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Realization of a negative refractive index in a three-level Λ system via spontaneously

generated coherence

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2010 J. Phys. B: At. Mol. Opt. Phys. 43 215503

(http://iopscience.iop.org/0953-4075/43/21/215503)

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IOP PUBLISHING JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS

J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 215503 (7pp) doi:10.1088/0953-4075/43/21/215503

Realization of a negative refractive indexin a three-level Λ system viaspontaneously generated coherence

Sulagna Dutta and Krishna Rai Dastidar

Department of Spectroscopy, Indian Association for the Cultivation of Science, Kolkata 700 032, India

E-mail: sulagna dutta@yahoo.co.in and krishna raidastidar@yahoo.co.in

Received 5 May 2010, in final form 30 July 2010Published 20 October 2010Online at stacks.iop.org/JPhysB/43/215503

Abstract

A new scheme to realize simultaneous negative permittivity and permeability in a three-levelclosed � system with incoherent pumping via spontaneously generated coherence has beenpresented. We have shown that the negative refractive index can be realized in a heteronuclearmolecule such as LiH. The negative refractive index can be achieved in a band of frequencyrange. The position and the band of the frequency region of negative refraction can bemanipulated by controlling the incoherent pump rate. We achieve a figure of merit(FOM) = |Re(n)/Im(n)| = 1.797 for the LiH molecule.

1. Introduction

Recently there has been considerable interest in the study of anew material called a left-handed medium (LHM) possessingnegative real parts of both the dielectric permittivity εr andthe magnetic permeability μr over a certain frequency band[1–11]. The most well-known feature belonging to the LHMis the negative refraction denoted by the real part of an indexn = −√

εrμr [1]. These left-handed media exhibit a numberof peculiar electromagnetic and optical effects including thereversal of both Doppler shift and Cherenkov radiation [1],anomalous refraction [1], amplification of evanescent waves[3], subwavelength focussing [3, 5], a negative Goos–Hanchenshift [6], a reversed circular Bragg phenomenon [7], photonhelicity inversion [8], some unusual photon tunnelling effects[9], reversed H field circulation patterns and inverted E fieldlines in propagating structures [10] and switched field intensitylocations in anisotropic transmission structures [11].

Based on the classical electromagnetic theory, there areseveral ways to realize the LHM, including artificial compositemetamaterials [12–15], photonic crystal structures [16–18],transmission line simulation [19] and chiral material [20–23].These various approaches for the fabrication of the LHMrequire delicate manufacturing of spatially periodic structures.There is another scenario based on quantum interferenceand coherence effect to realize the negative refractive index.Oktel et al and Shen et al first proposed that the negativereal parts of both the dielectric permittivity εr and the

magnetic permeability μr over a certain frequency band canbe simultaneously achieved in a coherently driven three-levelatomic vapour [24, 25]. Thommen and Kastel suggested howthe LHM can be realized in an atomic four-level system, whichmakes the scheme more realistic and applicable to any realatomic system [26, 27]. Recently Liu et al [28] have shownthat left-handed properties can be electromagnetically inducedin a �-type four-level scheme on the Er3+:YAlO3 crystal basedon the scheme proposed by Thommen et al and Kastel et al[26, 27]. Zhang et al proposed a scheme for realization ofthe negative refractive index in a V-type four-level atomicsystem [29]. They proposed a new scheme to switch amaterial from positive to negative index by modulating therelative phase of the applied fields in a four-level atomic system[30]. Recently Krowne et al presented a scheme to realize anegative refractive index employing dressed-state mixed paritytransitions of atoms [31].

Previously we have proposed a scheme for experimentalrealization of electromagnetically induced transparency(EIT) considering spontaneously generated coherence (SGC)[32–37] in a heteronuclear molecular system such as LiH[36]. In this paper, we show that the negative refractiveindex can be achieved in this molecular system under suitableconditions in the presence of SGC. We have shown beforethat one can achieve EIT in a three-level � system due tothe destructive interference of absorptions induced by SGCand dynamically induced coherence (DIC) [34]. We showhere that an electromagnetically induced transparent system

0953-4075/10/215503+07$30.00 1 © 2010 IOP Publishing Ltd Printed in the UK & the USA

J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 215503 S Dutta and K Rai Dastidar

3

2

1

CouplingLaser Ω

IncoherentPump 2Λ

Probe LaserGΔ1

Δ2

2γ132γ12

(a) (b)

pEcE

13d12d

θ

Figure 1. (a) Schematic diagram for a � transition scheme. (b) The arrangement of field polarization required for a single field driving onetransition if dipoles are non-orthogonal.

[38, 39] realized in dilute LiH medium can exhibitsimultaneously a negative dielectric permittivity εr and anegative magnetic permeability μr over a certain opticalfrequency band. Thus one can realize a double-negativenegative index medium (DN-NIM) in a dilute LiH gaseousmedium. The frequency band of the negative refractioncan be manipulated by controlling the incoherent pump rate.The figure of merit (FOM) (|Re(n)/Im(n)|) of the LiHmolecular system has been calculated for different values ofthe incoherent pump rate and FOM has been found to be greaterthan unity. To show the importance of the effect of SGC forachieving simultaneous negative values of Re(εr) and Re(μr),we have calculated Re(εr) and Re(μr) in the absence of SGCin the LiH molecule.

This paper is organized as follows: in section 2, we presentour model and the expressions for the electric permittivityand magnetic permeability. In section 3, we discuss how thenegative refractive index can be realized in the heteronuclearmolecule and also present the analytical results and theirdiscussions for the LiH molecule. Finally conclusions aredrawn in section 4.

2. Theoretical model

In the �-scheme (figure 1(a)), a coherent coupling laser offrequency ωc with Rabi frequency �0 is applied between thestates |1〉 and |2〉, while a probe laser of frequency ωp withRabi frequency G0 is applied between the states |1〉 and |3〉.A bidirectional incoherent field is applied between the level|3〉 and the level |1〉 at a rate 2�. Spontaneous decay ratesfrom the level |1〉 to the levels |2〉 and |3〉 respectively are 2γ12

and 2γ13. Since the existence of the SGC effect depends onthe nonorthogonality of the electric dipole transition moments−→d 13 and

−→d 12, we have to consider an arrangement shown

in figure 1(b) where one field acts on only one transition,where θ represents the angle between the two induced dipoletransition moments. Here the two lower levels have the sameparity so that there exists a non-zero magnetic dipole moment−→μ 23 = 〈2|μ|3〉.

The density matrix equation of motion in the rotating-wave approximation and electric dipole approximation can bewritten as

ρ11 = i�cρ21 − i�∗cρ12 + iGpρ31 − iG∗

pρ13

− 2(γ12 + γ13 + �)ρ11 + 2�ρ33 (1)

ρ22 = i�∗cρ12 − i�cρ21 + 2γ12ρ11 (2)

ρ33 = iG∗pρ13 − iGpρ31 + 2(γ13 + �)ρ11 − 2�ρ33 (3)

ρ12 = i�c(ρ22 − ρ11) + iGpρ32 − (γ13 + γ12 + � + ic)ρ12

(4)

ρ13 = iGp(ρ33 − ρ11) + i�cρ23 − (γ12 + γ13 + 2� + ip)ρ13

(5)

ρ23 = i�∗cρ13 − iGpρ21 + [i(c − p) − �]ρ23

+ 2√

γ13γ12 cos θη0ρ11. (6)

The above equations are constrained by ρij = ρ∗ij and

ρ11 +ρ22 +ρ33 = 1. Here Rabi frequencies are denoted as �c =−→d 12 · −→

E c/h = �0 sin θ and Gp = −→d 13 · −→

E p/h = G0 sin θ ,where �0 and G0 are the Rabi frequencies without SGC. Thedetunings are c = ω12 − ωc and p = ω13 − ωp. Here η0

will be zero (one) if the SGC effect is ignored (considered).Usually systems with well-separated levels do not depend

on the relative phase between the two applied fields. Butin the case of closely spaced levels, the � system becomesquite sensitive to phases of the two coupling fields due to theexistence of the SGC term [33]. So we have to treat the Rabifrequencies as complex parameters. If we define φp and φc asthe phases of the probe and the coherent field respectively, thenwe get �c = �e−iφc , Gp = Ge−iφp , ρii = ρii , ρ12 = ρ12eiφc ,ρ13 = ρ13eiφp and ρ23 = ρ23eiφ , where φ = φp − φc. Thuswe obtain equations in redefined density matrix elements ρij

which are found to be identical to equations (1)–(6), with theSGC parameter η0 replaced by ηφ = η0eiφ , Gp replaced byG and �c replaced by �, where G and � are treated as realparameters.

Here the electric-dipole transition (|3〉 ↔ |1〉) and themagnetic-dipole transition (|2〉 ↔ |3〉) will produce theelectric polarizability and the magnetic susceptibility for the

2

J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 215503 S Dutta and K Rai Dastidar

probe laser respectively. In this case the atomic/molecularelectric polarizability for the probe field is

αe = 2d13ρ13

ε0Ep

(7)

and the atomic/molecular magnetic polarizability for the probefield is

αm = 2μ23ρ23

Hp

. (8)

Usually in the atomic/molecular system, the effect of anelectromagnetic wave on the magnetic permeability is weakerthan its effect on the electric susceptibility (μ23/cd13 ≈ 10−2)[24, 25]. Thus we can neglect the magnetic dipole transitionon the wave propagation in such systems. But if the magnitudeof the polarization ρ23 can be tuned to be larger than that ofthe polarization ρ13 by inducing an additional coherence SGCbesides DIC, then there may be a possibility of having thesame order of electric permittivity and magnetic permeability.

If we consider a dilute gas, the electric and magneticsusceptibility can be defined as

χe,m = Nαe,m, (9)

where N denotes the atomic density. In the case ofdilute gas, local field effect can be neglected and hencewe have not considered the Clausius–Mossotti relation here[40].

The relative permittivity and relative permeability aredefined as

εr = 1 + χe (10)

μr = 1 + χm. (11)

In our previous work [34, 35], we have analyticallysolved the density matrix equations (1)–(6) in the steadystate limit without any approximation, i.e. keeping all theorders of both the probe field and coherent field Rabifrequencies, spontaneous decay rates and incoherent pumpingand obtaining the exact analytical steady state value of thepolarizations ρ13 and ρ23. Using exact analytical expressionsof Re(ρ13) and Im(ρ13) from equations (7) and (27) of [35],the analytical values of the real and imaginary part of εr canbe calculated.

The real part of εr is given by

Re(εr) = 1 +2Nd13Re(ρ13)

ε0Ep

(12)

and the imaginary part of εr is given by

Im(εr) = 2Nd13Im(ρ13)

ε0Ep

. (13)

Using Hp = √εrε0/μrμ0Ep, we obtain the relative

permeability as

μr = 1 + ξ

(u2

23 − v223

)(1 + χ ′

e) + 2u23v23χ′′e

(1 + χ ′e)

2 + χ ′′2e

+ iξ2u23v23(1 + χ ′

e) − (u2

23 − v223

)χ ′′

e

(1 + χ ′e)

2 + χ ′′2e

, (14)

where u23 = Re(ρ23), v23 = Im(ρ23), χ ′e = Re(χe), χ ′′

e =Im(χe) and ξ = 4μ0N

2μ223

/ε0E

2p.

Using the above relations, we will analyse theelectromagnetic property of the medium.

3. Realization of a negative refractive index in aheteronuclear molecule

Using the analytical solutions of density matrix equations inthe steady state limit as mentioned in the previous section, wehave calculated the dielectric permittivity (εr ), the magneticpermeability (μr ) and the refractive index (n) to demonstratehow the negative refractive index can be achieved in dilute LiHgases. In a comparative study on NIM, Otkel et al have shownthat for dense gases (N = 1024 m−3) local field effects playa significant role, whereas for lower density (N = 1020 m−3)these effects are negligible. Subsequently Shen et al studiedthe negative refractive index ignoring the local field effects forN = 1027 m−3. In this paper we have presented the resultsfor negative refraction in the dilute LiH molecular system(N ∼ 5 × 1022 m−3) neglecting the local field effects.

We have shown previously [36] how SGC in moleculescan be invoked by an external field and EIT can be achievedfrom the destructive interference of absorptions due to DICand SGC. Here we show that the presence of SGC is importantto realize negative refraction and the small absorption due toEIT facilitates the increase in the value of FOM.

Let us apply a laser field between X1�+(v = 0, j = 0)

and X1�+(v = 0, j = 1) states of the LiH molecule. It thenproduces dressed states

|+〉 = sin ψ |j = 0〉 + cos ψ |j = 1〉 (13a)

|−〉 = cos ψ |j = 0〉 − sin ψ |j = 1〉 (13b)

with cos ψ = 12 + L

2S ′ , where L is the detuning of thelaser frequency from the molecular transition frequency.The two dressed states are separated by an interval S ′h =h

√S2 + 2

L, where S is the Rabi frequency of the transition.For heteronuclear molecules, which have permanent electricdipole moments, one can couple two rovibrational levels ofthe same electronic state by a single coupling laser. Forhomonuclear molecules this coupling can be invoked byRaman transition through an excited electronic state.

To configure a three-level �-system these two dressedstates (|+〉 and |−〉) are used as two lower levels and coupledto an excited state A1�+(v = 0, j = 2) by coherent andprobe fields respectively (see figure 2). We find that the dipolematrix elements between dressed states and the upper excitedlevel are d2+ = d12 cos ψ and d2− = −d12 sin ψ , where d12 isthe dipole transition moment between A1�+(v = 0, j = 2)

and X1�+(v = 0, j = 1) states. Thus, the system withthe upper state A1�+(v = 0, j = 2) and the lower dressedstates |+〉, |−〉 behaves as a � system with antiparallel electricdipole transition moments. The two lower levels have the sameparity so that there exists a non-zero magnetic dipole moment−→μ +− = 〈+|μ|−〉.

Figure 3(a) shows the variation of real parts of relativepermittivity (dashed line) and relative permeability (solidline) with detuning p/γ− in the LiH molecule, keeping� = 0.906γ− (16.54 kHz), cos ψ = 0.5, cos φ = 1,γ− = 4.413 75 × 10−13 au (18.25 kHz), γ+ = 1.471 46 ×10−13 au (6.08 kHz), η = η0 cos θ = −1, c = 0, d12 =3.76 × 10−2 au (9.556 Debye), � = 0.2266γ− (4.13 kHz) and

3

J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 215503 S Dutta and K Rai Dastidar

ω12 ω2+

−

+

2=j

CouplingLaser Ec

IncoherentPump 2Λ

Probe LaserEp

Δc

Δp

2γ-2γ+

−

+

ω2−

)2,0(1 ==+ jvA

)1,0(1 ==+ jvX

)0,0(1 ==+ jvX

'S

(a) (b)

Figure 2. (a) Laser-induced � system with non-degenerate transitions. A laser field applied to the X1�+(v = 0, j = 0) andX1�+(v = 0, j = 1) transition creates non-degenerate dressed states separated by S ′h, where S ′ is the Rabi frequency for the abovetransition. (b) The system with the upper state A1�+(v = 0, j = 2) and the lower dressed states |+〉, |−〉 behaves as a � system withantiparallel dipole transition moments.

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

-1

0

1

2

3

4

5

Re(

μ r) an

d R

e(ε r)

Probe detuning-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8Im

(εr)

Probe detuning(a) (b)

Figure 3. (a) Dependence of real parts of εr (dashed line) and μr (solid line) upon p/γ− in the LiH molecule for � = 0.906γ−(16.54 kHz), cos ψ = 0.5, cos φ = 1, γ− = 4.413 75 × 10−13 au (18.25 kHz), γ+ = 1.471 46 × 10−13 au (6.08 kHz), η = η0 cos θ = −1,c = 0, d12 = 3.76 × 10−2 au (9.556 Debye), � = 0.2266γ− (4.13 kHz) and G = 0.391 96γ− (7.15 kHz), N ∼ 5 × 1022 molecules m−3.(b) Dependence of imaginary part of εr upon p/γ−. (1 au of permittivity = 1.112 65 × 10−10 Farad m−1, 1 au of permeability =6.692 × 10−4 Henry m−1.)

G = 0.391 96γ− (7.15 kHz), N ∼ 5 × 1022 molecules m−3.The real part of εr is negative for p � −0.145γ−. But the realparts of εr and μr are simultaneously negative in the detunedfrequency band [−0.187γ−,−0.145γ−] (see figure 3(a)). Inthis frequency band, the system would become a LHM. Infigure 3(b), the variation of Im(εr) with the probe detuninghas been shown. It shows the EIT profile which occurs duethe destructive interference of absorptions induced by DIC andSGC [34].

Figures 4(a) and (b) show the variation of real parts ofrelative permittivity and relative permeability as a functionof p/γ− respectively for different values of the incoherentpump rate in the LiH molecule, keeping other parameters thesame as in figure 3. In these two figures, the dotted line,the dashed line and the solid line represent the variation ofreal part of relative permittivity and relative permeability for

� = 0.68γ− au (12.40 kHz), � = 0.793γ− au (14.47 kHz)and � = 0.906γ− au (16.54 kHz) respectively. The results for� = 0.906γ− au shown in figure 3(a) have also been presentedin these figures for comparison. In figure 4(a), the real partof εr is negative for p � −0.0699γ−, p � −0.0991γ−for � = 0.68γ− au and � = 0.793γ− au respectively. Therelative permittivity becomes steeper for a smaller value ofthe incoherent pump rate. In the inset of figure 4(a), we plotthe variation of Re(εr) as a function of p/γ− for differentvalues of the incoherent pump rate with η = 0, i.e. in theabsence of SGC. The dotted line, the dashed line and thesolid line in the inset of figure 4(a) represent the variation ofrelative permittivity with probe detuning for � = 0.68γ− au,� = 0.793γ− au and � = 0.906γ− au respectively. The realpart of εr becomes negative for p � 0.858γ−, p � 0.999γ−

4

J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 215503 S Dutta and K Rai Dastidar

-0.3 -0.2 -0.1 0.0 0.1 0.2-3

-2

-1

0

1

2

3

4

-6 -4 -2 0 2 4 6-6

-4

-2

0

2

4

6

8R

e(ε r)

probe detuning

probe detuning

Re(

ε r)

-0.3 -0.2 -0.1 0.0 0.1-8

-6

-4

-2

0

2

4

6

8

10

12

-6 -4 -2 0 2 4 61.975

1.980

1.985

1.990

1.995

2.000

2.005

2.010

2.015

2.020

2.025

Re(

μ r)

probe detuning

Re(

μ r)

probe detuning(a) (b)

Figure 4. Dependence of (a) real part of εr and (b) real part of μr upon p/γ− for � = 0.68γ− au (12.40 kHz) (dotted line),� = 0.793γ− au (14.47 kHz) (dashed line) and � = 0.906γ− au (16.54 kHz) (solid line) in the LiH molecule. Other parameters are kept thesame as in figure 3. In the insets of (a) and (b) the dependence of the real parts of εr and μr upon p/γ− for η = 0 has been plotted.

and p � 1.187γ− for � = 0.68γ− au, � = 0.793γ− au and� = 0.906γ− au respectively.

From figure 4(b) it is evident that the frequency bandof the negative value of Re(μr) changes dramatically withthe incoherent pump rate. For � = 0.68γ− au and � =0.793γ− au, Re(μr) is negative within the detuned frequencyranges [−0.127γ−,−0.0669γ−] and [−0.155γ−,−0.0991γ−]respectively. It is found from figure 4(a) that in thesefrequency bands Re(εr) is also negative. Thus the realizationof simultaneous negative values of the real parts of both thedielectric permittivity εr and the magnetic permeability μr ispossible in the LiH molecular system and this system willbehave as a LHM in these frequency bands. These negativefrequency bands shift towards the resonance as the incoherentpump rate is decreased. In the inset of figure 4(b), we plotthe variation of Re(μr) as a function of p/γ− for differentvalues of the incoherent pump rate with η = 0, i.e. in theabsence of SGC. The dotted line, the dashed line and thesolid line in the inset of figure 4(b) represent the variation ofrelative permeability with probe detuning for � = 0.68γ− au,� = 0.793γ− au and � = 0.906γ− au respectively. For thesevalues of the incoherent pump rate, the relative permeabilityis always positive in the frequency range [−6γ−, 6γ−]. Itexhibits a steep variation over a narrow frequency band in thevicinity of the resonance and its value remains close to 2 forall other detuning values. Thus it can be inferred from theinsets of figures 4(a) and (b) that we cannot simultaneouslyrealize negative real parts of both the dielectric permittivityεr and the magnetic permeability μr for this LiH molecularsystem in the absence of SGC. However the SGC effect greatlyinfluences the magnetic response of the system and one canrealize a negative value of Re(μr) for the negative detuningas shown in figure 4(b). From figures 4(a) and (b), we havefound that the frequency band for the negative value of Re(μr)

is less than that of Re(εr) at each value of the incoherentpump rate. Therefore the frequency band for which bothRe(εr) and Re(μr) are negative is limited by the range of thefrequency band for which Re(μr) is negative at each value ofthe incoherent pump rate. It can be stated that the position andthe magnitude of the negative values of Re(μr) and Re(εr) canbe manipulated by controlling the incoherent pump rate in thepresence of SGC for the LiH molecule.

Figures 5(a) and (b) show the variation of the real andimaginary parts of the refractive index as a function ofp/γ− respectively for different values of the incoherentpump rate in the LiH molecule. In these two figures, thedotted line, the dashed line and the solid line represent thevariation of the real and imaginary parts of the refractive indexwith probe detuning for � = 0.68γ− au, � = 0.793γ−au and � = 0.906γ− au respectively. In figure 5(a),Re(n) becomes negative in the detuned frequency bands lyingbetween [−0.138γ13,−0.0669γ13], [−0.175γ13,−0.0991γ13]and [−0.215γ13,−0.145γ13] for � = 0.68γ− au, � =0.793γ− au and � = 0.906γ− au respectively. It isclear from figure 5(a) that the frequency band of negativerefraction increases with the increase in the incoherent pumprate, while its magnitude decreases. Thus one can realizea negative refractive index in the LiH molecular mediumin these frequency regions. Previously it has been shown[37] that one can control the EIT profile and the groupindex in the LiH molecule by controlling the strength ofthe incoherent pump rate which leads to control over theelectric susceptibility in this system. In this paper wehave explored the feasibility of negative refraction in theelectromagnetically induced transparent molecular system byutilizing the destructive interference of SGC and DIC. Herethe nature of variation of the refractive index with frequencydepends on the nature of variation of electric susceptibility

5

J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 215503 S Dutta and K Rai Dastidar

-0.25 -0.20 -0.15 -0.10 -0.05-1.0

-0.5

0.0

0.5

1.0

1.5

Re(

n)

probe detuning

-0.25 -0.20 -0.15 -0.10 -0.050.0

0.5

1.0

1.5

2.0

Im(n

)

probe detuning

(a) (b)

Figure 5. Dependence of (a) the real and (b) the imaginary parts of the refractive index (n) upon p/γ− for � = 0.68γ− au (12.40 kHz)(dotted line), � = 0.793γ− au (14.47 kHz) (dashed line) and � = 0.906γ− au (16.54 kHz) (solid line) in the LiH molecule. Otherparameters are kept the same as in figure 3.

and magnetic permeability. Since the electric susceptibilitycan be controlled by controlling the incoherent pump rate,it plays a significant role in determining the frequency bandof negative refraction in this system. It is to be noted herethat in the negative refractive index region, initially we getanomalous dispersion and after a maximum negative value ofthe refractive index, we obtain normal dispersion. Figure 5(b)displays that the imaginary part of the refractive index of theleft handed molecular system is positive. With the increasein negative detuning, Im(n) initially shows a peak, then itgradually decreases and shows a dip. Over the frequencyregion where we obtain the negative refractive index (seefigure 5(a)), Im(n) shows a minimum positive value (seefigure 5(b)). Thus the LiH molecular system exhibitsminimum absorption for the frequency band where thenegative refractive index is obtained.

The FOM, defined as FOM = |Re(n)/Im(n)|, is afundamental parameter to characterize the performance of thenegative index medium. A system with large FOM leading tonegative refraction with minimum loss is required for efficientperformance. For � = 0.68γ−, FOM is 1.797, which is largerthan 1. For � = 0.793γ− (� = 0.906γ−), FOM is 0.99 (0.46).Thus in the LiH molecular system for a incoherent pump rate� = 0.68γ−, one can realize a more efficient low-loss LHM.

In the density matrix equations we have neglected theDoppler broadening. But in a realistic scheme, one has toconsider the Doppler broadening which may wash out theinterference effect. This effect has been eliminated in themeasurement of group velocity in a gas of ultra-cold Naatoms, cooled to a temperature of the order of nanokelvin[41]. In the past few years many researchers have chosento pursue the creation and study of ultracold polar moleculargases. Cooling molecules is more difficult than cooling atoms.Despite hurdles, considerable progress has been made. The

first trapping of neutral polar molecules, CaH, in a magneticfield was reported by Weinstein et al [42]. Recently Skoff et aldemonstrated Doppler-free saturated absorption spectroscopyof cold molecular radicals by using a buffer gas coolingmechanism [43]. The cooling of the LiH molecule has not beenexplored so far. So for this scheme one can use the counter-propagating beam method to eliminate the Doppler broadeningeffect [44]. In this system one can choose coherent and probefield polarizations to be antiparallel (i.e. counter propagatinglaser beams) so that the effect of Doppler broadening can beneglected since the transition frequencies are almost the same[37].

4. Conclusion

In this paper, we have investigated the negative refractiveindex of a three-level � system realized in a heteronuclearmolecular system such as LiH in the presence of SGC. Wehave achieved the simultaneous negative real parts of boththe dielectric permittivity εr and the magnetic permeability μr

over a certain frequency band where destructive interference ofabsorption channels due to DIC and SGC occurs in a dilute LiHgas (neglecting the local field effect). We have demonstratedthe importance of SGC in achieving a negative value of thereal part of the relative magnetic permeability (μr ) in thismolecular system. The position and the band of the frequencyrange of negative refraction can be manipulated by controllingthe incoherent pump rate. We obtain a high FOM value in thenegative refractive index region which indicates low loss.

References

[1] Veselago V G 1968 Sov. Phys.—Usp. 10 509[2] Shelby R A, Smith D R and Shultz S 2001 Science 292 77

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J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 215503 S Dutta and K Rai Dastidar

[3] Pendry J B 2001 Phys. Rev. Lett. 85 3966[4] Lakhtakia A, McCall M W and Weiglhofer W S 2003

Introduction to Complex Mediums for Optics andElectromagnetics ed W S Weiglhofer and A Lakhtakia(Bellingham, WA: SPIE) p 347

[5] Chen L, He S and Shen L 2004 Phys. Rev. Lett. 92 107404[6] Lakhtakia A 2004 AEU, Int. J. Electron. Commun. 58 229[7] Lakhtakia A 2003 Opt. Express 11 716[8] Shen J Q 2005 Phys. Lett. A 344 144[9] Zhang Z M and Fu C J 2002 Appl. Phys. Lett. 80 1097

[10] Krowne C M 2004 Phys. Rev. Lett. 92 053901[11] Krowne C M 2006 J. Appl. Phys. 99 044914[12] Pendry J B, Holden A J, Robbins D J and Stewart W J 1998

J. Phys.: Condens. Matter 10 4785[13] Pendry J B, Holden A J, Stewart W J and Youngs I 1996 Phys.

Rev. Lett. 76 4773[14] Yannopapas V and Moroz A 2005 J. Phys.: Condens. Matter

17 3717[15] Wheeler M S, Aitchison J S and Mojahedi M 2005 Phys. Rev.

B 72 193103[16] Berrier A, Mulot M, Swillo M, Qiu M, Thylen L, Talneau A

and Anand S 2004 Phys. Rev. Lett. 93 073902[17] Thommen Q and Mandel P 2006 Opt. Lett. 31 1803[18] He S, Ruan Z, Chen L and Shen J 2004 Phys. Rev. B 70 115113[19] Eleftheriades G V, Iyer A K and Kremer P C 2002 IEEE

Trans. Microw. Theory Tech. 50 2702[20] Pendry J B 2004 Science 306 1353[21] Yannopapas V 2006 J. Phys.: Condens. Matter 18 6883[22] Tretyakov S et al 2005 Photon. Nanostruct. 3 107[23] Mackay T G 2005 Microw. Opt. Technol. Lett. 45 120

Mackay T G 2005 Microwave Opt. Technol. Lett. 47 406(Erratum)

[24] Oktel M O and Mustecaplioglu O E 2004 Phys. Rev.A 70 053806

[25] Shen J Q, Ruan Z C and He S 2004 J. Zhejiang Univ. Sci.5 1322

[26] Thommen Q and Mandel P 2006 Phys. Rev. Lett. 96 053601[27] Kastel J and Fleischhauer M 2007 Phys. Rev. Lett. 98 069301[28] Liu C, Zhang J, Liu J and Jin G 2009 J. Phys. B: At. Mol. Opt.

Phys. 42 095402[29] Zhang H, Niu Y and Gong S 2007 Phys. Lett. A 363 497[30] Zhang H, Niu Y, Sun H, Luo J and Gong S 2008 J. Phys. B:

At. Mol. Opt. Phys. 41 125503[31] Krowne C M and Shen J Q 2009 Phys. Rev. A 79 023818[32] Javanainen J 1992 Europhys. Lett. 17 407[33] Menon S and Agarwal G S 1998 Phys. Rev. A 57 4014[34] Dutta S and Rai Dastidar K 2006 J. Phys. B: At. Mol. Opt.

Phys. 39 4525–38[35] Dutta S and Rai Dastidar K 2007 J. Phys.: Conf. Ser.

80 012030[36] Dutta S and Rai Dastidar K 2007 J. Phys. B: At. Mol. Opt.

Phys. 40 4284[37] Rai Dastidar K and Dutta S 2008 Europhys. Lett. 82 54003[38] Harris S E 1997 Phys. Today 50 36[39] Marangos J P 1998 J. Mod. Opt. 45 471[40] Panofsky W K H and Philips M 1962 Classical Electricity and

Magnetism (Reading, MA: Addison-Wesley)[41] Hau L V, Harris S E, Dutton Z and Behroozi C H 1999 Nature

(London) 397 594[42] Weinstein J, deCarvalho R, Guillet T, Friedrich B and

Doyle J M 1998 Nature 395 148[43] Stoff S M et al 2009 New J. Phys. 11 123026[44] Ye C Y, Zibrov A S, Rostovstev Yu V and Scully M O 2002

Phys. Rev. A 65 043805

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