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Eur. Phys. J. D (2014) 68: 133DOI: 10.1140/epjd/e2014-50038-2
Regular Article
THE EUROPEANPHYSICAL JOURNAL D
Coherently induced shifting of group index across the tripleatomic transition lines in Double-N-type system
Saswata Ghosha
Department of Physics, Visva-Bharati, Santiniketan 731235, India
Received 14 January 2014 / Received in final form 7 March 2014Published online 28 May 2014 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2014
Abstract. The group index and its corresponding absorption of a propagating probe pulse across threeatomic transition lines is theoretically presented for a Double-N-type non-EIT scheme. We have shownthat, group index on the three transition lines can be shifted accordingly to control the status (superlumi-nal/subluminal) of the pulse propagation with reduced absorption/gain by controlling the relative strengthof coherent coupling. The analytical expression of effective detuning clearly exhibits the nature of linear ACStark shift responsible for shifting/switching of group index on three different probe transition frequencyregimes.
1 Introduction
Controlling the group velocity of light pulse from fast lightto slow light and its complementary process (so that fromsubluminal propagation to superluminal propagation) in asingle system has recently attracted a great deal of inter-est. Generally superluminal/subluminal propagation is ac-companied with considerable absorption/gain which leadsto a considerable amount of pulse distortion. However it iswell established that the absorption and dispersion prop-erties of a coherent atomic medium can be modified bythe atomic coherence and quantum interference [1–5]. Inorder to reduce the distortion of propagating pulse, differ-ent types of schemes of electromagnetically induced trans-parency (EIT) were used [6–9]. In fact EIT is the mecha-nism underlying the most experiments in ultra slow [10,11]or ultra fast propagation [12–14]. Although the ideal on-one-photon resonance EIT is applicable to three/four levelmodel. In reality the situation is more complicated, withthe presence of near by hyperfine states which often con-tribute significantly to pulse propagation, that leads toalter the response of the system to the wave propagation.Moreover under EIT condition there is always a chanceof significant broadening of pulse which is barely abovethe noise level. Therefore it is well argued that EIT isnot necessary for achieving significant group velocity re-duction [15,16]. In most of the cases the speed control inatomic systems involve the tuning of the frequency ampli-tude [17–20] or phase difference of the applied fields [21].Various schemes were proposed for the switching from sub-luminal to superluminal pulse propagation [18–22] andvice versa [23–26]. It was also noticeable that the dou-ble control scheme would be more convenient and efficientfor manipulation of the optical properties of the atomic
a e-mail: [email protected]
system and hence the propagation of light [27,28]. Physi-cally the bottom line regarding control of propagation ofprobe pulse is, it can be manipulated by modifying the in-ternal structure of atoms via coupling between the atomicstates with the use of coupling field.
So far, in most of the cases the switching of group ve-locity has been limited to single frequency regime. How-ever in practical application one needs to achieve theswitching effect in different frequency regimes in a singlesystem. In accordance to our knowledge, for atomic sys-tems, there are two reports available till date where con-trolling of the group velocity of probe pulse at double fre-quency regimes were shown [29,30]. Hu et al. [29] showedthe possibility to obtain double switching from normalto anomalous dispersion with equal group velocity in twoseparate frequency regimes by employing a trichromaticfield. For the first time we introduced the concept of ACStark shift to obtain coherent double-switching effect withnegligible absorption/gain [30]. In particular here we in-tend to control the group velocity status in three separatefrequency regimes corresponds to three probe transitionsin a single system. In accordance with our objectives weuse two coupling fields to control the coherent AC Starkshift in a Double-N configuration.
2 Formulation of group index and lineshapeparameters
Our theoretical model consists of three closely spacedlower levels |1〉, |2〉, |3〉 and three closely spaced upperlevels |4〉, |5〉, and |6〉. This type of level spacing cor-responds to the hyperfine components of various alkaliatoms. The six-level system is irradiated by three elec-tromagnetic fields (two coupling and a probe field) un-der Doppler free condition forms a Double-N geometry
Page 2 of 7 Eur. Phys. J. D (2014) 68: 133
Fig. 1. An open Double-N-type six level system with threeclosely spaced upper levels and three closely spaced lower levelscoupled to two coupling fields and a probe field. The allowedtransitions are considered for F / = F //. The first couplingfield (Ω1) couples level |1〉 to both the upper levels |4〉 and|5〉, where as the second coupling field (Ω2) couples level |3〉 toboth the upper levels |5〉 and |6〉. The probe field (Ωp) scansacross the three probe field induced transitions.
(Fig. 1). All the possible dipole allowed optical transi-tions for hyperfine components F / = F // are consid-ered. The energy gap between the |i〉th and |j〉th levelis given by Ei − Ej = �ωij . The driving frequencies ofthe two controlling fields are ω1 and ω2, while that of theprobe field is ωp. The probe detuning (δp) is defined asδp = ωp − (
ω24+ω25+ω263
), while the detuning of the two
coupling fields (δ1 and δ2) are given by δ1 = (ω1 − ω14),and δ2 = (ω2 − ω36). Probe field couples three dipole al-lowed transitions |2〉 −→ |4〉, |2〉 −→ |5〉 and |2〉 −→ |6〉simultaneously with Rabi frequencies Ωp, Ω
/p , and Ω
//P , re-
spectively. The first coupling field couples the level pairs|1〉 −→ |4〉 and |1〉 −→ |5〉 with equal Rabi frequency Ω1,on the other hand the second coupling field couples thelevel pairs |3〉 −→ |5〉 and |3〉 −→ |6〉 with equal Rabi fre-quency Ω2, respectively. The present Double-N -type sixlevel scheme can be easily organized in real alkali atomicsystems like rubidium (87Rb) (P −→ D transitions) withtransition frequency at around 1529 nm. So the hyperfinelevel spacings of the three closely spaced upper levels ω45,ω46 are considered to be equal to 0.4 GHz. Hence the op-tical Bloch equations for the open Double-N -type six levelsystem driven by three electromagnetic fields are given by:
˙ρ11 = −γ1ρ11 − 2iΩ1 cosω1t (ρ41 − ρ14 + ρ51 − ρ15) ,
˙ρ22 = −γ2ρ22 − 2iΩp cosωpt (ρ42 − ρ24)
− 2iΩ/p cosωpt (ρ52−ρ25)−2iΩ//
p cosωpt (ρ62−ρ26),
˙ρ33 = −γ3ρ33 − 2iΩ2 cosω2t (ρ53 − ρ35 + ρ63 − ρ36) ,
˙ρ44 = −γ4ρ44 − 2iΩp cosωpt (ρ24 − ρ42)− 2iΩ1 cosω1t (ρ14 − ρ41) ,
˙ρ55 = −γ5ρ55 − 2iΩ/p cosωpt (ρ25 − ρ52)
− 2iΩ1 cosω1t (ρ15−ρ51)−2iΩ2 cosω2t (ρ35−ρ53) ,
˙ρ66 = −γ6ρ66 − 2iΩ//p cosωpt (ρ26 − ρ62)
− 2iΩ2 cosω2t (ρ36 − ρ63) ,
˙ρ12 = ˙ρ∗21 = − (γ12 + iω12) ρ12 − 2iΩ1 cosω1t (ρ42 + ρ52)
+ 2iΩp cosωptρ14 + 2iΩ/p cosωptρ15
+ 2iΩ//p cosωptρ16,
˙ρ13 = ˙ρ∗31 = − (γ13 + iω13) ρ13 − 2iΩ1 cosω1t (ρ43 + ρ53)+ 2iΩ2 cosω2t (ρ15 + ρ16) ,
˙ρ14 = ˙ρ∗41 = − (γ14 + iω14) ρ14
+ 2iΩ1 cosω1t (ρ11 − ρ44 − ρ54) + 2iΩp cosωptρ12,
˙ρ15 = ˙ρ∗51 =− (γ15+iω15) ρ15+2iΩ1 cosω1t (ρ11−ρ55−ρ45)
+ 2iΩ/p cosωptρ12 + 2iΩ2 cosω2tρ13,
˙ρ16 = ˙ρ∗61 = − (γ16 + iω16) ρ16 − 2iΩ1 cosω1t (ρ46 + ρ56)
+ 2iΩ//p cosωptρ12 + 2iΩ2 cosω2tρ13,
˙ρ23 = ˙ρ∗32 = − (γ23 − iω23) ρ23 + 2iΩ2 cosω2t (ρ25 + ρ26)
− 2iΩp cosωptρ43 − 2iΩ/p cosωptρ53
− 2iΩ//p cosωptρ63,
˙ρ24 = ˙ρ∗42 = − (γ24 + iω24) ρ24 + 2iΩp cosωpt (ρ22 − ρ44)
− 2iΩ/p cosωptρ54 − 2iΩ//
p cosωptρ64
+ 2iΩ1 cosω1tρ21,
˙ρ25 = ˙ρ∗52 = − (γ25 + iω25) ρ25 + 2iΩ/p cosωpt (ρ22 − ρ55)
− 2iΩp cosωptρ45−2iΩ//p cosωptρ65+2iΩ1 cosω1tρ21
+ 2iΩ2 cosω2tρ23,
˙ρ26 = ˙ρ∗62 = − (γ26 + iω26) ρ26 + 2iΩ//p cosωpt (ρ22 − ρ66)
− 2iΩp cosωptρ46 − 2iΩ/p cosωptρ56
+ 2iΩ2 cosω2tρ23,
˙ρ34 = ˙ρ∗43 = − (γ34 + iω34) ρ34 − 2iΩ2 cosω2t (ρ54 + ρ64)+ 2iΩp cosωptρ32 + 2iΩ1 cosω1tρ31,
˙ρ35 = ˙ρ∗53 = − (γ35 + iω35) ρ35
+ 2iΩ2 cosω2t (ρ33 − ρ55 − ρ65)
+ 2iΩ/p cosωptρ32 + 2iΩ1 cosω1tρ31,
˙ρ36 = ˙ρ∗63 = − (γ36 + iω36) ρ36
+ 2iΩ2 cosω2t (ρ33−ρ66−ρ56) + 2iΩ//p cosωptρ32,
˙ρ45 = ˙ρ∗54 = − (γ45 + iω45) ρ45 − 2iΩp cosωptρ25
+ 2iΩ/p cosωptρ42 + 2iΩ1 cosω1t (ρ41 − ρ15)
+ 2iΩ2 cosω2tρ43,
˙ρ46 = ˙ρ∗64 = − (γ46 + iω46) ρ46 − 2iΩp cosωptρ26
− 2iΩ1 cosω1tρ16 + 2iΩ//p cosωptρ42
+ 2iΩ2 cosω2tρ43,
˙ρ56 = ˙ρ∗65 = − (γ56 + iω56) ρ56 − 2iΩ/p cosωptρ26
+ 2iΩ2 cosω2t (ρ53 − ρ36)
+ 2iΩ//p cosωptρ52 − 2iΩ1 cosω1tρ16. (1)
Eur. Phys. J. D (2014) 68: 133 Page 3 of 7
Re (χ) = K
[{Γ24
(ξSin24
)+ Δω24
(ξ24 + ξCos
24
)}
(Γ 224 + Δω2
24)+
{Γ25
(ξSin25
)+ Δω25
(ξ25 + ξCos
25
)}
(Γ 225 + Δω2
25)+
{Γ26
(ξSin26
)+ Δω26
(ξ26 + ξCos
26
)}
(Γ 226 + Δω2
26)
]
(3)
Im (χ) = K
[{Γ24
(ξ24 + ξCos
24
) − Δω24
(ξSin24
)}
(Γ 224 + Δω2
24)+
{Γ25
(ξ25 + ξCos
25
) − Δω25
(ξSin25
)}
(Γ 225 + Δω2
25)+
{Γ26
(ξ26 + ξCos
26
) − Δω26
(ξSin26
)}
(Γ 226 + Δω2
26)
]
(4)
The diagonal element ρii corresponds the population ofthe energy level |i〉, while the complex off-diagonal matrixelements ρij (i �= j) correspond the coherence between thelevels |i〉 and |j〉. The longitudinal decay constant for |i〉thlevel is γi and the transverse relaxational rate from level|i〉 to |j〉 is γij = 1
2 (γi + γj) + γph. In practice, we neglectthe phase relaxation width (i.e. γph = 0). To obtain theanalytical solutions to the coupled Bloch equations (1)we start with the time evolution of the density matrixelements associated with allowed transitions by using thefollowing ansatz [31]:
ρ1j = ρ1j exp−iω1t,
ρ2j = ρ2j exp−iωpt,
ρ3j = ρ3j exp−iω2t, (where j = 4, 5, 6), (2)
where ρij are slowly varying functions of time. The probeRabi frequencies are assumed to be equal i.e. Ωp = Ω
/p =
Ω//p . Now the exact analytical solutions for density ma-
trix elements corresponding to the above equations (1)are unattainable and the detailed perturbative approxi-mate analytical solution for similar M -type and N -typesystems can be found in our earlier reports [30,32,33] . Inthis report we want to skip the details of the analyticalsolutions of the coupled equations [1], however to makethe paper self contained we briefly discuss the algebraicsteps for approximate analytical solutions regarding line-shape and group index. The present solution is based onthe so-called rotating wave approximation where the fastrotating terms are neglected. In order to have the approx-imate analytical solutions (i.e perturbative solutions) ofthe rate equations (1), it is also necessary to have theweak field approximation. The weak field approximationgives rise to the conditions Ωi
γ < 1 and Ωp
γ < 1, so that theterms beyond (Ω
γ )4 can be neglected. Using slowly varyingenvelope approximation (SVEA), the induced susceptibil-ity is calculated over the time, long compared to the decayof the atom, and short compared to the steady state situ-ation. The corresponding situation is called as seeminglysteady state condition. The probe response can be writtenin a closed analytical form. The analytical expressions forreal and imaginary parts of the probe susceptibility giverise to the dispersive and absorptive lineshape which aregiven by:
see equations (3) and (4) above,
where χ = χ exp iωt and the prefactor K involve constant(does not affect the lineshape) parameters like injection
rate, square of the transition moment and probe field am-plitude. The parameters ξCos
ij , ξSinij are given by:
ξCos24 = −| ρ45(0) | cosφe√
γ245 + ω2
45
− | ρ46(0) | cosφe√γ246 + ω2
46
+
(Ω1Ωp
)| ρ12(0) | cosφl
√γ212 +
(δp − δ1 + 2ω45+ω56
3
)2, (5)
ξSin24 =
| ρ45(0) | sin φe√γ245 + ω2
45
+| ρ46(0) | sin φe√
γ246 + ω2
46
−(
Ω1Ωp
)| ρ12(0) | sin φl
√γ212 +
(δp − δ1 + 2ω45+ω56
3
)2, (6)
ξCos24 = −| ρ54(0) | cosφe√
γ245 + ω2
45
− | ρ56(0) | cosφe√γ256 + ω2
56
+
(Ω1Ωp
)| ρ12(0) | cosφl
√γ212 +
(δp − δ1 + 2ω45+ω56
3
)2
+
(Ω2Ωp
)| ρ32(0) | cosφl
√γ223 +
(δ2 − δp + ω45+2ω56
3
)2, (7)
ξSin25 = −| ρ54(0) | sin φe√
γ245 + ω2
45
+| ρ56(0) | sin φe√
γ256 + ω2
56
−(
Ω1Ωp
)| ρ12(0) | sin φl
√γ212 +
(δp − δ1 + 2ω45+ω56
3
)2
−(
Ω2Ωp
)| ρ32(0) | sin φl
√γ223 +
(δ2 − δp + ω45+2ω56
3
)2, (8)
ξCos24 = −| ρ64(0) | cosφe√
γ246 + ω2
46
− | ρ65(0) | cosφe√γ256 + ω2
56
+
(Ω2Ωp
)| ρ32(0) | cosφl
√γ223 +
(δ2 − δp + ω45+2ω56
3
)2, (9)
ξSin26 = −| ρ64(0) | sin φe√
γ246 + ω2
46
− | ρ65(0) | sin φe√γ256 + ω2
56
−(
Ω2Ωp
)| ρ32(0) | sin φl
√γ223 +
(δ2 − δp + ω45+2ω56
3
)2, (10)
Page 4 of 7 Eur. Phys. J. D (2014) 68: 133
and ξ2i = ρ22(0)γ2
− ρii(0)γi
(i = 4, 5, and 6) are real pa-rameters like the difference between population per decaycorresponds to level |2〉 and any other upper level. Thefield independent phases φl and φe are the measure of theatomic coherence between dipole forbidden (i.e. not cou-pled by dipole transitions) closely spaced lower levels andupper levels, respectively. In other words φl and φe areassociated with the closely spaced hyperfine componentsof lower (F /) and upper levels (F //). Hence, φl and φe
are termed as lower and upper hyperfine coherence, re-spectively. The parameters ξSin
2j and ξCos2j (j = 4, 5 and 6)
involve, the initial coherence for three closely spaced up-per levels
| ρij(0) | cosφe√γ2
ij + ω2ij
,| ρij(0) | sin φe√
γ2ij + ω2
ij
,
and the “driving contributions” [27] of two coupling fields
(Ωi
Ωp
)| ρij(0) | cosφl
√
γ2ij +
(δi ± δp + ωij+2ωji
3
)2,
(Ωi
Ωp
)| ρij(0) | sinφl
√
γ2ij +
(δi ± δp + ωij+2ωji
3
)2.
The magnitude of the off-diagonal density matrix elementsand hence the initial coherence between the |i〉th and |j〉thlevel | ρij(0) | can be related to the initial population ofthose levels as
| ρij(0) |=√
ρii(0)ρjj(0).
The coherence involving three upper levels contribute in-significantly since γij � ωij . Nevertheless, the quantumcoherence governed by the phase angle φl plays the impor-tant role for the determination of magnitude as well as thesign of susceptibility for on-resonance coupling. Physicallyit means that lower level hyperfine coherence is inducedby the coupling field to effect the probe response via co-herent interaction amongst the upper and lower levels.Therefore the last terms of equations (5), (6), (9), (10)and the last two terms of equations (7) and (8) exhibitthe double-control mechanism for these three Lorentziansand their corresponding dispersion. The constant prefac-tor K in the equation is assumed to be equal to –1 forlineshape scaling purpose. In the present investigation wekeep the closely spaced upper levels coherence, φe = 0.The complex matrix elements ρ2j are involving the phaseangle Φ and are given by ρ2j exp(iωt) = ρ2j =| ρ2j |exp iΦ,where j = 4, 5, and 6. The phase angle Φ is the outcomeof the interference between the probability amplitudes fordipole allowed levels coupled to the probe field. Thereforethe phase angle Φ represents the probe coherence resultingfrom the coherent interaction between the levels coupledby the probe field. In the present investigation we keep
the value of probe coherence Φ equal to 0. Now, the ana-lytical expressions for the effective probe line width Γ2j ,and effective probe detunings Δω2j can be written as:
Γ24 = γ24 +Ω2
pγ45
γ245 + ω2
45
+Ω2
pγ46
γ246 + ω2
46
+Ω2
1γ12
γ212 +
(δp − δ1 + 2ω45+ω56
3
)2 , (11)
Γ25 = γ25 +Ω2
pγ45
γ245 + ω2
45
+Ω2
pγ56
γ256 + ω2
56
+Ω2
1γ12
γ212 +
(δp − δ1 + 2ω45+ω56
3
)2
+Ω2
2γ23
γ223 +
(δ2 − δp + ω45+2ω56
3
)2 , (12)
Γ26 = γ26 +Ω2
pγ46
γ246 + ω2
46
+Ω2
pγ56
γ256 + ω2
56
+Ω2
2γ23
γ223 +
(δ2 − δp + ω45+2ω56
3
)2 , (13)
Δω24 = (ω − ω24) −Ω2
pω45
γ245 + ω2
45
− Ω2pω46
γ245 + ω2
46
− Ω21
(δp − δ1 + 2ω45+ω56
3
)
γ212 +
(δp − δ1 + 2ω45+ω56
3
)2 , (14)
Δω25 = (ω − ω25) +Ω2
pω45
γ245 + ω2
45
− Ω2pω56
γ256 + ω2
56
− Ω21
(δp − δ1 + 2ω45+ω56
3
)
γ212 +
(δp − δ1 + 2ω45+ω56
3
)2
− Ω22
(δ2 − δp + ω45+2ω56
3
)
γ223 +
(δ2 − δp + ω45+2ω56
3
)2 , (15)
Δω26 = (ω − ω26) +Ω2
pω46
γ246 + ω2
46
+Ω2
pω56
γ256 + ω2
56
− Ω22
(δ2 − δp + ω45+2ω56
3
)
γ223 +
(δ2 − δp + ω45+2ω56
3
)2 . (16)
Now group velocity of probe pulse across three probetransitions can be obtained by using equations (3), and(5)–(16), to the following standard relation:
vg =c
[1 + 2π Re (χ) + 2πωp Re
(∂(χ)∂ωp
)] . (17)
The details of line width analysis (11)–(13) can be foundin our earlier work [32]. In short it is easy to say from(11)–(13) that the line width of the Lorentzians in-creases significantly with the coupling field strength foron-resonance coupling. In the present context the impor-tant shifting terms can be found in the equations (14)–(16)for effective probe detuning Δω2j . The first term in theright hand side of equations (14)–(16) is simple probe de-tuning, while the other three/four terms are the important
Eur. Phys. J. D (2014) 68: 133 Page 5 of 7
AC Stark shifting parameters of the spectral line. The sec-ond and third terms of Δω2j indicate the dependence offrequency shift on the upper-level spacings ω45, ω56 andprobe Rabi frequency Ωp (i.e., strength of the probe field),while the fourth term in equations (14) and (16) andfourth and fifth terms in equation (15) heavily rely onthe relative detuning of the coupling and probe field, andon the Rabi frequencies of the coupling field (i.e., strengthof the coupling field). So the effective detuning for proberesponse contains two types of AC Stark shifting parame-ters; one is due to probe field while another is due to cou-pling field. It is also to be noted that the terms responsiblefor shifting of the probe transitions in equations (14)–(16)are directly proportional to the square of the Rabi fre-quencies of the coupling or probe fields. Therefore, in thepresent scheme, the Stark shift is linear in intensity.
3 Shifting of group index
Group velocity status of the probe pulse across three probefield induced atomic transitions can be investigated usingequation (17). Where as the real and imaginary part of thesusceptibility (3) and (4) gives rise to the correspondingdispersion and absorption, respectively. In order to exam-ine the group velocity status we define a quantity fromequation (17)
ng − 1 = 2π Re(χ) + 2πωpRe(
∂(χ)∂ωp
),
where ng = cvg
is the group index. For the sake of compu-tation, the decay rates (γi) of different levels in the presentDouble-N -type scheme are chosen to be equal to 0.04 GHz,while the initial population of three lower levels are con-sidered as ρ11(0) = 0.28, ρ22(0) = ρ33(0) = 0.27 andthat of three upper levels are taken to be equal, so thatρ44(0) = ρ55(0) = ρ66(0) = 0.06. So initially the system isin population without inversion situation, where popula-tion of level |1〉 is considered to be slightly high comparedto that of other two lower levels |2〉, and |3〉 to maintainthe condition
∑ρii(0) = 1. As the lower level hyperfine
coherence relaxation rate is generally very small comparedto the optical transition, we are interested particularly onthe lower hyperfine atomic coherence effect. In the presentanalysis we keep the value of lower level hyperfine coher-ence fixed with φl = π. So the phase difference betweenupper level hyperfine coherence (φe) and that of lowerlevels is π. This particular choice of phase difference isconsidered here for minimal absorption/gain.
In order to show the shifting of group index, we startwith the condition when both the coupling fields areswitched off (i.e. Ω1 = Ω2 = 0) (Fig. 2a). In absenceof the two coupling fields group index corresponding tousual superluminal propagation can be found along withstandard absorption (Fig. 2b) across three probe transi-tions for Rabi frequency of the probe field Ωp = 0.02 GHz.Now as the two coupling fields are switched on and tunedto on resonance correspond to |2〉 −→ |4〉, and |2〉 −→ |6〉,probe transitions, the AC Stark shifting of group index is
-1e+08
0
1e+08
-0.4 0 0.4
n g -
1
Probe Detuning (δp GHz)
(Ω1=Ω2=0)(Ω1=Ω2=0.01GHz, δ1=δ2=0)(Ω1=Ω2=0.03GHz, δ1=δ2=0)
(a)
-100
0
-0.4 0 0.4
Abs
orpt
ion
/ Gai
n (A
rb. U
nits
)
Probe Detuning (δp GHz)
(Ω1=Ω2=0)(Ω1=Ω2=0.01GHz, δ1=δ2=0)(Ω1=Ω2=0.03GHz, δ1=δ2=0)
(b)
Fig. 2. (a) Group index and (b) corresponding absorptionare plotted against probe detuning for different values of Rabifrequencies (Ωi) and detuning (δi) of the two coupling fields asindicated at the top/bottom of the curves.
observed on the transition lines at both side of the probetransition |2〉 −→ |5〉 (Fig. 2a). However the status of thegroup velocity remains superluminal at probe detuningδp = −0.4 GHz and δp = 0.4 GHz for the Rabi frequen-cies of the two coupling fields Ω1 = Ω2 = 0.01 GHz, alongwith significant reduction of absorption as shown in Fig-ure 2b. As we increase the value of Rabi frequencies of thetwo coupling fields under same coupling condition, furtherStark shifting of the group index takes place which leadsto the switching of probe pulse propagation from super-luminal to subluminal at two probe transitions. Thereforean interesting phenomenon occurs in this single systemwhere subluminal propagation of probe pulse can be ob-served on |2〉 −→ |4〉, and |2〉 −→ |6〉, probe transitionswhile on |2〉 −→ |5〉, probe transition the pulse propa-gation remains superluminal. It can be noticed from thecorresponding absorption curve (Fig. 2b) that the reducedabsorption peaks are shifted to slight gain for two probetransitions |2〉 −→ |4〉, and |2〉 −→ |5〉. Hence we see that
Page 6 of 7 Eur. Phys. J. D (2014) 68: 133
-1e+08
0
1e+08
-0.4 0 0.4
n g -
1
Probe Detuning (δp GHz)
(Ω1=0.01GHz, Ω2=0.03GHz, δ1=δ2=0)(Ω1=0.03GHz, Ω2=0.01GHz, δ1=δ2=0)
Fig. 3. Group index is plotted against probe detuning for dif-ferent values of Rabi frequencies (Ωi) and detuning (δi) of thetwo coupling fields as indicated at the bottom of the curves.
-1e+08
0
1e+08
-0.4 0 0.4
n g -
1
Probe Detuning (δp GHz)
(Ω1=Ω2=0.03GHz, δ1=0, δ2=−0.4GHz)(Ω1=Ω2=0.03GHz, δ1=0.4GHz, δ2=0)
Fig. 4. Group index is plotted against probe detuning for dif-ferent values of Rabi frequencies (Ωi) and detuning (δi) of thetwo coupling fields as indicated at the bottom of the curves.
for Ωi < Ωp the probe pulse propagation is superluminalwhere as for Ωi > Ωp it switches to subluminal propa-gation. This particular observation is verified in Figure 3where under same coupling condition, for Ω1 = 0.01 GHzwe observe superluminal propagation on |2〉 −→ |4〉 probetransition and for Ω2 = 0.03 GHz we find subluminalpropagation on |2〉 −→ |6〉 probe transition. Now as weinterchange the value of Rabi frequencies between Ω1 andΩ2, the propagation status of the probe pulse is alsointerchanged accordingly at those two probe transitions(Fig. 3), with reduced absorption/gain (not shown). Nowby keeping the value of Rabi frequencies of the couplingfields fixed (i.e. Ω1 = Ω2 = 0.03 GHz), we change the de-tuning of the second coupling field to |3〉 −→ |5〉 on reso-nance (i.e. δ2 = −0.4 GHz). Under this particular couplingsituation we observe the Stark shift of the group indexcorresponds to |2〉 −→ |4〉, and |2〉 −→ |5〉 probe transi-tions (Fig. 4). Hence the group velocity status of the probepulse on |2〉 −→ |4〉, and on |2〉 −→ |5〉 probe transitionshas been shifted from superluminal to subluminal, where
-2.5e+07
-1e+07
0
1e+07
0 0.015 0.02 0.03
n g -
1
(Ω1) GHz
(Ωp=0.02GHz, δ1=0)(Ωp=0.015GHz, δ1=0)
Fig. 5. Group index on |2〉 −→ |4〉 probe transition (underon-resonance coupling) is plotted against Rabi frequency ofthe first coupling field (Ω1) for two different values of Rabifrequencies of the probe field (Ωp) as indicated at the bottomof the curves.
as for |2〉 −→ |6〉 probe transition it remains superluminal(Fig. 4). Later we tune the frequency of the two couplingfields ω1, ω2 so that the detuning of the first coupling fieldδ1 = 0.4 GHz, and that of the second coupling field δ2 = 0.That means now the first coupling field is on resonance to|2〉 −→ |4〉 probe transition, while the second one is on res-onance to |2〉 −→ |6〉 probe transition. Under this condi-tion, superluminal propagation is observed on |2〉 −→ |4〉transition, while subluminal propagation is exhibited on|2〉 −→ |5〉, and |2〉 −→ |6〉 probe transitions due to linearcoherent AC Stark shift. Finally the variation of group in-dex on |2〉 −→ |4〉 probe transition is plotted against thestrength (Rabi frequency Ω1) of the first coupling field,under on-resonance coupling (δ1 = 0) (Fig. 5). An inter-esting result can be observed in Figure 5 where the probepulse propagates superluminally for Ωi < Ωp, while thepulse propagation is luminal for Ωi = Ωp, and it becomessubluminal for Ωi > Ωp. It is also noticeable that this ob-servation of Figure 5 corresponds to our earlier results asshown in Figures 2a, 3, and 4, respectively. Therefore inthe present system the propagation status, and hence theamount of AC Stark shift, are extremely sensitive to therelative intensity of the coupling and probe fields.
4 Conclusion
We obtain the analytical formulation of group index andthe corresponding absorption, of a Double-N -type six levelsystem with three closely spaced upper levels and lowerlevels. The choice of this particular type of six level sys-tem is made to observe the shifting/switching of groupindex on three separate atomic transition lines in a singlesystem. In fact we have shown for the first time the shift-ing/switching of group index across three atomic tran-sition lines under different coupling conditions in a singlesystem. Although it is a non-EIT scheme the shifting takesplace for reduced absorption/gain via coherent coupling.
Eur. Phys. J. D (2014) 68: 133 Page 7 of 7
The shifting of group index takes place due to linear ACStark shift as evident from the analytical expression ofeffective probe detuning (14)–(16). Moreover the amountof AC Stark shift due to coherent coupling is maximumfor on-resonance coupling. However the significant reduc-tion of absorption/gain for shifting of group index can befound due to lower level hyperfine atomic coherence andhence the driving contribution of the coherent coupling.Interestingly, under certain coherent coupling (i.e. for thephase difference π between lower level hyperfine coherenceand that of upper levels) it is found that for weak coupling(i.e. Ωi < Ωp) the propagation of probe pulse is superlu-minal, for (Ωi = Ωp) it is luminal, where as for strongcoupling (Ωi > Ωp) it is shifted to subluminal.
This work was supported by University Grants Commission,Government of India, through a major research project.
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