electromagnetically induced transparency and its dispersion properties in a four-level inverted-y...

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Physics Letters A 317 (2003) 370–377

www.elsevier.com/locate/pla

Electromagnetically induced transparency and its dispersioproperties in a four-level inverted-Y atomic system

Amitabh Joshi∗, Min Xiao

Department of Physics, University of Arkansas, Fayetteville, AR 72701, USA

Received 20 August 2003; accepted 7 September 2003

Communicated by P.R. Holland

Abstract

We consider a four-level atomic system in inverted-Y configuration and study the phenomenon of electromagneticallytransparency (EIT) in this system under various parametric conditions in order to demonstrate controllability of the EITdispersion properties. 2003 Elsevier B.V. All rights reserved.

PACS: 42.50.Hz; 32.80.Wr; 42.65.Ky

Keywords: Electromagnetically induced transparency; Quantum interference

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1. Introduction

In recent past a great deal of attention has beento observe and understand the phenomenon of elemagnetically induced transparency (EIT) and relaeffects in multi-level atomic systems [1–3] interactiwith two or more electromagnetic fields. The closerelated phenomenon with EIT is coherent populattrapping (CPT) [1]. In an usual CPT situation the twelectromagnetic fields interacting with the atomof almost equal strength and the quantum interfere(QI) arises from these two fields. On the other handusual EIT situation, one of the fields is much stronthan the other so that the QI is mostly governedthe stronger field. In the terminology of EIT [1–3], th

* Corresponding author.E-mail address: [email protected] (A. Joshi).

0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserveddoi:10.1016/j.physleta.2003.09.010

-

weaker field is calledprobe field and the stronger fielis called thecoupling field. The EIT systems are potentially useful in applications such as electro-optidevices, sharp dispersion to control speed of lignonlinear optics, and lasing without inversion. Theduced atomic coherence in multi-level atomic systecannot only modify the linear absorption and dispsion properties, but also enhance the nonlinear opprocesses such as four-wave mixing [4,5], harmogeneration [6,7], and two-photon absorption [8] aall optical switching [9]. The QI can also modify threfractive index properties of a medium. It is due toKramers–Kronig relation, any change in absorptionmedium will be accompanied by change in the dispsion of the medium and that leads to phenomenaelectromagnetically induced focusing [10] and sllight [11]. By making use of atomic coherence, tnonlinear optical properties of an atomic medium cbe modified and controlled. Recently, enhanced th

.

http://www.elsevier.com/locate/pla

A. Joshi, M. Xiao / Physics Letters A 317 (2003) 370–377 371

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order Kerr-nonlinear index of refraction (n2, in theexpression ofn = n0 + n2|E|2) was measured neathe EIT condition, as well as near the more geneCPT conditions in three-levelΛ-type rubidium atomsinside an optical cavity [12]. Also, it has been oserved that the EIT can lead to suppression of nlinear response of the system. In a ladder-type folevel atomic system the suppression of two-photonsorption has been predicted theoretically and lateobserved experimentally [13]. In another experimewhich also utilized a four-level EIT atomic systemcold Rb atoms, the complete suppression of both ophoton and two-photon absorption was demonstra[14]. Also, photon switching by quantum interferenin a four-level atomic system has been proposedobserved experimentally [15]. These experiments hmotivated us to further explore the EIT characterisin the four-level system with an inverted-Y configurtion. This inverted-Y configuration is consisted of twdifferent EIT sub-systems so it becomes interestinlook at collective behaviors of these sub-systemswell as to control the collective EIT characteristicsvarious parameters.

The Letter is organized as follows. In Sectionwe present our model and its approximate solutionSection 3, we present exact numerical results and tdiscussion. This is followed by concluding remarksSection 4.

2. The model

In this work, we consider a closed four-level atomsystem in inverted-Y configuration as shown in Figwhich has been experimentally realized in Rb atoLevels|1〉, |2〉 and|3〉 are in a usual three-level laddetype configuration and level|0〉 together with levels|1〉 and |2〉 forms a three-levelΛ-type configurationSo, this composite system consists of two sub-systei.e., one ladder-type and otherΛ-type. The descriptionof the optically allowed transitions in this systeis as follows. The transition|1〉 to |2〉 (transitionfrequency ω12) interacts with a weak probe fielE1 (frequencyω1) having Rabi frequency 2Ω1 =E1d12/h. A coupling fieldE0 (frequencyω0) drivesthe transition|0〉 to |2〉 (with transition frequencyω02) with the Rabi frequency of 2Ω0 = E0d02/h

while a pumping fieldE2 (frequencyω2) is acting on

Fig. 1. Schematic diagram of a four-level atomic system inverted-Y configuration. Here,ω1, ω0, and ω2 are frequencies oprobe, coupling and pumping fields, respectively.

the transition|2〉 and |3〉 (with transition frequencyω23) and have a Rabi frequency equals to 2Ω2 =E2d23/h. The radiative decay constants from levels|3〉to |2〉, |2〉 to |0〉, and |2〉 to |1〉 are γ3, γ2, and γ1,respectively. The corresponding atomic detuningsthese transitions are∆2 = ω2 − ω23, ∆0 = ω0 − ω02,and∆1 = ω1 − ω12, respectively.

If there is no coupling field then this configuratioreduces to a standard ladder-type three-levelsystem driven by the probe and the pumping fields.the other hand if we set the pumping fieldE2 to zerothen this configuration along with probe and couplfields forms a standardΛ-type three-level EIT systemThe density-matrix equations of motion in dipole arotating-wave approximations for this system canwritten as follows:

˙ρ11 = 2γ1ρ22 + 2γ0ρ00 + iΩ1(ρ12 − ρ21),

˙ρ22 = −2(γ1 + γ2)ρ22 + 2γ3ρ33 − iΩ1(ρ12 − ρ21)

− iΩ0(ρ02 − ρ20) + iΩ2(ρ23 − ρ32),

˙ρ33 = −2γ3ρ33 − iΩ2(ρ23 − ρ32),

˙ρ00 = 2γ2ρ22 − 2γ0ρ00 + iΩ0(ρ02 − ρ20),

˙ρ12 = −(γ1 + γ2 − i∆1)ρ12 + iΩ0ρ10

+ iΩ2ρ13 + iΩ1(ρ11 − ρ22),

˙ρ13 = −(γ3 − i(∆1 + ∆2)

)ρ13 + iΩ2ρ12 − iΩ1ρ23,

˙ρ23 = −(γ1 + γ2 + γ3 − i∆2)ρ23 − iΩ1ρ13

− iΩ0ρ03 + iΩ2(ρ22 − ρ33),

372 A. Joshi, M. Xiao / Physics Letters A 317 (2003) 370–377

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˙ρ10 = −(γ0 − i(∆1 − ∆0)

)ρ10 + iΩ0ρ12 − iΩ1ρ20,

˙ρ02 = −(γ1 + γ2 − i∆0)ρ02

+ iΩ1ρ01 + iΩ2ρ03 + iΩ0(ρ00 − ρ22),

˙ρ03 = −(γ0 + γ3 − i(∆0 + ∆2)

)ρ03

(1)+ iΩ2ρ02 − iΩ0ρ23,

in which γ0 is related to non-radiative relaxationstate |0〉 and the trace condition

∑i ρii = 1. For

obtaining linear susceptibility, we need to solve tdensity matrix equations (1) under the steady-scondition. Under the assumption that the couplfield (E0) and the pumping field (E2) are muchstronger than the probe field (E1), it can be assumethat almost all the atoms are in the ground st|1〉. Under the weak probe field approximation, texpression of theρ12 in the first-order goes as

(2)

ρ12 = iΩ1(ρ11 − ρ22)

(γ1 + γ2 − i∆1) + Ω20

γ0−i(∆1−∆0)+ Ω2

2γ3−i(∆1+∆2)

.

While writing down Eq. (2), we assumed that initialthe values ofρ20 and ρ23 are approximately zeroEq. (2) gives some ideas of how coupling field (E0)and the pumping field (E2) affect the absorption andispersion properties of the probe fieldE1, whichgive rise to the EIT conditions. WhenE2 = 0, weget the EIT equation (from Eq. (2)) for a three-levsystem in aΛ configuration. On the other hand wheE0 = 0, then Eq. (2) reduces to EIT equation fa three-level system in a ladder configuration. Nthat Eq. (2) is strictly valid only under the weaprobe field approximation so we will concentrategeneral results using numerical solutions of Eq. (1the steady-state limit without invoking the weak-profield approximation.

3. Numerical results and discussion

We now discuss the results for this composite Esystem under consideration by numerical integraof Eq. (1) under the steady-state condition. For tpurpose we will examine the coherence termρ12 forthe probe transition in terms of its real and imaginparts as a function of∆1/γ1. The imaginary (real) parof ρ12 versus∆1/γ1 represents the probe absorpti(dispersion) spectrum upto all orders.

We first concentrate on the situation whenEIT is most prominent, i.e., under the condition∆0/γ1 = ∆2/γ1 = 0. In Fig. 2, the plots of Im(ρ12)

as a function of∆1/γ1 are shown withγ1 = γ2 =γ3 = 1.0, γ0 = 0 andΩ1/γ1 = 0.01. Curve (a) is forΩ0/γ1 = 0.5, Ω2/γ1 = 0.0, which reminds us theideal EIT for a three-level system inΛ configuration.The QI and the dark state play the role for observEIT in this sub-system. Curve (b) is forΩ0/γ1 = 0.0,Ω2/γ1 = 1.5, which depicts the EIT in a three-levsystem in ladder configuration. In a three-level atomsystem in ladder configuration with probe field aplied to the lower transition, the value of the radtive decay constant (γ3) of the upper most level calead to a significant degradation in the depth of EITIm(ρ12) [3]. If γ3 is smaller thanγ1 we observe deepeEIT in comparison to the situation whenγ3 γ1[3]. The combined effects of these two sub-syste(theΛ-sub-system and the ladder-sub-system) caseen in curve (c) where we keepΩ0/γ1 = 0.5 andΩ2/γ1 = 1.5, which shows a double EIT effect havinfeatures of both Fig. 2(a) and (b). In this situation,the steady-state limit, both coherences termsρ01 andρ31 get developed which is reflected in the probesorption spectrum in terms of double EIT. This sytem can be analyzed in terms of the dressed stcreated by the coupling field and the pumping fieIt is the destructive interference between two exction pathways to these dressed states from the grostate|1〉 that gives rise to the EIT. We can see thin a more transparent manner by a simple analyIn the resonant condition, the interaction Hamiltonis H = hΩ1|1〉〈2| + hΩ0|0〉〈2| + hΩ2|2〉〈3| + h.c.,where |1〉, |0〉, |2〉, and |3〉 are the discrete atomistates (Fig. 1). The eigenvalues of this Hamilton

are 0, 0,±√

Ω21 + Ω2

0 + Ω22, respectively. The corre

sponding eigenstates to the zero eigenvalues areear combination of the states|1〉, |0〉, and |3〉. Thus,we have two dark states or the degenerated dark sin this case. If the atoms are in any or both of tdark states then EIT is exhibited. The interestingpect of this EIT situation is its controllability by appropriately selecting the field amplitudes for the couplifield and the pumping field. This has been showncurve (d) where we keepΩ0/γ1 = 0.5, Ω2/γ1 = 3.0,and the EIT dip-width and amplitude are controllas desired. The real part ofρ12 or Re(ρ12) is plot-


Fig. 2. Im(ρ12) as a function of∆1/γ1 for the parametric conditions:γ1 = γ2 = γ3 =1.0;∆0/γ1 = ∆2/γ1 = 0.0; Ω1/γ1 = 0.01. Curves (a),(b), (c), and (d) are forΩ0/γ1 = 0.5, Ω2/γ1 = 0.0; Ω0/γ1 = 0.0, Ω2/γ1 = 1.5; Ω0/γ1 = 0.5, Ω2/γ1 = 1.5; andΩ0/γ1 = 0.5, Ω2/γ1 = 3.0,respectively.

Fig. 3. Re(ρ12) as a function of∆1/γ1 with all other conditions same as in Fig. 2.


Fig. 4. Im(ρ12) as a function of∆1/γ1 for the parametric conditions:γ1 = γ2 = γ3 =1.0; Ω1/γ1 = 0.01, Ω0/γ1 = 0.5, Ω2/γ1 = 1.5.Curves (a), (b), (c), and (d) are for∆0/γ1 = −1.0,∆2/γ1 = 0.0; ∆0/γ1 = −2.5,∆2/γ1 = 0.0; ∆0/γ1 = 0.0,∆2/γ1 = 2.0; and∆0/γ1 = −2.0,∆2/γ1 = 2.0, respectively.

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ted in Fig. 3(a), (b), (c), (d) for the same parameconditions of Fig. 2(a), (b), (c), (d). Fig. 3(a) showthe typical EIT dispersion curve for the three-levsystem inΛ configuration and Fig. 3(b), on the cotrary, represents EIT dispersion for the three-level stem in a ladder configuration [3]. Fig. 3(c) accounfor the combined EIT dispersion due to the preseof Λ and ladder sub-systems in this four-level stem. The most prominent effect can be seen near∆1/γ1 = 0 region where most of the changes ocur due to the double EIT condition. The contrlability of this dispersion by manipulating the copling and the pumping laser fields is clearly sein Fig. 3(d).

After studying EIT at exact resonant conditionsthe coupling and the pumping laser fields with atomtransition frequencies (e.g.,∆0/γ1 = 0, ∆2/γ1 = 0),we next study how these detunings bring changethe EIT characteristics of the system. In Fig. 4 we pIm(ρ12) as a function of∆1/γ1 with γ1 = γ2 = γ3 =1.0 andΩ1/γ1 = 0.01, Ω0/γ1 = 0.5, Ω2/γ1 = 1.5.Curves (a) and (b) are for non-zero coupling detunbut pumping field detuning zero, e.g.,∆0/γ1 = −1.0,

∆2/γ1 = 0.0 and ∆0/γ1 = −2.5, ∆2/γ1 = 0.0, re-spectively. We again obtain the double EIT but Edips shifted with each other or in other words siated at different values of∆1/γ1 in Figs. 4(a) and (b)As expected the EIT due toΛ-sub-system changinits location because only the coupling field detunis non-zero in these two curves. If the coupling fiedetuning is kept zero (∆0/γ1 = 0,0) and the pumping field is detuned (say,∆2/γ1 = 2) then there is ashift in the EIT resonance of the ladder-sub-systas shown in Fig. 4(c). Under the two-photon resoncondition, e.g.,∆0/γ1 = −2.0, ∆2/γ1 = 2.0, there isa merging of the two EIT peaks into a single peand there is a loss of symmetry in the resulting sinEIT peak (Fig. 4(d)). One side of this single EIT peis very sharply changing but the other side is slowchanging. This implies that controlling detuningsthe coupling laser and the pumping laser fieldslead to controlling of double EIT width, shape, andcation very effectively—something difficult to achievwith one of the sub-systems alone. The real part ofρ12(Reρ12) is shown in Fig. 5(a), (b), (c), (d) for the identical conditions of the parameters as in Fig. 4(a),



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(c), (d). The dispersion properties represented by thcurves are again the combined response of thethree-level sub-systems. The locations of peaksdips are sensitive functions of the values of∆0/γ1 and∆2/γ1, which can be clearly observed in Fig. 5(a), ((c). At the two-photon resonant condition (Fig. 5(dthere is merger of two dips which are related totwo sub-systems and there is an asymmetry in thisalso. This means that we can also control the dissive part very easily by changing the detuning pameters.

We further study our this EIT system under sodifferent parametric conditions. In Fig. 6, we plIm(ρ12) as a function of∆1/γ1 with γ1 = γ2 =γ3 = 1.0. In curve (a) we keep∆0/γ1 = −2.0,∆2/γ1 = 0.0, Ω0/γ1 = 0.5, Ω2/γ1 = 3.0, Ω1/γ1 =0.01. Double EIT appears as expected, however,EIT dip due toΛ-sub-system situated near∆1/γ1 =−2 is showing dispersion like features. Interestingthe Re(ρ12) under the same parametric conditionsshown in Fig. 7(a)) has a peak-like structure. Whwe set the two-photon resonant condition forladder-sub-system, e.g.,∆0/γ1 = −3.0, ∆2/γ1 = 3.0,with Ω0/γ1 = 1.0, Ω2/γ1 = 1.0, Ω1/γ1 = 0.1, thetwo EIT peaks are not only merging into one bproducing a true dispersion-like profile for Im(ρ12) in

Fig. 6(b) and a perfect dip-like feature in Re(ρ12) inFig. 7(b) situated near the region of∆1/γ1 = −3.0.This dispersion-like EIT profile in Im(ρ12) (or the dip-like feature in Re(ρ12)) can easily be controlled iwidth, amplitude, and position by suitably selectithe system parameters. For example, we can obthe dispersion-like EIT feature exactly at∆1/γ1 = 0,which is not easy to obtain with the any of the singsub-systems. By selecting∆0/γ1 = 0.0, ∆2/γ1 = 2.0,with Ω0/γ1 = 1.0, Ω2/γ1 = 4.0, Ω1/γ1 = 0.1, wehave shown how to control the location as wellwidth/height/shape of EIT profile for Im(ρ12) andRe(ρ12) in Fig. 6(c) and Fig. 7(c), respectively. Othe other hand we can completely suppress thedue to one of the sub-system by suitably adjustingmagnitude of control field and/or the pumping field.demonstrate this we keep∆0/γ1 = 0.0, ∆2/γ1 = 0.0,with Ω0/γ1 = 2.5, Ω2/γ1 = 6.0 andΩ1/γ1 = 0.01,and plot Im(ρ12) [Re(ρ12)] in Fig. 6(d) [Fig. 7(d)].Since the magnitude of the pumping field is largcompared to the coupling field, we get a considerasuppression of the EIT due to theΛ-sub-system in thissituation. However, converse will be true, i.e., the Edue to ladder-sub-system can also be suppressedmagnitude of the coupling field is larger with respeto the pumping field.


Fig. 6. Im(ρ12) as a function of∆1/γ1 for the parametric conditions:γ1 = γ2 = γ3 = 1.0. Curves (a), (b), (c), and (d) are for∆0/γ1 = −2.0,∆2/γ1 = 0, Ω1/γ1 = 0.01, Ω0/γ1 = 0.5, Ω2/γ1 = 3.0; ∆0/γ1 = −3.0,∆2/γ1 = 3.0, Ω1/γ1 = 0.1, Ω0/γ1 = 1.0,Ω2/γ1 = 1.0; ∆0/γ1 = 0.0,∆2/γ1 = 2.0, Ω1/γ1 = 0.1, Ω0/γ1 = 1.0, Ω2/γ1 = 4.0; and ∆0/γ1 = 0.0,∆2/γ1 = 0.0, Ω1/γ1 = 0.01,Ω0/γ1 = 2.5, Ω2/γ1 = 6.0, respectively.



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4. Conclusions

We have studied the phenomenon of EIT anddispersion properties in a four-level atomic systhaving inverted-Y configuration. Such a system cbe considered to be composed of two three-lesub-systems inΛ and ladder configurations. ThEIT characteristics of both sub-systems persistthis four-level system as the coherences generby the coupling field and the pumping field do nmutually destroy each other. By manipulating tsystem parameters it is possible to control the Echaracteristics of the system from a double EITa single EIT situation as well as from a absorptdip-like EIT feature to dispersion-like EIT feature(situated at∆1/γ1 = 0) in both real and imaginarparts of the susceptibilities. Also, we can supprthe EIT due to one of the subsystems as desiredchanging parameters. This kind of system is certauseful in any optical switching devices which abased on the phenomenon of EIT because conand variation of EIT are the main features of thsystem. The four-level system discussed here haseasily realized in a recent experiment [14]. Alsdispersion control can find important applicationsmanipulating group velocity of light pulses and opticbeam shaping.

Acknowledgements

We acknowledge the funding support from tNational Science Foundation and the Office of NaResearch.

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electromagnetically induced transparency and its dispersion properties in a four-level inverted-y...

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