806 J. Opt. Soc. Am. B/Vol. 21, No. 4 /April 2004 Zhu et al.

Steady-state population inversion by multiphotonelectromagnetically induced transparency

Yifu Zhu, Joseph Saldana, and Lingling Wen

Department of Physics, Florida International University, Miami, Florida 33199

Ying Wu

National Key Laboratory for Laser Technique and Department of Physics, Huazhong University of Science andTechnology, Wuhan 430074, China

Received June 27, 2003; revised manuscript received October 28, 2003; accepted December 4, 2003

We show that electromagnetically induced transparency suppresses nonlinear absorption of all orders in amultilevel atomic system and leads to selective, multiphoton excitation of resonantly coupled atomic states.Under appropriate conditions, higher-order nonlinear absorption becomes dominant and selective steady-statepopulation inversion is created among the resonantly coupled states. © 2004 Optical Society of America

OCIS codes: 270.1670, 020.4180, 190.5650.

1. INTRODUCTIONExploration of population transfer and population inver-sion in atomic systems in both transient and steady statesis important for understanding of atom–photon interac-tions as well as applications in laser physics and nonlin-ear optics. Atomic coherence and population distributionmay be manipulated in multilevel systems coupled bymultiple laser fields. Many studies of adiabatic popula-tion transfer with overlapping laser pulses have beenpublished, and adiabatic population inversion was per-formed efficiently in multilevel quantum systems.1–4

Later, precise control of population transfer by stimulatedadiabatic Raman processes was developed, and completepopulation transfer was achieved by use of partially over-lapping Raman pulses.5 However, it is more difficult toachieve steady-state population inversion. It has beenshown that absorptive and dispersive properties of atomicmedia can be manipulated by coherent population trap-ping and electromagnetically induced transparency(EIT).6–10 Coherent population trapping and EIT can beused for the control of population and coherence of atomicsystems in both transient and steady-state regimes,which leads to interesting phenomena such as laserswithout population inversion11,12 and large dispersionwithout absorption.13,14 In contrast, it has been shownthat population inversion without lasing can be created inthree-level systems with a quantum beat configuration.15

In recent years there have been many studies of multi-photon coherence in multilevel systems and their applica-tions in various nonlinear optical processes.16–19 Herewe analyze a multilevel ladder-type atomic system coher-ently coupled by multiple laser fields. We show that, in athree-level system, EIT is created, and nonlinear absorp-tion of all orders is suppressed by destructive interfer-ence. Such suppression of nonlinear absorption of all or-ders may be referred to as multiphoton EIT and can bemodified by two-photon Raman detuning of the first pump

0740-3224/2004/040806-05$15.00 ©

laser and the control laser. We show explicitly that se-lective nonlinear excitation leads to steady-state popula-tion inversion in five- and six-level systems. In particu-lar, population inversion may be generated betweenstates coupled by resonant three-photon excitation andmay be useful for amplification and generation of short-wavelength coherent light. The multiphoton EIT in co-herently coupled atomic systems may be also used forstudies of nonlinear processes such as four-wave mixingand coherent hyper-Raman scattering20,21 for efficientgeneration of light at low pump intensities.

2. COHERENTLY COUPLED MULTILEVELSYSTEMA general treatment of the multiphoton excitation in amultilevel atomic system based on the wave-function ap-proach has been discussed in the literature.22 We con-sider the coherently coupled multilevel system depicted inFig. 1(a). Three atomic states (ground states u0& and uc&and excited state u1&), a control laser (frequency vc andRabi frequency 2Vc), and a pump laser (frequency v1 andRabi frequency 2V1) form a standard L-type configura-tion and exhibit EIT. A ladder of N 1 1 atomic states,designated u0&, u1&,...uN&, are coupled by N laser fields.We consider the single-photon coupling between twoneighboring states [the ith field with frequency v i andRabi frequency 2V i couples states ui 2 1& and ui& (i5 1, 2,...N)] as shown in Fig. 1(a). But, for more-generaldiscussions, multiphoton coupling between neighboringstates is allowed, and the results will be the same if onereplaces the single-photon Rabi frequency with theequivalent multiphoton Rabi frequency, as shown by anexample of two-photon coupling between states u2& and u3&in Fig. 1(b).

We define the atomic-state vector as uC& 5 A0u0&1 Ac exp@i(k1 2 kc) – r#uc& 1 ( j51

N Aj exp@i(k1 1 ¯ 1 kj)

2004 Optical Society of America

Zhu et al. Vol. 21, No. 4 /April 2004 /J. Opt. Soc. Am. B 807

• r#u j&. In the rotating-wave approximation, the time-dependent Schrodinger equation for probability ampli-tudes Aj is given by

S d

dt1 gc 2 iDcDAc 5 iVc* A1 , (1a)

S d

dt1 g1 2 iD1DA1 5 iV1A0 1 iVcAc 1 iV2* A2 ,

(1b)

S d

dt1 gn 2 iDnDAn 5 iVnAn21 1 iVn11* An11 ,

AN11 [ 0, n 5 2,...,N, (1c)

where D j 5 (v1 1 v2 1 ¯ 1 v j) 2 e j ( j 5 1, 2...N) rep-resents the j photon detuning, Dc 5 (v1 2 vc) 2 ec

Fig. 1. (a) Resonantly coupled N 1 1-level system with single-photon coupling between neighboring states; i.e., states ui 2 1&and ui& are coupled by coherent field v i with Rabi frequency2V i (i 5 1 2 N). (b) Resonantly coupled N 1 1-level systemwith single-photon as well as multiphoton coupling betweenneighboring states.

5 D1 2 D1c (D1c 5 vc 2 ec) is the two-photon Raman de-tuning, and e j is the energy of atomic state u j& (e0[ 0 is taken for ground state u0&). g j ( j 5 1, 2,...N) isthe decay rate of state u j&. The steady-state solutions ofEqs. (1) can be readily derived. It is straightforward toshow that, when Dc 5 gc 5 0, all Ai 5 0 (i 5 1, 2,...,N)and Ac 5 2(V1 /Vc)A0 . That is, all linear and nonlinearabsorptions vanish, irrespective of the other frequency de-tunings. This phenomenon may be referred to as multi-photon (N-photon) EIT (for N 5 2, this leads to two-photon EIT, as was discussed in Refs. 23–25), which is asimple extension of three-level EIT and coherent popula-tion trapping.6,7 In a different context in which the Rabifrequencies of the coupling field and the pump field (Vc; V1) are comparable, three-photon EIT in an N-typefour-level system was observed recently.26 We show forthe multilevel system depicted in Fig. 1 that varying two-photon detuning Dc will modify the multiphoton EIT andlead to the possibility of selective nonlinear excitation andthe creation of selective population inversion among thecoupled atomic states in the coupled N 1 1 level system.The general analytical solutions are readily attainable,but their derivation is tedious and they offer no clearphysical insight.

To show explicitly that the modified multiphoton EITresults in selective nonlinear excitation and in the cre-ation of population inversion in the coupled ladder-typeatomic system, we present analytical solutions for a five-level EIT system (N 5 3). Furthermore, we present nu-merical calculations for a six-level EIT system (N 5 4).The similarity of the results for the five- and six-level sys-tems implies that selective steady-state population inver-sion may be extended to an arbitrary N-level system.

3. ANALYSIS OF THE FIVE-LEVEL SYSTEMWith the standard EIT condition (Vc @ V1), one has A0' 1, and the steady-state solutions for the five-level (N5 3) system are given by

Ac 5 @~D2D3 2 uV3u2!Vc* V1#/S, (2a)

A1 5 2Dc~D2D3 2 uV3u2!V1 /S, (2b)

A2 5 V1V2DcD3 /S, (2c)

A3 5 2DcV1V2V3 /S, (2d)

where S 5 (D1Dc 2 uVcu2)(D2D3 2 uV3u2) 2 uV2u2DcD3 @Dj[ D j 1 ig j ( j 5 c,1, 2, 3)#. Neglecting gc and assumingthat all lasers except coupling laser vc and first pump la-ser v1 are on resonance with their respective transitions,one has D i 5 D1 (i 5 1, 2, 3), and the population prob-abilities are given by

P1 5 Dc2V1

2@~ uV3u2 2 D12 1 g2g3!2

1 D12~g2 1 g3!2#/uSu2, (3a)

Pc 5 Vc2V1

2@~ uV3u2 2 D12 1 g2g3!2

1 D12~g2 1 g3!2#/uSu2, (3b)

808 J. Opt. Soc. Am. B/Vol. 21, No. 4 /April 2004 Zhu et al.

P2 5 V12V2

2Dc2~D1

2 1 g32!/uSu2, (3c)

P3 5 V12V2

2V32Dc

2/uSu2, (3d)

where Pi (i 5 c, 1, 2, 3) is the population probability instate ui&. The numerical calculations of population prob-abilities Pi (i 5 1 –3) and relative absorption rates Im(rij)for the three pump fields in the five-level system (N5 3) are plotted in Fig. 2 versus two-photon detuningDc 5 D1 . The identical spectral line shapes in Figs. 2(a)and 2(b) show that P1 is proportional to the single-photonabsorption (u0& → u1&), P2 is proportional to the two-photonabsorption (u0& → u1& → u2&), and P3 is proportional to thethree-photon absorption (u0& → u1& → u2& → u3&). For two-photon detuning Dc 5 0, one has Pi 5 0 (i 5 1,2,3), andthere is no atomic population in any excited state ui&. Alllinear and nonlinear absorptions are suppressed. Thecontrol field creates destructive interference for multipho-ton transitions, leading to multiphoton EIT (where P1represents the single-photon EIT, P2 represents the two-photon EIT, and P3 represents three-photon EIT, whichare all centered at Dc 5 0). When the two-photon detun-ing Dc 5 0, the atoms are driven into the excited states by

Fig. 2. (a) Relative absorption rates for pump laser v i @i5 1, 2, 3, corresponding to Im( p01), Im( p12), and Im( p23), re-spectively] and (b) population probabilities Pi (i 5 1 –3) in thefive-level system (N 5 3) versus frequency detuning Dc 5 D1 ofthe first pump laser [all other lasers are on resonance with theirrespective transitions, i.e., D j 5 0 (i 5 2, 3), D1c 5 0]. Single-photon absorption rate Im( p01) is proportional to P1 , two-photonabsorption rate Im( p12) is proportional to P2 , and three-photonabsorption rate Im( p23) is proportional to P3 . Other parametervalues are Vc 5 0.5g1 , V1 5 V2 5 V2 5 0.02g1 , g2 5 0.2g1 ,g4 5 0.1g2 , and g1 5 0.001g1 .

various orders of linear and nonlinear absorption. Whenthe coupling laser is on resonance (D1c 5 0), two-photondetuning Dc 5 D1 and the multiphoton EIT exhibit sym-metrical line profiles across D1 5 0 (Fig. 2). The band-widths of the multiphoton EITs are different for the dif-ferent orders, and the width of the higher-order EIT issmaller than that of the lower-order EIT. Therefore thepeak nonlinear absorption of higher orders occurs at asmaller D1 value than lower-order absorption, as shownin Fig. 2. Therefore, one achieves selective multiphotonexcitation by varying detuning D1 . For an N-level sys-tem it is reasonable to assume that if D1(n) (n 5 1 2 N)is the detuning D1 for the maximum nth-order absorptionin the system, one has D1(1) 5 D1(2) 5 D1(3) 5 ...5 D1(N) . This fact can be used to optimize a specific or-der of nonlinear absorption.

Selective excitation of different orders of nonlinear ab-sorption makes it possible to create selective steady-statepopulation inversion among various pairs of atomic statesin a five-level system. For example, when two-photon ab-sorption u0& → u1& → u2& is optimized [with appropriate de-tuning D1 5 Dc (D1c 5 0)], excited state u2& is preferen-tially populated because single-photon absorption u0& →u1& at this value of Dc is still largely suppressed. If Rabifrequency V2 is reasonably large, such that the stimu-lated two-photon absorption rate is greater than sponta-neous decay g2 , then a steady-state population inversionbetween states u2& and u1&, P2 . P1 , may be created. Thesame physical mechanism can be applied to other nonlin-ear excitations (such as two-photon excitation u1& → u2& →u3& and three-photon excitation u0& → u1& → u2& → u3&).Therefore the steady-state population inversion of variousorders can be created in the five-level atomic system withmoderate pump Rabi frequencies V2 and V3 . From Eqs.(3), the conditions for steady-state population inversionfor various pairs of atomic states are given by

V32 . g3

2 1 D12 ~P3 . P2!, (4a)

V22 .

@~ uV3u2 2 D12 1 g2g3!2 1 D1

2~g2 1 g3!2#

D12 1 g3

2

~P2 . P1!, (4b)

D12 . Vc

2 ~P1 . Pc!, (4c)

V22 .

Vc2@~ uV3u2 2 D1

2 1 g2g3!2 1 D12~g2 1 g3!2#

V32Dc

2

~P3 . Pc!. (4d)

We note that population inversion P3 2 Pc . 0 is cre-ated by three-photon excitation u0&–u1&–u2&u3&, which canbe used for amplification of shorter-wavelength lightthrough the dipole allowed transition u3&–uc& at frequencyvm 5 vc 1 v2 1 v3 . In real atomic systems, typicallyg1 . g2 . g3 , the above conditions can be selectively sat-isfied in a parameter range of V2 . g2 , D1 ; g2 , g2, V3 , Vc , and Vc ; g1 @ V1 . Experimental candi-dates for such a system can be found in alkaline atoms.For example, we consider 87Rb atoms with the designatedstates chosen as follows: 5S1/2 , F 5 1 as u0&; 5S1/2 , F5 2 as uc&; 5P1/2 as u1&; 5D3/2 as u2&; and nP1/2 as u3&. The

Zhu et al. Vol. 21, No. 4 /April 2004 /J. Opt. Soc. Am. B 809

corresponding transitions are u0&–u1& and uc&– u1& at 795nm, u1&–u2& at 762 nm, and u2&–u3& at ;1500 nm. The rel-evant decay rates are then g1 ; 5.6 MHz, g2; 0.7 MHz, g3 ; 0.08 MHz, and gc ; 0.005 MHz. InFig. 3 we show the calculated population probability Pi[Fig. 3(a)] and the normalized population inversion (Pi2 Pj)/(Pi 1 Pj) (i . j) [Fig. 3(b)], with parameterstaken for 87Rb atoms. Note that, because of the EIT con-dition, Vc @ V1 , the absolute probabilities of the exis-tence of excited state populations are very small @Pi! P0 ; 1 (i 5 c, 1–3)#. However, the normalized popu-lation inversion can become very large, approaching 100%near the peak. Therefore, almost complete populationinversion may be selectively created for the coherentlycoupled atomic system. It is interesting to note thatsteady-state population inversions P3 . Pc , P3 . P2 ,and P2 . P1 can be created simultaneously. The calcu-lations demonstrate that steady-state population inver-sion in a coherently coupled atomic system can be selec-tively obtained with moderate pump Rabi frequencies V2and V3 when the control laser Rabi frequency is set atVc ; g1 @ V1 .

Fig. 3. (a), (b) Population probabilities Pi (i 5 c, 1–3) and (c),(d) normalized population inversion i 2 j 5 (Pi 2 Pj)/(Pi 1 Pj)in the five-level system (N 5 3) versus frequency detuning Dc5 D1 of the first pump laser [all other lasers are on resonancewith their respective transitions; i.e., D j 5 0 (i 5 2, 3), D1c5 0]. The parameter values are Vc 5 g1 , V1 5 0.05g1 , V25 V3 5 0.5g1 , g2 5 0.14g1 , g3 5 0.015g1 , and gc 5 0.001g1 .

4. ANALYSIS OF THE SIX-LEVEL SYSTEMTo show that suppression of light absorption in all ordersby multiphoton EIT and selective population inversioncan be extended to other multilevel atomic systems (Fig.1) with N . 3, we consider next a six-level atomic system(N 5 4). Because the analytical solutions are tediousand offer no physical insight, we have chosen to presentnumerical calculations as in Fig. 3, for which the calcu-lated population probabilities Pi (i 5 c, 1, 2,...,4) are plot-ted in Figs. 4(a) and 4(b) and population inversion Pi2 Pj (i . j) are plotted in Figs. 4(b) and 4(c). The rel-evant parameters are chosen to be different but consis-tent with those of the five-level (N 5 3) system. The cal-culations show that, at the EIT center (all lasers are onresonance), all absorptions are suppressed @Pi 5 0 (i5 1, 2,...,4)# and the multiphoton EIT is created. As thefirst pump laser is detuned from the EIT center, selectiveexcitation of the nonlinear absorption occurs, and thepopulation inversion is then created. As an independentcheck of the validity of the results related above, we alsonumerically solved the density matrix equations for thefive- and six-level systems. The calculations show that,for the identical parameters, the numerical results fromthe density matrix equations demonstrate the samesteady-state population inversion and are consistent withsolutions of Eqs. (1).

Although we have considered only atomic systems withneighboring states resonantly coupled by direct single-photon transitions, the analysis and the results presentedhere can easily be extended to similar atomic systemswith neighboring states coupled by resonant multiphotontransitions through virtual intermediate states or far-detuned real states [Fig. 1(b)]. One can derive similar re-sults by simply replacing single-photon coupling Rabi fre-quency V i with multiphoton coupling Rabi frequency

Fig. 4. (a) Population probabilities Pi (i 5 c, 1–4) and (b) nor-malized population inversion i 2 j 5 (Pi 2 Pj)/(Pi 1 Pj) in thesix-level system (N 5 5) versus frequency detuning D1 of thefirst pump laser [all other lasers are on resonance with their re-spective transitions, i.e., D j 5 0 (i 5 2, 3, 4), D1c 5 0]. The pa-rameter values are Vc 5 g1 , V1 5 0.05g1 , V2 5 V3 5 V45 0.6g1 , g2 5 0.2g1 , g3 5 g4 5 0.1g1 , and gc 5 0.001g1 .

810 J. Opt. Soc. Am. B/Vol. 21, No. 4 /April 2004 Zhu et al.

Vnj . Because the frequency detuning of the resonantmultiphoton coupling from the real intermediate states isusually large, the required values of Rabi frequencies V jcan be obtained only with much stronger pump lasers.For example, the equivalent two-photon Rabi frequency isV (2)j 5 V j

2/D j8 (D j8 is the detuning from the intermedi-ate state and V i is the single-photon Rabi frequency).The required pump laser intensity is }V j

2 5 V (2)jD j8,which will be much greater because of the typically largevalue of detuning D j8 from the real intermediate states.

5. CONCLUSIONSIn conclusion, we have shown that multiphoton electroni-cally induced transparency suppresses linear and nonlin-ear absorption to all orders in a ladder-type multilevelatomic system. Selective excitation of different orders ofnonlinear absorption may be achieved simply by manipu-lation of two-photon Raman detuning Dc . Selectivepopulation inversion among various pairs of atomic statesin five- and six-level systems can be attained with moder-ate coupling Rabi frequencies and is expected to be appli-cable to an arbitrary N-level system. The multiphotonEIT may be useful for nonlinear spectroscopic measure-ments and generation of short-wavelength light based onmultiphoton nonlinear optical processes such as four-wave mixing and hyper-Raman generation.20,21 The gen-eral analysis above is not expected to be applicable to aDoppler-broadened multilevel system because it is diffi-cult to eliminate the Doppler shifts for multiphoton pro-cesses that involve odd numbers of photons. Neverthe-less, it is possible to explore the multiphoton EIT effectsby carrying out experiments with cold atoms confined inmagneto-optical traps in which the Doppler effect can beneglected.

ACKNOWLEDGMENTSThe research of Y. Zhu is supported by the National Sci-ence Foundation (grant 0140032) and the U.S. Office ofNaval Research (grant N00014-01-1-0754). That of Y. W.is supported in part by the National Natural ScienceFoundation of China through grants 90108026, 60078023,and 10125419 and by the Chinese Academy of Sciencesthrough the 100 Talents Project and grant KJCX2-W0-4.

Y. Wu’s e-mail address is yingwu2@eyou.com.

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