic medium arriving at thesteady state is prolonged. In contrast to Fig. 4, the periodic gain-absorption profiles in zero-phase cases (left panels) of Figs. 5and 6, exhibit prominent signature of loop structures (specially in

cases for the first combination) having two or more loops withlarger magnitudes. With regard to the contributions of Reðρ23Þ andReðρ12Þ in Imðs13Þ as shown in Fig. 4 for the three combinationswith ϕ¼ 0 or π, it is worth mentioning that the difference inamplitudes of the two coherence terms is reduced in the

P. Panchadhyayee / Optics Communications 309 (2013) 95–102 101

corresponding cases when γ31 becomes 0:1γ instead of γ. In thezero-phase condition for the first combinations in Fig. 5 and 6the transient population dynamics in the levels j1⟩ and j3⟩with thepassage of time exhibits the continuous interchange of populationin between two states follows the generation of multiple loops(not shown). As compared to the transient absorption spectrumfor ϕ¼ π=2 in Fig. 4a for the first combination, the steady-statevalue in the corresponding spectrum in Fig. 5a indicates theincreasing probe absorption. The gain-absorption profile of thethird combination in Fig. 6b for ϕ¼ π=2 exhibits the samequalitative transient and steady-state (gain) features as shown inFig. 5b. It is verified that, in such parametric variation for decayrates, the gain-absorption characteristics of the second combina-tion for ϕ¼ π=2 show absorption in both the transient and steadystates, which is similar to that in corresponding profile in Fig. 4b.

With a view to above discussions a pertinent question naturallycomes how the Rabi-frequencies of the coupling fields affect theonset and prominence of loop structures with multiple loops. Wehave presented the related features in Fig. 7 where we usethe increasing set of equal Rabi-frequencies (rc ¼ rm ¼ γ for a;rc ¼ rm ¼ 4γ for b; rc ¼ rm ¼ 60γ for c) of the coupling fields keepingthe other parameters same as mentioned for the zero-phasecondition of Fig. 5a. For very small Rabi-frequency values of thecoupling fields (rc ¼ rm ¼ γ) no trace of loop structure is found inFig. 7a. With the increase both in the Rabi-frequencies of couplingfields, the feature of coherence induced multiple loops sets in, asshown in Fig. 7b for rc ¼ rm ¼ 4γ accompanied by the subsequentincrease in oscillatory amplitudes of atomic response of probelaser. As the applied coupling laser fields become stronger(rc ¼ rm ¼ 60γ), we observe that the steady-state transparency getsmodulated in the transient process with the notable evolution ofmultiple loops in atomic response. We have checked that, at thatcondition, the competition between Reðρ23Þ and Reðρ12Þ yields a

-0.45

-0.30

-0.15

0.00

0.15

0.30

γt

Im (ρ

13)

-0.36

-0.18

0.00

0.18

0.36

0 2 4 6 8 10 0 4

Im (ρ

13)

2

Fig. 7. Evolution of coherence controlled loop structure in atomic response, Imðρ13Þ forrp ¼ 0:1γ, γ21 ¼ γ, γ31 ¼ 0:1γ, γ32 ¼ 0, ϕ¼ 0, Δc ¼ Δm ¼ Δp ¼ 0.

-0.50

-0.25

0.00

0.25

0.50

γt

Im (ρ

13)

0 2 4 6 8 10

Fig. 8. The transient evolution of atomic response, Imðρ13Þ under non-resonant conditionset as rc ¼ rm ¼ 40γ, rp ¼ 0:8γ, γ21 ¼ γ, γ31 ¼ 0:1γ, γ32 ¼ 0, ϕ¼ 0.

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

3.2. Non-resonant case

Next, we address the issues of using probe and optical drivingfields in non-resonant condition. We have chosen the first combi-nation (rc ¼ rm ¼ 40γ) to show the impact of probe detuning (Δp)and coupling field detuning (Δc) on the transient response Imðρ13Þin Fig. 8. Other parameters are set as rp ¼ 0:8γ, γ21 ¼ γ, γ31 ¼ 0:1γ,ϕ¼ 0. In reference to the Fig. 5a for ϕ¼ 0, it is observed in Fig. 8athat, for non-zero detuning of probe (Δp ¼ 10γ) using the resonantcoupling laser, the number of loops in temporal dynamicsincreases within the same range of normalized time. In the reverseconditions of probe and coupling detuning (Δp ¼ 0, Δc ¼ 10γ) thesame feature replicates as is seen for Fig. 8a. Further, when we setboth the detunings as equal (Δp ¼ Δc ¼ 10γ), the appearance of themulti-looped structure of atomic response in Fig. 8b remainsalmost same as that shown in Fig. 5a for ϕ¼ 0. But it is interestingto note that the consequent reduction of time interval (withcomplete zero-value in atomic response) between two successiveloops is observed with the application of non-zero detuning of theprobe and coupling lasers.

4. Conclusion

In summary, we analyze the impact of different relevantparameters like the Rabi frequency, detuning and the relativephase of the control fields on transient evolutions of gain,absorption and populations in a closed three-level V-type atomicsystem for both resonant and non-resonant conditions. Differentcombinations of Rabi-frequencies of control fields are chosen to

γt6 8 10 0 2 4 6 8 10

-0.4

-0.2

0.0

0.2

0.4

γt

Im (ρ

13)

(a) rc ¼ rm ¼ γ; (b) rc ¼ rm ¼ 4γ; and (c) rc ¼ rm ¼ 60γ. Other parameters are set as

0 2 4 6 8 10-0.4

-0.2

0.0

0.2

0.4

γt

Im (ρ

13)

of applied fields – (a) Δp ¼ 10γ, Δc ¼ 0 and (b) Δp ¼ Δc ¼ 10γ. Other parameters are

P. Panchadhyayee / Optics Communications 309 (2013) 95–102102

present the nature of transient variation of atomic response. Underresonant condition of probe and control lasers the probe responseshows transparency in steady state after the transient periodicvariation accompanied by the signature of loop-structure. For thethird Rabi frequency-combination the role of the Raman coher-ence is significant in the transient gain-absorption response toachieve inversionless gain in bare atomic state basis. In every caseof ϕ¼ 0 at the condition of exact probe resonance, we mentionthat the loop patterns in atomic response are due to the contribu-tions of two coherence terms (Reðρ23Þ and Reðρ12Þ) associated withthe two-step excitation pathways, while for ϕ¼ π=2, Imðρ23Þ andImðρ12Þ play the vital role in behind. As a whole, the collectivephase modulates the transient dynamics showing transient andstationary values of gain, absorption and transparency withoutinvoking any restriction to the dynamical variation of populations.On controlling radiative properties of the atomic system it isshown that the response time for the atomic medium arriving atthe steady state is prolonged with the decrease in decay rates. Theimpact of probe Rabi frequency on the evolution of the coherenceinduced multi-loop patterns at resonant condition is also exam-ined with equal Rabi frequencies of coupling lasers. In comparisonwith purely resonant case, we can infer that the time interval(with complete zero-value in atomic response) between twosuccessive loops gets reduced by the non-zero detuning appliedonly in any one of the lasers involved in the system.

Acknowledgements

The author gratefully acknowledges Prof. Hong Guo for somehelpful discussion and valuable suggestions. The author alsoacknowledges Prof. Prasanta Kumar Mahapatra and Dr. BibhasKumar Dutta for their valuable suggestions.

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