Slow and fast light dynamics in a chiral cold and hot atomic medium

1.1 MB·

by  Bakht A Bacha

Slow and fast light dynamics in a chiral cold and hot atomic medium
Bakht A Bacha
Department of Physics, Hazara University, Pakistan
Fazal Ghafoor
Department of Physics, COMSATS Institute of Information Technology, Islamabad, Pakistan
Rashid G Nazmidinov
Laboratory of Theoretical Physics JINR, Moscow Region, Dubna, Russia
WestudyChiralBasedElectromagneticallyInducedTransparency(CBEIT)ofalightpulseandits
associatedsubluminalandsuperluminalbehaviorthroughacoldandahotmediumof4-leveldouble-
4
1 Lambda type atomic system. The dynamical behavior of this chiral based system is temperature
0 dependent. The magnetic field based chirality and dispersion is always opposite as compared with
2 the electric field ones. Contrastingly, the response of the chiral effect along with the incoherence
Dopplerbroadeningmechanismenhancesthesuperluminalbehaviorascomparedwithitstraditional
n
degrading effect. Nevertheless, the intensity of a coupled microwave field destroys the coherence
a
of the medium and degrade superluminality and subluminality of the sysmtem. The undistorted
J
retrieved pulse from a hot chiral medium delays by 896ns than from a cold chiral medium under
9 same set of parameters. Nevertheless, it advances by −31ns in the cold chiral medium when a
1
suitably different spectroscopic parameters are selected. The corresponding group index of the

medium and thetime delay/advance,are studied and analyzed explicitly [Note: A revise version is
]
h underpreparation]
p
-
nt Quantum coherence is widely initiated through strong (a) (b)
a laser fields in multi-level atomic or molecular system 4
u interacting with electromagnetic fields. The coherent p 3 2 p kpv
q atomic or molecular states, which are prepared by the 3 3 +αkv
[ laser field, may then generates phenomenon of quantum g1 b 1 g4 +α21k21v

1 interference among various transition amplitudes. The g2 g3 3+α3kbv
v control over the response function of the media is then b
+αkv
7 the resultofthe controlofquantumcoherenceandinter- 1 2 b b b
8
ference effects. In the recent two decades, Electromag-
6
neticallyinducedtransparencyEITandrelatedquantum
4 FIG.1: (a)SchematicsoftheDoubleLambdaAtomicsystem.
interference effect in various atomic schemes has been
. (b) Doppler-broadened system
1 studiedg theoretically as well as experimentally [1]. The
0
modification of optical properties of quantum material
4
1 media remained a hot area due its expected large num- large. Nevertheless, some related works can be found in
: ber of useful applications [2, 15]. [16, 17, 17–29] and in the references therein. Further,
v
One ofthe modifiedmedia is the Left-handed medium Moti and his co-worker have been presented an exper-
i
X which posses simultaneously negative electric permittiv- imental demonstration for temporal cloaking while us-
r ity and magnetic permeability [3–5]. Earlier explored ing the concepts of time-space duality between diffrac-
a left-handed media were anisotropic in nature. Neverthe- tion and dispersive broadening. Their experiment may
less, later with the use of new techniques isotropic left be significant effort toward the development of complete
handed media were also realized[6, 7]. Both the permit- spatio-temporalcloakswhileusingnegativerefractivein-
tivity and permeability were required negative (ǫ,µ<0) dices [30]. Glasser et al [31] used four-wave mixing in
for a negative refractive index with the magmatic dipole hot atomic vapors and experimentally measured multi-
moments very smaller than the electric one. Neverthe- spacial-mode images with the use of negative group ve-
less, due to the above rigorous condition, media with locity. Inthis experimentthe degreeoftemporalreshap-
negative permeability in the optical frequencies domain ing was quantified and increased with the an increase
met very rarely. Even with these complications, mate- of the pulse-time-advancement. Furthermore, to impart
rialmedia of negative refractiveindex basedonEIT and the information of images in an optical pulse propagat-
photonic resonance were also studied. ingthrougharegionofanomalousdispersionmayhavea
Obviously,the quantumcoherenceandinterferenceef- largenumberofpotentialapplicationswhileavoidingthe
fecthasremarkableimpactonthespeedofaprobepulse. pulse reshaping. Four-waves mixing in double-Lambda
Consequently, high degree control over the group veloc- scheme of the Rubidium atom has also been shown to
ity has been made possible both theocratically and ex- exhibit multi-spatial-mode entanglement [32]. Neverthe-
perimentally. The detail study in this direction is very less, a system exhibiting less losses or gain may lead to
2
an ideal multi-spatial-mode entanglement. The spatial The general form of density matrix equation is given by
properties of the multi-spatial-mode entanglement may the following relation:
also be precessed in the presence of fast light medium if
dρ i 1
the system is constrained to less losses or gains. t = − [ρt,Ht] Γij (σ†σρ+ρσ†σ 2σρσ†) (2)
In this paper we investigate the effect of a chiral cold dt ¯h − 2 X −
and hot medium of double Lambda type atoms on a
where σ is the raising operatorandσ is loweringopera-
propagating pulse. The chiral medium is a medium in †
torfor the four decaysprocesses. Here,inthe dynamical
whichtheelectricpolarizationiscoupledtothemagnetic
equationweusethetransformationequationoffastvary-
field component of the incident electromagnetic fields in
ingtransitionmatrixelementstotheslowlyvaryingtran-
free space while the magnetization is coupled to the in-
sition matrix element through the ρ = ρ exp[ i∆ ],
cident electric field components. These mechanisms of ij ij − ij
to remove the time dependent exponential factors. Fi-
the atom-field interaction ensures the proposed medium e
nally,weobtainedthethreecoupleddynamicalequations
to posses the characteristics of negative refractive index
for our system as:
even with no need of negative permittivity and perme-
ability[14]. Inthisconnection,weselectachiralcoldand
· 1 i
hot atomic medium and investigate the effect of chiral- ∼ρ14 = [i∆p− 2(γ1+γ2)]ρ14+ 2Ωp(ρ11−ρ44)
ity on EIT and consequently on subluminal and super-
i ei ei e
luminal behavior of the propagating pulse. Unlike the + Ω exp[iϕ]ρ + Ω ρ Ω ρ , (3)
3 13 2 12 b 34
2 2 − 2
traditionaldegradingbehavioroftheDoppler broadened
e e e
medium,hereweexploredcontrastingly,anenhancement
in the superluminal behavior of the propagating probe · 1 i i
pulse for the hot chiral medium. Furthermore, we note ∼ρ13 = [i∆b− 2(γ1+γ2)]ρ13+ 2Ωbρ11+ 2Ω1ρ12
thatthenatureofthepropagationisdifferentforsuitably i i e i e e
different set of parameters. Our explored undistorted Ωbρ33 Ωpρ43+ Ω3exp[ iϕ]ρ14, (4)
−2 − 2 2 −
retrieved pulse from a hot chiral medium delays signifi-
e e e
cantlyascomparedwithacoldchiralmediumwithasim-
ilar conditions. Nevertheless, the pulse advances enor- · 1 i
mously with a suitably different spectroscopic parame- ∼ρ12 = [i(∆p−∆2)− 2(γ1+γ2+γ3+γ4)]ρ12− 2Ω1ρ13
ters. Evidently, the behavior of the propagating pulse i i i e e
+ Ω ρ Ω ρ Ω ρ , (5)
is modified when the medium is reverted from Doppler 2 2 14− 2 b 32− 2 p 42
broadened (hot) to the Doppler-free (cold) mode and e e e
In the derivation of the above dynamical equation we
vice versa. The group index of the medium, time de-
considered Ω , and Ω in the first order, while Ω , Ω
lay/advance is also studied and analyzed explicitly. p b 1 2
and Ω are assumed in all order of the perturbations.
Wechooseafour-levelatomicsysteminDouble-lambda 3
Next, we assume the atoms initially in the ground state
configuration as shown in Fig. 1. The lower hyperfine
1 . Therefore,thepopulationinitiallyintheotherstates
ground levels 1 and 2 are coupled with the upper ex- | i
| i | i is zero i.e., ρ = 1, ρ = ρ = 0, ρ = ρ = 0,
cited level 4 , by a control field having Rabi frequency 11 44 34 32 33
| i ρ = ρ = 0. Next, we assume the temperature of the
Ω , andwitha probefieldhavingRabifrequencyΩ ,re- 42 43
2 p e e e e e
medium hot and assume the system Doppler-broadened.
spectively. Theexcitedhyperfinestate 3 iscoupledwith
e e
| i To incorporate the broadening effect in the system we
thesamelowergroundlevelsbyamagneticfieldwiththe
replacedthe detuning parametersby: ∆ =∆ +α k v,
Rabifrequency Ω , anda controlfield with the Rabifre- 1 1 1 1
b
∆ = ∆ +α k v, ∆ = ∆ + α k v ∆ = ∆ + kv.
quencyΩ ,respectively. Wealsocoupledthetwoexcited 2 2 2 2 b b 3 b p p
1
In the above replacement If we consider α = 1,
hyperfine levels 3 and 4 by a microwavefieldwith the i=1,2,3
| i | i then the coherent fields are co-propagating with the
Rabi frequency Ω . Further, to keep the nature of in-
3
probe field while α = 1 represent their corre-
teraction general we consider these fields detuned from i=1,2,3 −
sponding counter propagation. Here k , k , k ,k , k
their respectivecouplingenergylevelsandaredefinedas 1 2 3 p b
are the wave vectors of coherent fields and the probes
under: ∆ =ω ω ,∆ =ω ω and∆ =ω ω ,
1 13 1 2 14 2 p 14 p
− − − electric and magnetic fields. Nevertheless, we assumed
∆ = ω ω , ∆ = ω ω . Next to drive the equa-
b 13 b 3 34 3
− − k = k = k = k = k = k, in our analysis for a sim-
tions of motion and analyze optical response functions 1 2 3 b p
for the system, we proceed with the following interac- plicity. To evaluate ρ(1) and ρ(1) in its steady state limit
14 13
tionpictureHamiltonianinthedipoleandrotatingwave we used the following expression.
e e
approximations as:
t
¯h ¯h Y(t)= e−M(t−t,)Xdt, = M−1X, (6)
H(t) = Ω exp[ i∆ t] 1 4 Ω exp[ i∆ t] 1 3 Z −
p p b b
−2 − | ih |− 2 − | ih | −∞
¯h ¯h where Y(t) and X are column matrices while M is a 3x3
Ω [ i∆ t] 2 3 Ω exp[ i∆ t] 2 4
−2 1 − 1 | ih |− 2 2 − 2 | ih | matrix. The solutions obtained are given bellow
¯h
−2Ω3exp[−i∆2t+iϕ]|3ih4|+H.c. (1) ρ(114) =ΩpβEE +ΩbβEB, (7)
e
3
ρ(1) =Ω β +Ω β , (8) M = ξHEE+χ H,respectively. Theseelectricandmag-
13 p BE b BB µ0c m
neticsusceptibilities forthe hotatomicsystemiswritten
where e
as:
βEB =βB∗E = 2[Ω1Ω2Ω2s−in(ϕiΩ−1ΩΩ212A+12+ΩΩ3A22A32e+xp(Ω[iϕ23+])4A1A2)A3(]9,) χe(kv)= N2σ14µ0µ213βBǫE0h¯β(Eh¯B−+NNµσ012µ421β3EβEBB(h¯)−Nµ0µ213βBB()17)
and and
βEE,(BB) = 2[Ω1Ω2Ω2si(n−ϕi−)iΩ(Ω2121A(1++)−Ω224AA22+(1()ΩA233+)4A1A2)A3(]1,0) χm(kv)= ¯h NNµ0µµ21µ3β2BβB , (18)
− 0 13 BB
while
respectively. The associated chiral coefficients are pre-
1 sented in the following single equation:
A =[i(∆ +kv) (γ +γ )], (11)
1 p 1 2
− 2
Ncσ µ µ β
14 0 13 EB,(BE)
ξ (kv)= . (19)
EH,(HE) (h¯ Nµ µ2 β )
1 − 0 13 BB
A =[i(∆ +α kv) (γ +γ )], (12)
3 b 3 1 2
− 2 Here, the electric dipole moment is defined as σ =
14
3γ ǫ ¯hc3/2ω3 . We select v = 0 when the medium
and 1 0 14
ips cold and obviously the Doppler broadening effect can
1 be minimized. The medium is then called as a cold chi-
A =[i(∆ +α kv ∆ α kv) (γ +γ +γ +γ )].
2 1 1 b 3 1 2 3 4
− − − 2 ralmedium. Nevertheless,Ifahotmediumisconsidered,
(13)
where the Doppler shift is dominant. we denote it as a
In Eqs. (7) and (8) β and β , appear for electric
EE BB hot chiral atomic medium. For cold atomic medium, we
andmagneticpolarizabilities,whileβ andβ ,repre-
EB BE representsusceptibilitiesbyχ andχ ,whileforthechi-
e m
sents the chirality coefficients. The electric polarization
ral these are ξ and ξ , respectively. These results
EH HE
is defined by P = Nσ ρ , and magnetization can be
14 14 areobtainfromEqs. (18)-(21],whenoneputv =0. The
measured from M = Nµ ρ , where σ , is the electric
13 13 14 Doppler susceptibilities are the average of χ (kv), and
e
and µ , is the magnetic dipole moments and N, repre-
13 χ (kv), overthe Maxwelliandistribution. The averaged
m
sents atomic number density. The Rabi frequencies are
electric and magnetic susceptibilities can be estimated
related to electric and magnetic fields through the rela-
separately from the following combined expression
tions, Ω = σ .E/¯h and Ω = µ .B/¯h, respectively.
p 14 b 13
Tgihveenebleyctariscimapnldifiemdafgonremticaspuonladreizrations collectively is χe(d,()m) = V 1√π Z ∞ χe,(m)(kv)e−(kVvD2)2d(kv) (20)
D
−∞
Nσ2 β Nσ µ β
P,(M)= 14 EE,(BB)E,(B)+ 14 13 EB,(BE)B,(E), where VD = √KBTω2Mc2, is the Doppler width. Here,
¯h ¯h χ(d) and χ(d) are the Doppler broadened electric and
(14) e m
magnetic susceptibilities for hot atomic system. In the
where B = µ (H +M). Substituting the value of B in
0
similar way we can estimate the chiral coefficient terms
magnetic polarization, and rearranging the equation we
ofthe coupledelectric -magneticfields averagedoverthe
obtained the magnetization as:
Maxwellian distribution of the atomic velocity as
Nµ µ2 β Nµ σ β
M(kv)= ¯h−N0µ01µ3213BβBBBH + ¯h−N13µ01µ4213BβEBBE(15) ξE(dH),(HE) = V 1√π Z ∞ ξEH(kv)[ξEH(kv)]e−(kVvD2)2d(kv()21)
D
−∞
Substituting the value of M, and B = µ (H +M), in
0
We know that the refractive index of a medium is n =
P,weobtainedtheexpressionforelectricpolarizationas r
under √ǫµ, where ǫ = 1+ χe and µ = 1 + χm. The terms
ξ and ξ is the conurbation from chirality to the
EH HE
G+Nσ2 β (h¯ Nµ µ2 β ) refractiveindex. The correspondingchiraldependent re-
P(kv) = 14 EE − 0 13 BB E
¯h(h¯ Nµ µ2 β ) fractive then gets its shape as:
− 0 13 BB
Nσ14µ0µ13βEB 1
+ H, (16) n = Re[((1+χ )(1+χ ) (ξ +ξ )2)1/2
(h¯ −Nµ0µ213βBB) r e m − 4 EH HE
i
+ (ξ ξ )] (22)
EH HE
G=N2σ µ µ2 β β . 2 −
14 0 13 BE EB
The group refractive index and Group velocity are also
Thesusceptibilityisaresponsefunctionofthemedium
presented here as:
duetoanappliedelectricfield. Theelectricandmagnetic
polarizationsintheformofchiral-basedelectricandmag- ∂nr
N =Re[n +(ω ∆ ) ], (23)
neticsusceptibilityaredefinedbyP =ǫ0χeE+ξEcHH and g r 14− p ∂∆p
4
and
[a] [c]
c
v = , (24)
g Re[n +(ω ∆ )∂nr]
r 14− p ∂∆p
respectively. The corresponding time delay/advance is
defined as τd = L(Ncg−1). These are the main results [b] [d]
which will be analyzed and discussed in details in the
last section.
Further we used the transfer function for observa-
tion of out put pulse shape. The output pulse shape
E (ω), after propagating through the medium can be
out
related to the input pulse shape E (ω) by the expres-
in
sion E (ω) = H(ω)E (ω), where H(ω) = e ik(ω)L is [e]
out in − [f]
thetransferfunctionfortotallytransmittingmedium. we
select a Gaussian input pulse of the form:
E (t)=exp[ t2/τ2]exp[i(ω +δ)t], (25)
in − 0 0
whereδ,istheupshiftedfrequencyfromtheemptycavity. FIG. 2: Electric and magnetic Real as well as imaginary
The Fourier transforms of this function is then written susceptibilitiesvsprobedetuning∆p/γ,suchthatγ =1GHz,
bcaylcEuilna(tωed)a=s √12π R−∞∞Ein(t)eiωtdt. The input signal is γ0.11=γ,γΩ12==γ13γ=, VγD4==00..15γγ,,∆αi1==±0γ1,,∆µ123==00γ.0,0∆00b5=3σ01γ4,mΩs1−=1,
λ=1nm[a,c] Ω3 =0.7γ[b,d] Ω3 =1γ, ϕ=π/2 [e,f]Ω2 =4γ,
E (ω) = τ /√2exp (ω ω δ)2τ2. /4 (26) Ω3 =0.7γ, VD =0.1γ
in 0 0 0
h− − − i
byTthheeocuontpvuotluEtioount(tth)ecoarnembeaosb:tainfromtheinputpulse lIinm Eqs. ∂N(g32w)-h(3il5e),τ na0pp=eareldimfωo−r→tωh0eNign,puGtvdpuls=e
widωt−h→iωn0tch∂eωtime dom0ain. Also, G is the group ve-
Eout(t)= √12π Z ∞ Eint(ω)H(ω)eiωtdω (27) locity dispersion and ω0 is the centrvadl frequency of the
pulse. Furthermore, the resonances appears at the loca-
−∞
tion ω = ω . It is the frequency of the pulse at which
For simplicity the output pulse shape is calculated in to 0 14
the delay or advancement time is maximum and the dis-
the first order derivative of group index. It is reported
tortion is minimum.
as:
We explain our main results of the Eqs. (18)-(21)] as-
τ √c [2i(Ln ct) δτ2c]2 sociated with the Real and imaginary parts of electric
E (t) = 0 exp[ 0− − 0 ]
out 2iLcGvd+cτ02 4c(2iLcGvd+cτ02) and, magnetic susceptibilities and their corresponding
chiralities,for both Doppler broadenedand Doppler-free
p n L δ2τ2
exp[i(t 0 )ω 0]: (28) systems. The discussion is then extended to group in-
0
× − c − 4 dex, time delay and pulse shape distortion using their
analytical expressions presented in the earlier text. In
The inverse fourier transform of E (t) is E (ω) and
out out Fig. 2, the plots are traced for electric and magnetic
is written by:
susceptibilitiesforsomespectroscopicparameters. Obvi-
√2π Q2 ously the behavior of electric susceptibility is in contrast
E (ω) = exp[ 1 ] with The electric and magnetic susceptibility. There-
out ∆w −4c(2iLcG + 4cπ2 )
vd (∆w)2 fore,thedispersionassociatedwithelectricsusceptibility
1 4π2δ2 4iω Ln is contrastingly opposite to the anomalous dispersion of
0 0
exp[ ( +Q2 )],(29) the magnetic one with a less absorption in comparison.
× 4 − ∆w − c
Evidently, the absorption profile enhances with the hot
where medium as compared with the cold one.
Intriguingly, the behavior of chirality in case of elec-
4cπ2(ω ω δ)
Q =2iL(ω ω )cG + − 0− +2iLn(30) tric field coupling is the mirror inversion as compared
1 − 0 vd (∆w)2 0 withthebehaviorofmagneticcouplingalongwithanen-
hanced profile for Doppler-free system overthe Doppler-
and
broadened. Nevertheless, the intensity of the microwave
[4cπ2δ 2iLn ]2 fieldgeneratesincoherenceprocessesanddisturbsthein-
Q = (∆w)2 − 0 . (31) terference mechanism of the system. The light propa-
2 c(2iLcGvd+ (∆4cwπ2)2) gation in former corresponds to slow while later it cor-
5
[a] [c] [a] [c]
[b]
[d]
[b]
[d]
[e] [f]
FIG. 5: Real and imaginary parts of chirality vs probe de-
tuning ∆p/γ, such that γ =1GHz, γ1 =γ1 =γ3 =γ4 =2γ,
∆1 = 0γ, ∆2 = 0γ, ∆b = 0γ, Ω1 = 2γ, Ω2 = 2γ,
VD = 1.5γ, αi = ±1, µ13 = 0.000053σ14ms−1, λ = 1nm[a,c]
Ω3 =1.5γ[b,d] Ω3 =5γ, ϕ=π/2
FIG. 3: Real and imaginary parts of chiralities vs probe
detuning ∆p/γ, such that γ = 1GHz, γ1 = γ1 = γ3 = γ4 =
0.1γ, ∆1 = 0γ, ∆2 = 0γ, ∆b = 0γ, Ω1 = 0.1γ, Ω2 = 1γ,
VD = 0.5γ, αi = ±1, µ13 = 0.000053σ14ms−1, λ = 1nm[a,c] [a] [c]
Ω3 = 0.7γ[b,d] Ω3 = 1γ, ϕ = π/2 [e,f]Ω2 = 4γ, Ω3 = 0.7γ,
VD =0.1γ
[d-1]
[a]
[c-1] [b] [d]
[d-2]
[c-2]
FIG. 6: Electric and magnetic Real as well as imaginary
[b] susceptibilitiesvsprobedetuning∆p/γ,suchthatγ =1GHz,
γ1 = γ1 = γ3 = γ4 = 0.1γ, ∆1 = 0γ, ∆2 = 0γ, ∆b = 0γ,
Ω1 = 0.1γ, αi = ±1, µ13 = 0.000053σ14ms−1, λ = 1nm,
ϕ = π/2, Ω2 = 4γ, Ω3 = 0.7γ, VD = 0γ(solid), 0.1γ(red
dashed),0.2γ(green dashed), 0.3γ(blue dashed)
FIG. 4: Electric and magnetic Real as well as imaginary
susceptibilitiesvsprobedetuning∆p/γ,suchthatγ =1GHz,
γ1 =γ1 =γ3 =γ4 =2γ, ∆1 =0γ, ∆2 =0γ, ∆b =0γ, Ω1 =
2γ, Ω2 = 2γ, VD = 1.5γ, αi = ±1, µ13 = 0.000053σ14ms−1, The absorption at resonance ∆p = 0, is reduced in the
λ=1nm[a,c] Ω3 =1.5γ[b,d] Ω3 =5γ, ϕ=π/2 electric susceptibility, while there is increase in the mag-
netic susceptibility. The slope of dispersion is normal in
the electric susceptibility, while anomalous in the mag-
responds to fast light. Amazingly, in the hot medium netic susceptibility. Nevertheless, the magnetic effect on
the Doppler shift developscoherence andthe the dipbe- absorptionanddispersionisverysmallascomparedwith
tween the two peak absorptionlines becomes more obvi- the electric effecton absorptionanddispersion. Further,
ous. Consequently, unlike in the traditional superlumi- with the control field Ω =0.7γ, the absorption at reso-
3
nalitywheretheDopplershiftdegradethesuperluminal nance in both the cases (cold/hot) are 0.2au. Neverthe-
fast light, here it enhances significantly. less, the absorption reduces in the Doppler free medium
Different spectral behaviors are observed, when there to 0.05au when the control field intensity is increased
is no Doppler broadening (cold system) as well as, when to Ω = 1γ and have small gaps in both the absorp-
3
there is Doppler broadening (hot system) in the system. tion and dispersion. The modification at ∆ =0 of slow
p
The solid line is for Doppler free susceptibility and the lightpropagationattheresonancepointfromcoldtohot
dashed line is for Doppler broadened susceptibility. The atomic medium is also obvious. Further, in Fig .3[a-d]
electric and magnetic susceptibilities have opposite ab- the real and imaginary parts of chiralities ξ and ξ
HE EH
sorptionanddispersionbehaviorasshowninFig .2[a-d]. under similar condition of Fig 2 are traced which also
6
ity ξ becomes vanished while it is increased for the
HE
[a] magnetic susceptibility and chirality ξ . The sym-
EH
[c] [e] metries in χ and ξ , and in χ and ξ is ideal.
e HE m EH
These interesting results are observed at the parameters
γ = 0.1γ, Ω = 0.1γ, Ω = 4γ, V = 0.1γ and
i=1,2,3,4 1 2 D
Ω =0.7γ, α = 1 and ϕ=π/2, as shown in Fig .2[e,f]
3 i
±
and Fig .3[e,f].
In Figs .(4)-(5) the real and imaginary parts of chi-
[b]
[d] [f] ralities ξ and ξ , are shown with the same parame-
EH EH
ters. The absorption of cold medium is less (large) than
the absorption of hot medium at low (large) intensity
of the control field Ω = 1.5γ, at the resonance point.
3
The real and imaginary parts of the chirality ξ have
HE
FIG. 7: [a,b,c,d] Group index vers probe detuning ∆p/γ, similar spectral behavior as compared with χ for low
e
such that γ = 1GHz, γ1 = γ1 = γ3 = γ4 = 2γ, ∆1 = 0γ, intensity of the control field for both the cold and hot
∆2 = 0γ, ∆b =0γ, Ω1 = 2γ, Ω2 = 2γ, VD = 1.5γ, αi = ±1, atomic medium. Nevertheless, for high intensity of the
µ13 =0.000053σ14ms−1, λ=1nm, ω14 =104γ,L=6cm[a,c]
control field the spectral behaviors are inverted for both
Ω3 =1.5γ[b,d] Ω3 =5γ,ϕ=π/2[e,f] Group indexandgroup
the cold and hot atomic medium as shown in Fig 4[a,b]
velocity versΩ3/γ, usingthe same parameters but ∆p =0γ
and Fig 5[a,b]. Furthermore, similar characteristics is
also true for ξ for its real and imaginary parts of the
EH
Input magnetic when the the intensity of the control field is
Output
[a] dfroepepler [a] high as shown in Fig 4[d] and Fig 5[d]. In Fig .6 the
Odouptppuletr electric and magnetic susceptibilities and the chiralities
aretracedwiththeDopplerwidths,aparametercontrol-
lablewithtemperatureofthe medium. Averydominant
response is seen about the time delay and advance when
the Doppler width is increased stepwise. Consequently,
[c] [d] goingcoldchiralmedium to a hotchiralmedium the be-
havior of the system can be reverted.
The group index and time delay/advancement for the
parametersofFig.2isshowninFig. 7. Attheresonance
point ∆ /γ = 0 sharp positive peaks of group indices
p
and times delays/advancement are observed. When the
intensityofthecontrolfieldΩ is0.5γ,thevalueofgroup
FIG.8: [a,b]ThenormalizedGaussianpulseintensityofinput 3
and outputs at the parameters given Fig2, Vs t and ω−ω0, index for cold medium is Ng =1415.65 and the value of
δ = 2GHz, τ0 = 5.50ns, ∆w = 2π/τ0, c = 3×τ1008m/s,∆Lw= groupindex for hot medium is Ng(d) =1618.15. The cor-
01.80.96m2/,c,nω00==1ω41145.=651,0n4d0γ[=c,d1]6T1h8e.1n5o,rGmvadliz=ed75G9a.4u4s/scia,nGpdvudls=e rIfestphoenidnitnegnsgirtoyuopfdceolnatyrsoalrfieetldd=Ω12is1ninscarneadste(ddd)fr=om9107.n7sγ.
3
intensity of input and outputs at the parameters given Fig4,
to 1γ. Then the group index are N = 110.96 and
Vs τt0 and ω∆−wω0, n0 = −2023.81, nd0 = −1487.22, Gvd = N(d) =164.013,whiledelaysintimesarget =219nsand
−9006.67/c, Gd =−344.57/c g d
vd t(d) = 997ns, respectively. Correspondingly, the group
d
velocities vary from v = c/1415.65 to v = c/110.96
g g
showoppositebehaviorforthe realandimaginaryparts, and vg(d) = c/1618.15 to vg(d) = c/164.013. The above
respectively. resultsarespatiallyimportantto the preservationoftwo
light pulses. When two light pulses reach at same times
The real and imaginary parts of chirality ξ (ξ )
HE EH
to the instrument which store its information. The in-
shows similarity with electric (magmatic) susceptibility
strument stored the information of one of the pulse and
in real as well as imaginary parts of the susceptibility
lost the information of the other pulse. There delays
with inverted behaviors from cold to hot medium at res-
in times recovered this difficulty and the information of
onance ∆ = 0 as shown in Fig 2[c,d] and Fig 3[a,b].
p
both the pulses will be preserved. In Fig. 7 (c,d), for
Consequently, their dynamical characteristics have con-
∆ =0γ, the values of group indices are N = 2023.81
trast behavior for the dispersion and absorption for the p g
−
two cases, respectively. [see Fig 2[a,b] and Fig 3[c,d]]. and Ng(d) = 1487.22 at the intensity of the control
−
The dominant chiral effect on magnetic susceptibility is field Ω = 1.5γ. The corresponding advance times are
3
also obvious as compared with the electric one. t = 404ns and t(d) = 297ns. In this case the cold
ad − ad −
Curiositly,controllingtheintensityassociatedwithΩ , medium is more superluminal, and the advance time is
2
the absorption in the electric susceptibility and chiral- largerthenthehotmediumby107ns. Whentheintensity
7
of the controlfield is increasefrom 1.5γ to 5γ, the group shapes for cold and hot medium, if the coherent fields
index of the cold medium is N = 595.818, while the counter propagate to the probe field or co-propagate to
g
−
groupindex ofthe hotmedium is N(d) = 751.666. The theprobefielduptofirstordergroupvelocitydispersion.
g
corresponding advance times are then t− = 119.36s Thepulsesarefullydistortion-lesstothefirstordergroup
ad
and t(d) = 150.53ns. The advance time diff−erence is indexderivativewithrespecttoangularfrequency. How-
ad − ever if one counts higher order derivative then the pulse
31.17ns. In this case the Doppler broadened medium is
is slowly distorted.
more superluminal than the Doppler free medium.
In the previous related studies, it has been observed In conclusion, we study Chiral Based Electromagnet-
thattheDopplerbroadenedmediumislesssuperluminal, ically Induced Transparency (CBEIT) of a light pulse
thanaDopplerfreemedium. Hereachiralmediumshows and its associated subluminal and superluminal behav-
contrastinglydifferentbehavior,that it may enhance su- ior through a cold and a hot medium of 4-level double-
perluminality at certain conditions of the coherent con- Lambda type atomic system. The dynamical behavior of
trol field shown in Fig 7[d]. This fact is more clearly ob- this chiral based system is temperature dependent. The
served in the Fig 7[e,f]. It is clear from these plots that magnetic field based chirality and dispersion is always
below the intensity of the control field Ω = 3.6γ, the opposite as compared with the electric field ones. Con-
3
groupindex of cold chiralmedium is more negative with trastingly,theresponseofthechiraleffectalongwiththe
a larger group velocity as compared with the hot chiral incoherenceDopplerbroadeningmechanismenhancesthe
medium. Thereforethechiralcoldmediumismoresuper- superluminal behavior as compared with its traditional
luminal than the chiral hot medium below the intensity degradingeffect. Nevertheless,the intensity ofacoupled
of control field of 3.6γ. Nevertheless, for Ω > 3.6γ, the microwave field destroys the coherence of the medium
3
group index of the chiral hot medium is more negative and degrade superluminality and subluminality of the
and larger negative group velocity as compared to the sysmtem. The undistorted retrieved pulse from a hot
chiral cold medium, Hence chiral hot medium is more chiral medium delays by 896ns than from a cold chiral
superluminal above the intensity of the control field of medium under same set of parameters. Nevertheless, it
3.6γ. Furthermore, at high intensity of the control field advances by 31ns in the cold chiral medium when a
−
the group index and group velocity becomes saturated suitably different spectroscopic parameters are selected.
(not shown). The corresponding group index of the medium and the
Fig .8 shows Gaussian pulse shapes of input and out- time delay/advance, are studied and analyzed explicitly
put of the chiral medium. All the pulses are the same [Note: A revise version is under preparation
[1] M.Fleischhauer,A.Imamoglu andJ.P.Marangos, Rev. [16] K. J. Boller, A. Imamoglu, and S. E. Harris, Phys. Rev.
Mod. Phys. 77, 633 (2005). Lett. 66, 2593(1991).
[2] M.D.LukinandA.Imamoglu,Nature,413,273(2001). [17] A. Kasapi, M. Jain, G. Y. Yin, and S. E. Harris, Phys.
[3] Pendry,J.B,A.J.Holden,W.J.Stewart,andI.Youngs, Rev. Lett.74, 2447 (1995).
, Phys.Rev.Lett. 76, 4773-4776, (1996). [18] L.V. Hau, S. E. Harris, Z. Dutton and C.H. Behroozi,
[4] Pendry, J. B., A. J. Holden, D. J. Robbins, and W. J. Nature (London) 397, 594 (1999 ).
Stewart,J.Phys.Condens.Matter10,4785-4809,(1998). [19] D.Budker, D. F. Kimball, S. M. Rochester, and V. V.
[5] Pendry, J. B., A. J. Holden, D. J. Robbins, and W. J. Yashchuk,Phys.Rev.Lett. 83, 1767 (1999).
Stewart,MicrowaveTheoryTech.47,2075-2084, (1999). [20] M. M. Kash, Scully,Phys. Rev.Lett. 82, 5229 (1999).
[6] Shelby,R.A,D.R.Smith,andS.Schultz,Science,292, [21] O. Schmidt,R.Wynands,Z.Hussein, and D.Meschede,
77-79, (2001). Phys. Rev.A 53, R27 (1996).
[7] Shelby, R. A, D. R. Smith, S. C. Nemat-Nasser, and S. [22] C. Garrett and D. McCumber, Phys. Rev. A 1, 305313
Schultz,Appl.Phys.Lett,78, 489-491, (2001). (1970).
[8] Koschny, T. L. Zhang, and C. M. Soukoulis, Phys. Rev. [23] S.ChuandS.Wong,Phys.Rev.Lett.48,738741(1982).
B, 71, 121103(2005). [24] B.SegardandB.Macke,Phys.Lett.109,213216(1985).
[9] Guney, D. O., T. Koschny, and C. M. Soukoulis, Opt. [25] B. Segard, B. Macke, and F. Wielonsky, Phys. Rev. E
Express, 18, 12348-12353, (2010). 72, 035601 (2005).
[10] N.Engheta, Propag. Lett. 1, 10-13, (2002). [26] R. Y. Chiao and P. W. Milonni, Fast light, slow light,
[11] Shen, L., S. He, and S. Xiao, Phys. Rev. B, 69, 115111, Opt.Photon. News13, 2630 (2002). R.J. McLean, Fast
(2004). light in atomic media, J. Opt.12,104001(2010).
[12] Schurig, D., J. J. Mock, B. J. Justice, S. A. Cummer, [27] Bakht Amin Bacha, Iftikhar Ahmad, Arif Ullah and
J. B. Pendry,A.F. Starr, and D. R.Smith,Science314, Hazrat Ali,phys,Scr,88,045402(2013).
977-980, (2006). [28] G.S.Agarwal and T.N Dey, phys Rev. A 68,
[13] J. B. Pendry,Phys. Rev.Lett. 85, 3966 (2000). 063816(2003).
[14] J. B. Pendry,Science 306, 1353 (2004). [29] shang-qi kuang,peng Du,Ren-gang, optics express
[15] R.W.Boyd andD.J.Gauthier, Science326, 1074-1077 16,11605(2008).
(2009). [30] M. Fridman, A. Farsi, Y. Okawachi, Nature, 481, 62-65
8
(2012). [32] V. Boyer, A. M. Marino, R. C. Pooser, Science 321,
[31] RYan.T.Glasser, Ulrich,Vogl, Poul. D.Lett ,Optic. Ex- 544547 (2008).
press.20,13702-13710 (2012).