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Optical properties of an inhomogeneously broadenedΛV-system with multiple excited statesParamjit Kaura, Vineet Bhartib & Ajay Wasana

a Department of Physics, Indian Institute of Technology, Roorkee, India.b Department of Physics, Indian Institute of Science, Bangalore, India.Published online: 26 Jun 2014.

To cite this article: Paramjit Kaur, Vineet Bharti & Ajay Wasan (2014) Optical properties of an inhomogeneously broadenedΛV-system with multiple excited states, Journal of Modern Optics, 61:16, 1339-1347, DOI: 10.1080/09500340.2014.931479

To link to this article: http://dx.doi.org/10.1080/09500340.2014.931479

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Journal of Modern Optics, 2014Vol. 61, No. 16, 1339–1347, http://dx.doi.org/10.1080/09500340.2014.931479

Optical properties of an inhomogeneously broadened �V-system with multiple excited states

Paramjit Kaura∗, Vineet Bhartib and Ajay Wasana

aDepartment of Physics, Indian Institute of Technology, Roorkee, India; bDepartment of Physics, Indian Institute of Science,Bangalore, India

(Received 23 April 2014; accepted 27 May 2014)

We present a theoretical model using a density matrix approach to show the influence of multiple excited states on theoptical properties of an inhomogeneously broadened �V-system of the 87Rb D2 line. These closely spaced multipleexcited states cause asymmetry in absorption and dispersion profiles. We observe the reduced absorption profiles, dueto dressed state interactions of the applied electromagnetic fields, which results the Mollow sideband-like transparencywindows. In a room temperature vapor, we obtain a narrow enhanced absorption and steep positive dispersion at theline center when the strengths of control and pump fields are equal. Here, we show how the probe transmittance varieswhen it passes through the atomic medium. We also discuss the transient behavior of our system which agrees well withthe corresponding absorption and dispersion profiles. This study has potential applications in controllability of groupvelocity, and for optical and quantum information processing.

Keywords: electromagnetically induced transparency; Doppler-broadened system; �V-system; transient properties;absorption and dispersion

1. Introduction

In the last few decades, the phenomenon of electromagnet-ically induced transparency (EIT) has attracted substantialattention in several theoretical and experimental studiesin the three-level atomic systems. EIT is a phenomenon,in which the absorption properties of probe field’s transi-tion modify if a strong control field is applied on anothertransition that shared one of the levels with first transition[1–3]. EIT techniques have many interesting and practicalapplications such as lasing without inversion [4,5],precision magnetometers [6], probe amplification [7],high-resolution spectroscopy [8], optical switching [9,10],enhancement of second- and third-order nonlinearprocesses [11,12], quantum correlations [13,14], reductionof quantum noise [15], storage of light [16], enhancement ofelectro-optical effect [17] and quantum memories [3]. EIThas also a remarkable feature to control the group velocity oflight in a material medium. The group velocity of light canbe made small [18] or large than the speed of light in vacuumor even become negative [19,20]. The slow group velocityor subluminal propagation of light occurs in optical mediawith normal dispersion whereas superluminal propagationof light occurs with anomalous dispersion [21].

Besides the basic three-level EIT configurations, i.e.�-, �- and V-schemes, more complicated EIT systems are

∗Corresponding author. Email: paramdph@iitr.ac.in

possible by the addition of further electromagnetic fields(em) and/or atomic levels. For example, the light propa-gation can be changed from subluminal to superluminal inthe three-level system with three em fields [22], a giantenhancement of the Kerr nonlinearity [23] and a double darkstate [24] arise when a four-level atomic system is drivenby three em fields. In addition to these, there are differenttypes of four-level systems which can be Y-type, invertedY-type and tripod-type systems. These systems have a widerange of applications such as EIT transparency enhance-ment [25], coherent control [26], vacuum-induced interfer-ence [27], multi-wave mixing due to atomic coherence andinterference [28] and realization of quantum phase gate [29].Another type of a four-level system is the N-system whichis formed by applying an extra em field between one of theground states of the three-level �-system and an additionalexcited state. In such a system, an additional dressed state iscreated that results an electromagnetic induced absorption(EIA) peak inside the transparency window of the �-system[30,31].

In this work, we have consider the closed �V-type EITsystem in which two ground states and two excited statesare coupled by three em fields. This �V-type EIT systemis formed by the combination of �- and V- systems witha common probe field. The EIA-like enhanced absorption

© 2014 Taylor & Francis

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1340 P. Kaur et al.

is obtained in a four-level system both theoretically andexperimentally which is resulted from degeneracy of thedressed states [32,33]. The similar type of EIA is obtainedin an open four-level system within a degenerate two-statesystem due to transfer coherence or transfer of populationfrom the excited state to the ground state [34,35]. We areinterested in the �V-type EIT system because it has numer-ous applications, for example, high-resolution spectroscopy[33,36], lasing without population inversion or lasing withinversion [37], controlling the light group velocity [33],marking a slow light signal with a fast optical precursor [38],quantum information processing [39], spatial and temporalcloaking devices and image reshaping [40].

Until now, several theoretical and experimental workson EIT have been done in alkali atoms by assuming onlythree-level system. But, in fact, there are presence of closelyspaced hyperfine levels in these atoms. We cannot ignorethese nearby levels as applied field interacts with them too. Ithas already been studied that the presence of these hyperfinelevels affects the absorption profiles significantly in thevarious configurations of EIT [41–43].

In the present paper, we propose an extensive study tosee the influence of a multiple excited states on the opticalproperties of the �V-type EIT system and it has been com-pared with a similar four-level system. We derive Hamil-tonian and write corresponding density matrix equationsfor this system by using the semi-classical approach. Inthis scheme, we use a fixed control field Rabi frequencywith a variable pump field Rabi frequency. We considerthe case of a stationary as well as a moving atoms andperform thermal averaging of velocities at room temper-ature atomic vapor using Maxwell-Boltzmann distributionfunction. The probe transmittance through medium has alsobeen calculated by using Beer’s law. Further, we have solvedthe time-dependent density matrix equations to investigatethe transient behavior of a six-level EIT system.

This paper is organized as follows: in Section 2, a theoret-ical model is presented by using a density matrix approach.The numerical results of the six-level system are discussedand compared with the similar four-level system in Section3. Finally, Section 4 gives a conclusion of this theoreticalwork.

2. Theoretical model for the �V-type six-level system

We consider a six-level �V-type system in 87Rb interactingwith three em fields as depicted in Figure 1. Our six-levelsystem is formed by considering two hyperfine levels of aground state 52S1/2 and four hyperfine levelsof the excited state 52P3/2. The strong σ+

c - polarized controlfield with a frequency ωc couples the transition |g2〉 ≡|F = 1, m F = −1〉 → |e1〉 ≡ |F = 0, m F = 0〉 with a de-tuning �c = ωc−ωe1g2 .Another weak σ−

p - polarized probefield with a frequency ωp couples the transition |g1〉 ≡|F = 1, m F = 1〉 → |e1〉 ≡ |F = 0, m F = 0〉 with a

6.835 GHz

1

157 MHz

267 MHz

5 2S1/2

5 2P3/2

F3

2

0

1

2

mF values

72 MHz

0 2-1

e1

e2

e3 es

g1g2

}

}

Δc}Δp

Δs

σc+ σs

+σp

_

1

Figure 1. A multilevel �V-system of 87Rb-D2 line with only theconsidered mF values. In the six-level system, |g1〉 and |g2〉 are theground states and |e1〉, |e2〉, |e3〉 and |es〉 are the excited states.The four-level approximation involves only |g1〉, |g2〉, |e1〉 and|es〉 states. (The color version of this figure is included in theonline version of the journal.)

detuning �p = ωp −ωe1g1 . We apply a third σ+s - polarized

pump field with a frequency ωs that drives the transition|g1〉 ≡ |F = 1, m F = 1〉 → |es〉 ≡ |F = 2, m F = 2〉 witha detuning �s = ωs − ωes g1 . Here ω jk = (E j − Ek)/� isdefined as the atomic transition frequency between levels jand k ( j > k) without any perturbation in the ground stateE j .

In an actual atomic system, there are presence of closelyspaced hyperfine levels. So in this system, the ground states|g1〉 and |g2〉 also couple with excited states |e2〉 ≡|F = 1, m F = 0〉 and |e3〉 ≡ |F = 2, m F = 0〉 by σ−

p - po-larized probe and σ+

c - polarized control fields, respectively,due to electric dipole allowed selection rules. This �V-typesystem can also be obtained by combining �- and V- typesystems with a common probe field. Our six-level systemcan be reduced to a simple four-level system if only |g1〉,|g2〉, |e1〉 and |es〉 states are present. This four-level �V-typesystem can be reduced to a three-level � (or V)-system ifwe σ+

s - polarized pump (or σ+c - polarized control) field is

absent.To present the model and equations of motion for our six-

level system, we first write the Hamiltonian for this system.The Hamiltonian of the six-level system under the electricdipole and rotating wave approximation is written as

H = −�

((�P − �c) |g2〉 〈g2| + �s |es〉 〈es |

+3∑

j=1

(�p − ωe j e1)∣∣e j

⟩ ⟨e j

∣∣ )

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Journal of Modern Optics 1341

+ �

2

(�sg1es |g1〉 〈es | +

3∑j=1

(�pg1e j |g1〉⟨e j

∣∣

+ �cg2e j |g2〉⟨e j

∣∣) + c.c

)(1)

where c.c denotes the complex conjugate of the off-diagonalterms of the Hamiltonian and the terms �x jk denotes theRabi frequencies of the applied fields for the transitions| j〉 ↔ |k〉 (with x = p, c and s for probe, control and pumpfields, respectively) [43]. The �∗

x jk is the complex conjugateof the Rabi frequency �x jk .

In the framework of semiclassical theory, the time evolu-tion of the system is governed by the density matrix equa-tions of motion, given by [44]

ρ = − i

�[H, ρ] − 1

2{, ρ} (2)

Here the decay rates are included into Equation (2) throughdecay matrix . By substituting expression (1) into Equa-tion (2), we obtain the following set of coupled densitymatrix equations:

ρg1g1 = −ρg1g1

τd+ 1

2τd+ be1

3e1ρe1e1 + be2

2e2ρe2e2

+ be3

3e3ρe3e3 + bes es ρes es

+3∑

j=1

i

2(�∗

pg1e jρg1e j − �pg1e j ρe j g1 )

+ i

2(�∗

sg1esρg1es − �sg1es ρes g1); (3a)

ρg2g2 = −ρg2g2

τd+ 1

2τd+ be1

3e1ρe1e1 + be2

2e2ρe2e2

+ be3

3e3ρe3e3 +

3∑j=1

i

2(�∗

cg2e jρg2e j − �cg2e j ρe j g2);

(3b)

ρes es = −es ρes es + i

2(�sg1es ρes g1 − �∗

sg1esρg1es ); (3c)

ρe j e j = −e j ρe j e j + i

2(�pg1e j ρe j g1 − �∗

pg1e jρg1e j )

+ i

2(�cg2e j ρe j g2 − �∗

cg2e jρg2e j ); (3d)

ρg1g2 = − (γ + i(�p − �c)

)ρg1g2

+3∑

j=1

i

2(�∗

cg2e jρg1e j − �pg1e j ρe j g2) − i

2�sg1es ρes g2 ;

(3e)

ρg1es =(

−es

2− i�s

)ρg1es + i

2�sg1es (ρg1g1 − ρes es )

−3∑

j=1

i

2(�pg1e j ρe j es ); (3f )

ρg1e j =(

−e j

2− i(�p − ωe j e1)

)ρg1e j

+ i

2�pg1e j (ρg1g1 − ρe j e j ) −

3∑k �= j, k=1

i

2(�pg1ek ρek e j

+ �sg1es ρes e j − �cg2e j ρg1g2); (3g)

ρg2es =(

−es

2+ i(�p − �c − �s)

)ρg2es

−3∑

j=1

i

2(�cg2e j ρe j es ) + i

2�sg1es ρg2g1 ; (3h)

ρg2e j =(

−e j

2− i(�c − ωe j e1)

)ρg2e j

+ i

2�cg2e j (ρg2g2 − ρe j e j ) −

3∑k �= j, k=1

i

2(�cg2ek ρek e j

− �pg1e j ρg2g1); (3i)

ρes e j =(

− (es + e j )

2− i(�p − �s − ωe j e1)

)ρes e j

+ i

2(�pg1e j ρes g1 + �cg2e j ρes g2 − �∗

sg1esρg1e j ); (3j)

ρe j ek =(

− (e j + ek )

2+ iωe j ek

)ρe j ek

− i

2(�∗

pg1e jρg1ek − �pg1ek ρe j g1 + �∗

cg2e jρg2ek

− �cg2ek ρe j g2). (3k)

In above Equations (3a)–(3k), ρ jk is the density matrixelement between a state | j〉 and a state |k〉, and describes thecoherence between these states. Also, ρ j j denotes the popu-lation in the state | j〉. The complex conjugate of coherenceis ρ jk = ρ∗

jk . The branching ratio of the j th level is givenby b j . The spontaneous decay rate of the excited states |e1〉,|e2〉, |e3〉, |es〉 are e1 = e2 = e3 = es = 6.1 MHz,respectively. We have also incorporated the ground-statedecay due to the finite interaction time τd of the atoms withthe applied em fields. Here, we assume that the ground-state decoherence rate is γ = 0.0001e1 which is muchsmaller than the decay rate of the excited states. For ournumerical calculations, we consider that the atom’s lossrate is much smaller than the radiative decay rates of theexcited states, τ−1

d e1,e2,e3,es . Therefore, we assumeτd ≈ 300 µs which corresponds to the interaction time ofatom with applied fields of a few millimeters in diameter forall temperature [42,43]. We also assume that rate at whichnew atoms are entering the interaction region is equal to therate at which atoms are leaving the interaction region. Thisrate is included by the term 1/2τd in Equations (3a) and (3b).The matrix elements of dipole moments and correspondingRabi frequencies of all the transitions in our six-level systemare calculated by using the Clebsch-Gordan coefficient forhyperfine levels [43,44].

We have considered that our six-level system is closedsince all of atom from 52P3/2 decay to the ground state52S1/2 with unit branching ratio. So, closure of the systemrequires that

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1342 P. Kaur et al.

ρg1g1 + ρg2g2 + ρes es +3∑

j=1

ρe j e j = 1. (4)

We have solved the above density-matrix equations nu-merically for a steady-state solution (ρ = 0) using theweak probe, strong control and pump fields. Initially, allthe sub-levels of ground state are equally populated. Butunder the weak probe conditions, all the atoms will trap to

state |g1〉, i.e. ρg1g1 ≈ 1 and ρg2g2 , ρes es ,

3∑j=1

ρe j e j ≈ 0.

These trapped atoms can only be excited to the state |es〉and form a closed (or cyclic) transition between states |g1〉and |es〉. The dispersion and absorption coefficients for theprobe field couple the transition |g1〉 → |e1〉 is proportionalto Re(ρg1e1) and Im(ρg1e1), respectively.

3. Results and discussion

In this section, we discuss our numerical results for theprobe absorption and dispersion of the six-level �V-EITsystem. We show these numerical results for a stationary aswell as for a moving atoms and compare these results withsimilar four-level system. For a moving atom, we performthermal averaging at the room temperature to see the effectof all velocities groups on probe absorption and dispersionprofiles. Finally, we also discuss the effect of length ofmedium on the probe transmittance and transient behaviorof our system.

3.1. Stationary atom

We show the numerical solution of the imaginary and realparts of ρg1e1 versus probe field detuning �p for four- andsix-level systems with fixed value of control and variousvalues of pump Rabi frequencies in Figure 2. The solidcurves are for the six-level system and the dashed curvesare for the respective four-level system.

First, we discuss our numerical results for four-level sys-tem. If we apply a control field that is resonant with thecorresponding transition |g2〉 → |e1〉 by considering thepump field Rabi frequency zero (�sg1es = 0), our four-level system reduces to a well-known three-level � system.Our theoretical calculations show that the absorption splitsinto an Autler-Townes doublet [45] and shows a classictransparency window at the line center as shown in Figure2(a). These doublet peaks are two symmetric dressed statescreated by the control field [46]. Now if we further increasethe control field Rabi frequency (�cg2e1 ), the separationbetween these peaks, i.e. the linewidth of transparency win-dow increases. In this case, the doublet is generated becausethe resonant control field dresses levels |g2〉 and |e1〉 intothe superpositions of these two states. The correspondingslope of the probe dispersion curve is positive near �p = 0.The positive slope of the probe dispersion shows slow light

propagation or subluminal light as shown in Figure 2(b).Now we switch on the pump field and apply a fixed controlfield Rabi frequency, i.e. �cg2e1 = 12 MHz, each absorptionpeak further splits into a symmetric pair (or four absorptionpeaks appear), shown in Figure 2(c). These symmetric pairof reduced absorption across the line center are knownas Mollow sideband-like transparency windows [32]. Theexistence of these transparency windows may be usefulfor optical and quantum information processing such asquantum gate operation and multichannel all-optical buffermemory [32]. The creation of these four absorption peakscan be understood with dressed state picture of the four-level system. The interaction Hamiltonian for the four-levelsystem can be written using Equation (1) as:

HI = �

2

(�sg1es |g1〉 〈es | + �cg2e1 |g2〉 〈e1| + c.c

)(5)

The eigenvalues of the interaction Hamiltonian are

λc± = ± �cg2e1

2(6a)

λs± = ± �sg1es

2(6b)

These eigenvalues are corresponding to four dressedstates and the energy separation (λc± − λs±) between thesestates by the interaction of the control and pump fields isalso known as dressed state transitions. The four absorp-tion peaks are the four dressed states transitions. When weapply �sg1es = 12 MHz which is equal to the control fieldRabi frequency, our results show the enhanced absorptionat the zero probe detuning as shown in Figure 2(e). Thisenhanced absorption attributed to the fact that separationbetween two center peaks is zero, i.e. two dressed states be-come degenerate at the line center. This EIA-like enhancedabsorption arises because of the degenerate dressed stateinteractions. In this case (�cg2e1 = �sg1es ), our results showthe negative slope with enhanced absorption which leadsto fast light propagation or superluminal light as shown inFigure 2(f).As the pump field increases further, i.e. �sg1es =18 MHz, the enhanced probe absorption at the line centersplits into a doublet and we again obtain a transparency win-dow. The corresponding probe dispersion becomes normaland shows a positive slope near �p = 0. Hence, there is anormal dispersion which leads to a subluminal light. Furtherincreasing the pump field successively, the linewidth ofthe induced EIT window increases and concurrently, theprobe absorption becomes more positive and less steeper at�p = 0. Our calculations show that as the Rabi frequencyof the pump field increases, the subluminal phenomenonbecome more obvious.

Now we extend our discussion to the numerical results ofthe six-level system. Our numerical calculations show thatthe absorption and dispersion profiles become asymmetricdue to the presence of multiple excited states, as shown inFigure 2. These absorption and dispersion profiles are alsoshifted to the negative probe detuning. This asymmetric

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Journal of Modern Optics 1343

behavior and shift of profiles arise because our six-levelsystem has three �V channels with different dipole mo-ments for various transitions.

3.2. Thermal averaging for a moving atom

The analysis discussed in the previous section is correct fora stationary atom. As we know that most of the EIT experi-ments are performed in the room temperature atomic vaporso we have to account for the thermal velocity distribution.Previously, it has been shown for a three-level�-system thatthe overall transparency window shrinks and the effective

linewidth decreases after carrying out the thermal averagingat some temperature [47,48]. It has also been shown thatthe transparency window for a �-system disappears totally[42] and it increases a bit more for a �- system [43], if themultiple excited states have taken into account in these sys-tems. Recently, we have seen that the overall transparencywindow in V-type EIT system remains broad even aftercarrying out thermal averaging at room temperature [49].In another study, Bharti et al have theoretically shown thatin the room temperature atomic vapor, the enhanced absorp-tion of an N-system at the line centre starts to disappear with

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 2. The probe absorption Im(ρg1e1 ) and dispersion Re(ρg1e1 ) coefficients for atoms as a function of probe detuning (�p) in thefour-level (blue-dashed line) and six-level (red solid line) systems for various values of �sg1es with fixed value �cg2e1 . In this calculation,the control and pump field detunings are �c = �s = 0. (The color version of this figure is included in the online version of the journal.)

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1344 P. Kaur et al.

(a) (b)

(c) (d)

Figure 3. The Doppler-broadened probe absorption and dispersion as a function of probe detuning (�p) at room temperature 297 K. Thered solid and blue-dashed lines are for the six-level and four-level systems, respectively. All the graphs are normalized by the same factor.The reduction in the overall scale of absorption and dispersion at room temperature is due to the absorption spreads over all the velocityclasses. (The color version of this figure is included in the online version of the journal.)

a decrease in hyperfine separation [31]. In this section, weare interested to see the effects of a finite temperature on theprobe absorption and dispersion of the �V-type EIT systemwith multiple excited states.

To make our calculations Doppler-free, we consider anatom with velocity v interacting with the co-propagatingprobe, control and pump fields [32]. The Doppler shifts(�D) for probe, control and pump fields are given by±v/λp,±v/λc and ±v/λs , where λp, λc and λs are the wavelengthsof the probe, control and pump fields, respectively. The sign± depends on whether the atom is moving away from ortoward the applied fields. To perform thermal averagingover all the velocities, we use a Maxwell-Boltzman veloc-ity distribution at the room temperature (297 K). The mostprobable velocity corresponding to the room temperature is237.52 ms−1.

Figure 3 shows the corresponding absorption and dis-persion curves versus probe field detuning �p with thefixed value of control Rabi frequency and different valuesof pump Rabi frequencies. The solid curve is for the six-level system and the dashed curve is for the four-levelsystem. First we consider the four-level system, when thepump field is turned off, i.e. �sg1es = 0 and the controlfield is on resonance (�c = 0), then this system reducesto a three-level �-type EIT system as we have alreadydiscussed. Our result shows a narrowing of transparencywindow at the line center as shown in Figure 3(a). The

slope of the positive dispersion becomes steep at �p = 0due to summing over all atom’s velocities contributions asshown in Figure 3(b). When the pump field is turned onand the Rabi frequencies of the pump and control fields areequal, the enhanced absorption at the line center becomesnarrower as shown in Figure 3(c). The corresponding slopeof the dispersion curve becomes steep and negative at thezero probe detuning as shown in Figure 3(d). One clearlysees that the slope of the dispersion curve changes frompositive to negative and the normal dispersion turns to ananomalous dispersion as the Rabi frequency of the pumpfield increases.

In case of six-level system, when �sg1es = 0 and�cg2e1 = 12 MHz, the transparency window and corre-sponding positive slope of the dispersion curve disappearas shown in Figure 3(a) and (b). Further, when the Rabi fre-quencies of both the fields are equal, i.e. �sg1es = �cg2e1 =12 MHz, the asymmetric enhanced absorption and negativeslope of the dispersion curve at the zero probe detuningdrop and are less than the four-level system as shown inFigure 3(c) and (d). Besides this, both the probe absorp-tion and dispersion curves shift towards the negative probedetuning. This significant modifications in the absorptionand dispersion profiles are due to the presence of multipleexcited states in our system. Our numerical calculationsshow that both the absorption and dispersion are reducedsignificantly after carrying out the thermal averaging.

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Journal of Modern Optics 1345

(a)

(b)

Figure 4. Probe field transmittance t for a Doppler-broadened (a)four-level system and (b) six-level system at room temperaturewith equal strength of the control and pump fields i.e.�cg2e1 = �sg1es = 12 MHz. The red solid, blue-dashed and blackdash-dotted curves are for the length of the medium 50 mm,100 mm and 150 mm, respectively. In this calculation, the controland pump field detunings are �c = �s = 0. (The color version ofthis figure is included in the online version of the journal.)

3.3. Probe transmittance through the medium

In this section, we discuss the probe transmittance throughthe medium of a particular length. The transmittance (t) ofa weak probe field is given by Beer’s law [42]:

t = exp

[−8π2L

λpI m(ρg1e1)

](7)

where L is the length of the medium. We have assumedthat the absorption line shape has a Gaussian form. Figure4 shows the probe transmittance t as a function of probedetuning �p for various values of L with fixed values of�sg1es and �cg2e1 . The red solid, blue-dashed and blackdash-dotted curves are for the length of the medium 50 mm,100 mm and 150 mm, respectively. Our results for four-level system show that the maximum probe transmittance isobtained for L = 50 mm as shown in Figure 4(a). When wefurther increase the length of medium (L = 100 and 150 mm),the probe transmittance decreases. Our calculations demon-strate that as the length of the medium increases, the probetransmittance decreases and the transparency windowbecomes broader.

In case of six-level system, the peak transmittance be-comes asymmetric as compared to four-level system and

Figure 5. Time evolution of (a) Im(ρg1e1 ) and (b) Re(ρg1e1 )for six-level system. The strength of control field is �cg2e1 =12 MHz. The blue-dashed, black dash-dotted, red solid andgreen-dotted curves are for �sg1es = 0, 6, 12 and 18 MHz,respectively. (The color version of this figure is included in theonline version of the journal.)

slightly shifts toward the negative probe detuning shown inFigure 4(b). Our calculations show that to obtain the narrowand maximum transmittance, we should optimize the lengthof vapor cell.

3.4. Transient behavior

We now discuss transient behavior of the six-level EITmedium in the presence of three em fields. The transientproperties are very useful in the applications of the op-tical switching of EIT-based systems and have potentialapplications in optical information processing and trans-mission [50]. Figure 5 gives the time-dependent probe fieldabsorption and dispersion profiles of the considered six-level system for various values of �sg1es with fixed valueof �cg2e1 = 12 MHz. We solve our time-dependent densitymatrix equations to show the time evolution of Im(ρg1e1)

and Re(ρg1e1). First, we discuss the transient behavior ofthe Im(ρg1e1), i.e. the probe absorption as shown in Figure5(a). For a six-level system, in the presence of the controlfield, when the pump field is switched off, the probe ab-sorption has a transient oscillatory behavior in initial time,of the order of few microseconds, and it decays rapidlyto zero. The physical reason behind this vanishing probe

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1346 P. Kaur et al.

absorption is the creation of Aulter-Townes doublet thatlargely suppresses the probe field absorption. When thepump field is on, the probe absorption becomes steady in afew microseconds. Now if both the Rabi frequencies of thecontrol and pump fields are equal, then there is a maximumenhanced absorption that reaches the steady state in a fewmicroseconds.

Now we investigate the transient behavior of the Re(ρg1e1),i.e. the probe dispersion as shown in Figure 5(b). The dis-persion properties of the probe field are very useful for itspotential applications of controlling the group velocity oflight from subluminal to superluminal. When only controlfield is on, then the positive dispersion has a oscillatingbehavior in small time interval and it finally reaches to asteady state as the time increases. When both the appliedfields are on, i.e. �sg1es = �cg2e1 = 12 MHz, the dispersionhas oscillatory behavior for a few microseconds and reachesto a negative value at the steady state as time increases. Thistransient behavior can be explained by the probe absorptionand dispersion profiles shown in Figure 2. Our numericalcalculation for the transient response shows that our six-level EIT medium exhibits normal dispersion (or sublumi-nal light) when �sg1es �= �cg2e1 and exhibits anomalousdispersion (or superluminal light) when �sg1es = �cg2e1 .

4. Conclusion

We have theoretically investigated the influence of multipleexcited states on the optical properties of �V-type EITsystem. Our numerical calculations demonstrate that thesemultiple excited states lead to a asymmetric absorption anddispersion profiles as compare with a four-level system.

In the case of a stationary atom, we observe a sym-metric pair of reduced absorption across the line centerwhich are known as Mollow sideband-like transparencywindows. These transparency windows may be useful in op-tical and quantum information processing, e.g. multichan-nel all-optical buffer memory and quantum gate operation.When Rabi frequencies of both the control and pump fieldsare equal, the enhanced absorption is obtained at zero probedetuning, i.e. our system shifts from the transparency win-dow to the enhanced absorption. This enhanced absorptionarises due to the degenerate dressed states.

In a Doppler-broadened system, an enhanced absorptionbecomes narrow after carrying thermal averaging at roomtemperature vapor. Concurrently, the dispersion curve be-comes steep and negative due to summing over all thevelocities contributions. In addition to this, it is found thatthe probe transmittance through medium reduces as thelength of the medium becomes large.

We have also discussed the transient behavior of the six-level system by solving the time-dependent density matrixequations. Our numerical results of the transient behaviorshow that maximum absorption arises for balanced Rabifrequencies of the pump and control fields. We have also

demonstrated that EIT medium exhibits normal dispersionwhich leads to a subluminal of light when �sg1es �= �cg2e1

and anomalous dispersion which leads to a superluminal oflight when �sg1es = �cg2e1 . This variation of the dispersionwith the strength of pump field may be used for controllingthe group velocities of light.

AcknowledgementsPK is thankful to the Ministry of Human Resource Development(MHRD), India for the financial assistance.

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