PHYSICAL REVIEW A, VOLUME 62, 043407

Coherent population trapping under bichromatic fields

R.G. Unanyan,* S. Guerin,† and H.R. JauslinLaboratoire de Physique de l’Universite´ de Bourgogne, Centre National de la Recherche Scientifique, Boıˆte Postale 47870,

21078 Dijon, France~Received 5 May 2000; published 15 September 2000!

We explore extendedL systems driven by two resonant laser fields in a strong regime characterized by thefact that the Rabi frequencies are much bigger than the difference between the two field frequencies. We showthat a specific coherent superposition of the lower states is trapped during the dynamics. This extended trappedstate is different from the trapped state usually encountered in the rotating-wave-approximation regime. Weshow the robustness of this trapping in the sense that it is stable with respect to significant deviations of thefield parameters~frequencies and amplitudes!.

PACS number~s!: 42.50.Hz, 03.65.2w, 73.40.Gk

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I. INTRODUCTION

The dynamics induced by two resonant~or quasiresonant!laser fields interacting with a three-stateL system exhibits arich variety of phenomena and processes, such as lawithout inversion@1#, laser cooling@2#, population transfer@3#, and loss-free pulse propagation@4,5#. The underlyingmechanism of these coherent phenomena is the existena so-calledtrapped superposition wave function that doenot involve the upper bare state~denotedu2&, which in gen-eral induces incoherent losses of population, e.g., throspontaneous emission! @6#. This trapped~or dark! state is infact astationarystate of the dressed system atom-plus-fieMost of these processes can be adequately treated withilimits imposed by the rotating-wave approximation~RWA!@7#. One of the essential ingredients of this approximationthe requirement that each of the laser fields interact only wone pair of levels, which is satisfied either because theference of the transitions frequencies is sufficiently largebecause optical selection rules prevent additional couplinIn such cases the dynamics is dominated by two sinphoton resonances, each associated with the exchangesingle photon between one laser field and the atom or mecule. This results in an effective dressed Hamiltonian whthe trapped state~for equal field amplitudes! is given by

uFDRWA~ t !&5

1

A2@ u1&2u3& exp~2 idt !], ~1!

where we have denotedu1& andu3& the two lower bare stateand

d5vP2vS ~2!

the difference between the pump~coupling the 1-2 transi-tion! and Stokes~coupling the 2-3 transition! laser frequen-cies, as diagrammed in Fig. 1. This state involves the

*Permanent address: Institute for Physical Research of ArmeNational Academy of Sciences, Ashtarak-2, 378410, ArmeEmail address: unanyan@physik.uni-kl.de

†Email address: guerin@jupiter.u-bourgogne.fr

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lower bare states with a time-dependent phase that indquantum oscillations between these states with the beatquencyd. The preparation of this RWA trapped state cresult from the initial conditionuf(2`)&5u1& by switchingon the Stokes laser~suddenly or adiabatically! before thepump laser is switched on adiabatically~with pulse-shapedfields, for example!.

In the present work we explore the natural extensionthe trapped state beyond RWA in a strong-field regimefined by the condition that the Rabi frequencies are mubigger than the differenced of the two frequencies. In thisregime, each of the two lasers interacts with each of theof states. We refer to this system as anextendedL system.We obtain anextended trapped statethat has the property osuppressing the quantum oscillationsbetween the two lowerbare states. We will show that it reads

uFDq &5

1

A11uqu2~qu1&2u3&), ~3!

with

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FIG. 1. Diagram of linkage patterns among three atomic sta~full horizntal lines!, showing pump~P! and Stokes~S! transitions.

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R. G. UNANYAN, S. GUERIN, AND H. R. JAUSLIN PHYSICAL REVIEW A62 043407

q[^2ud•ePu3&/^1ud•ePu2&5^2ud•eSu3&/^1ud•eSu2& ~4!

the ratio of the dipole couplings, and has the desired propof being a stationary state of the dressed system atom-pfield for strong enough unequal field amplitudes. This stawhen chosen as an initial condition, stays trapped whenfields are suddenly switched on. In the dressed represetion, the RWA trapped state@Eq. ~1!# involves the stateu1&and the stateu3& dressed with one Stokes photon minus opump photon, while the extended trapped state discushere@Eq. ~3!# involves the statesu1& and u3& with no dress-ing.

It may be noted that such coupling of levels inL systemswere studied in the context of adiabatic passage@8# and ofoptical bistability @9#. Grishanin et al. demonstrated themechanism that gives rise to novel spectral structuresresonance fluorescence@10#. Xia et al.showed the possibilityof electromagnetically induced transparency in the preseof common coupling@11#. One example of such coupling habeen explored in the context of tunneling@12#. We willbriefly discuss this application to tunneling in Sec. IV.

In the next section we show numerical results exhibitthe extended trapped state. Section III is devoted to thelytic explanation of this result.

II. NUMERICAL RESULTS

The Hamiltonian with classical fields for the procereads

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a-

H~vt1u!5H01d• (j 5P,S

ejEjcos~v j t1u j !, ~5!

whereH0 is the Hamiltonian of the free three-level syste$u1&,u2&,u3&% of energiesE1 ,E2 ,E3 , d is the dipole momentoperator~coupling transitions 1-2 and 2-3, but not 1-3!. TheHamiltonian acts on the Hilbert spaceH5C3. The total elec-tric field containing the two main frequenciesv5(vp ,vs) ischaracterized by unit vectorsej , real-valued amplitudesE5(Ep ,Es), and the initial phasesu5(up ,us). The solutionstate vectoruf(t)& satisfies the time-dependent Schro¨dingerequation

i\]

]tuf~ t !&5H~vt1u!uf~ t !&. ~6!

We consider the following regime:~i! resonant fields\vP5E22E1 , \vS5E22E3, so that the pump-Stokes combnation maintains the two-photon Raman resonance betwthe statesu1& and u3&, ~ii ! fields weak enough so that thRabi frequencies are much smaller than the Bohr frequenassociated with the transitions 1-2 and 2-3, and~iii ! fieldsstrong enough so that the Rabi frequencies are not negligin comparison with the beat frequencyd5vP2vS . Underthese conditions, the effective Hamiltonian reduces to@8#

Heff~u1dt !5\

2 F 0 VP1VSe2 i(u1dt) 0

VP1VSei(u1dt) 0 q@VS1VPe2 i(u1dt)#

0 q@VS1VPei(u1dt)# 0G , ~7!

-b--

i-

in

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where we have introduced the Rabi frequencies\VP52^1ud•ePu2&EP and\VS52^1ud•eSu2&ES ~note the differ-ent definition ofVS compared to Ref.@8#! andu5uP2uS ,the difference between the initial phases of the classfields. Without loss of generality, we considerq51 and nor-malize the timet and the Rabi frequencies with respectd: t 5dt andV5V/d. We definet50 as the time when thefields are switched on. The trapped state~3! is denoteduFD

1&.In Fig. 2 we have plotted for various field amplitudes

PD~ t !5 z^FD1uf~ t !&Hz2, ~8!

the projection of the state vector solution on the trapped sand

P2~ t !5E0

2p

du z^2uf~ t !&Hz2, ~9!

the population history on stateu2& ~averaged over the classcal phase in order to remove the rapid oscillations!, where

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the scalar product inH is denoted u&H . Figure 2~a! showsthat uFD

1& can beapproximately trappedfor strong equalfield amplitudes, such thatVP5VS@d and for particularvalues, e.g.,VP5VS519.72d. We will show that this resultis due to zeros of Bessel functions~see next section for details! and that it is related to the tunneling stabilization otained by Grossmannet al. @13# ~see the Conclusion for details!. This mechanism is not robust, as shown in Fig. 2~b!,where the trapping is lost for slightly different field ampltudesVP5VS520d. Figure 2~c! diplays therobust trappingfor strong unequal field amplitudes (VP ,VS@d) VP520dand VS510d. The robustness of this trapping is shownFig. 3, which displays mintPD(t), the minimum over time (tP@0,100d#) of the projection of the state vector solution othe trapped state, as a function of various field amplitudRegions of robust trapping~white areas! can be seen forstrong unequal field amplitudes. They lie symmetrically oside a zone centered around the lineVP5VS . When thisline VP5VS is approached the trapping is suppressedspecific ranges of amplitudes, which become wider: the tr

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COHERENT POPULATION TRAPPING UNDER . . . PHYSICAL REVIEW A62 043407

ping is less and less robust. The white zone of trappingtends up to zero amplitude for one of the fields. This trappfor one field has been analyzed by Kilinet al. @12# andPaspalakis@14#. We have checked that this trapping is alrobust with respect to various detunings of the order ofd.

III. ANALYTIC RESULTS

The HamiltonianH(vt1u) being quasiperiodic with thetwo frequenciesvP andvS , a natural way to study its properties is to invoke Floquet theory. We denote the timindependent quasienergy operator~or dressed Hamiltonian!,which allows us to take into account the full Hamiltonian fthe atom-plus-field@15–18#, as

K~u!5H~u!2 i (j 5P,S

\v j

]

]u j. ~10!

It acts in the enlarged spaceK5H^ L2(dup/2p)^ L2(dus/2p) where eachL2(du j /2p) is the Hilbert spaceof square-integrable functions of the angleu j . We denote thefull dressed basisun;kP ,kS& wheren51,2,3 stands for thebare states and the positive or negative integerskP ,kS for therelative photon numbers. The solution state vector canwritten as uf(t)&5(mcm exp(2i lmt/\)uCm(u1vt)& withlm the dressed eigenvalues~the quasienergies! associated

FIG. 2. State vector solution projected on the trapped sPD(t) and population history on state 2~averaged over the classica

phase! P2(t), as a function of normalized timedt for q51, and~a!VP5VS519.72d @numerical, full lines; forPD , formula ~20a!,dotted line; forP2, formula ~20b!, dotted line#, ~b! VP5VS520d,and ~c! VP520d andVS510d.

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with the Floquet eigenvectorsuCm& and cm5^Cmuf(t50)&K . The dressed Hamiltonian of our problem canapproximated by aneffectiveoperator @19# Keff'R0

†KR0 ,with R05diag@1,e2 iuP,e2 i(uP2uS)#, which reads

Keff~u!52 i\d]

]u1

\

2Heff~u!. ~11!

The quasienergy operator~11! can be seen as the effectivdressed operator written in the dressed basis$u1;0,0&,u2;21,0&,u3;21,1&%.

The stateuFD1& has the necessary desired condition

have in general no matrix element with levelu2& for anypositive or negative integerskP and kS :^2;kP ,kSuKuFD

1&K50. We note that this condition is not sufficient to havetrapped state in the strong regime considered here. Infollowing we prove that this stateuFD

1&, under the conditionsof strong fields of sufficiently different amplitudes, issta-tionary for the full atom-plus-field HamiltonianKeff , mean-ing that this state, chosen as the initial condition, is suchz^FD

1uf(t)&Hz2'1.It proves to be convenient to introduce the unitary tra

formation

S~u!51

A2F e2 i u/2 0 e2 i u/2

0 A2 0

2eiu/2 0 eiu/2G , ~12!

leading to the transformed Hamiltonian

te

FIG. 3. Minimum over time of the projection of the state-vectsolution on the trapped state mintPD(t) as a function of various fieldamplitudes forq51. The crosses labeleda, b, and c refer to theparameters relevant for Figs. 2~a!, 2~b!, and 2~c!, respectively.

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R. G. UNANYAN, S. GUERIN, AND H. R. JAUSLIN PHYSICAL REVIEW A62 043407

S†KeffS52 i\d]

]u1\F 0 0 0

0 0 S cos~u/2!

0 S cos~u/2! 0G

2\F 0 0 d/2

0 0 iD sin~u/2!

d/2 2 i D sin~u/2! 0G , ~13!

where we have decomposed the Hamiltonian in realimaginary parts using the notationsS5(VP1VS)/A2 andD5(VP2VS)/A2. The idea to solve the eigenvalue prolem is to first diagonalize the Floquet Hamiltonian, whiincludes the term proportional toS ~the dominant term! andto treat the rest~proportional toD and tod) as perturbationsWe denote byT the matrix that diagonalizes the dominaterm

T~u!51

A2F A2 0 0

0 e2 i g(u) 2e1 ig(u)

0 e2 i g(u) eig(u)G ~14!

with

g~u!5x sin~u/2!, x52S/d. ~15!

To treat the resonant terms of the resulting Hamilton(ST)†KeffST ~which are proportional toD andd) in the limitof strong field such thatS@D,d, we keepu-independentterms~as in the RWA! using Bessel functions:

sin~u/2!e62ig56 iJ1~2x!, e6 ig5J0~x!, ~16!

which gives the following approximate Hamiltonian

Heff'2\30

d

2A2J0~x!

d

2A2J0~x!

d

2A2J0~x! 0 DJ1~2x!

d

2A2J0~x! DJ1~2x! 0

4 ,

~17!

satisfyingKeff[(ST)†KeffST'2 i\d]/]u1Heff . We obtainthree families of eigenvectors ofKeff , which we denote asC0eiku and C6eiku with C0 and C6 the eigenvectors oHeff .

When the field amplitudes areequal ~i.e., D50!, we ob-tain the eigenvalueslm and the eigenvectors ofHeff :

l050, l656\dJ0~x!/2, ~18!

uC0&51

A2 F 0

21

1G , uC6&5

1

2F 7A2

1

1G . ~19!

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The approximate projections on the trapped state and on2, respectively, can thus be written as

PD~ t !' cos2@l1t/\#, ~20a!

P2~ t !'2F(k

~J2k11~x!!2Gsin2@l1t/\#. ~20b!

The populationoscillatesbetweenuFD1& and the orthogona

states involving the intermediate stateu2&. We remark thatthis oscillation occurs with smaller frequency for strongfield amplitudes. Thusthe stateuFD

1& is generally not atrapped state when Rabi frequencies are equal. We showthis result in Fig. 2~b!, which comparesPD(t) calculatednumerically with formula~20a! and numericalP2(t) withformula ~20b!. They fit quite well~better for smaller times!.This oscillation ~and consequent loss! is suppressed ifl1

50, which occurs if the equal field amplitudes are such t2S/d is a zero of the Bessel functionJ0. We can thus ob-serve an approximate trapping~for small enough times! forthese isolated field amplitudes. This is shown in Fig. 2~a!,where the field amplitudes have been chosen such that 2S/dis approximately a zero of the Bessel functionJ0 . This ap-proximate trapping can also be observed in Fig. 3 as isolapoints on the lineVP5VS . We remark that this is related tthe nonrobust stabilization demonstrated by Grossmannet al.@13# ~see the Conclusion for details!.

For different field amplitudes, such that

d!uDu!S, ~21!

we obtain the eigenvalues@at first order indJ0(x)/DJ 1(2x)#

l0'0, l6'6\DJ1~2x! ~22!

and the associated eigenvectors@at zeroth order indJ0(x)/DJ1(2x)#

uC0&'1

A2 F 1

0

0G , uC6&'

1

A2 F 0

71

1G . ~23!

The projection can thus be written for allt as

PD~ t !'1, ~24!

which means that the population stays on the stateuFD1& with

a very small contribution on levelu2&, as shown by the nu-merical simulation of Fig. 2~c!. Losses to levelu2& aresmaller for strongerD : The stateuFD

1& is thus a trappedstate when field amplitudes are sufficiently different. The ap-proximations~22! and~23! are not valid whenDJ1(2x) anddJ0(x) are of the same order. This case has to be treateif the field amplitudes were equal. Thus the robustness oftrapped state is limited by ranges of field amplitudes srounding 4S/d being zeros of the Bessel functionJ1. Theseintervals are smaller for strongerD compared tod. They canbe observed in Fig. 3 as parallel lines emerging very appromately perpendicular to the lineVS5VP .

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COHERENT POPULATION TRAPPING UNDER . . . PHYSICAL REVIEW A62 043407

For field amplitudes such that

d!uDu&S, ~25!

meaning that one field amplitude~say VS! is much smallerthan the other one, we can use perturbation theory to treMore precisely, we define K15R1

†KeffR1 with R1

5diag@1,1,eiu# and obtain

K1~u!52 i\d]

]u1

\

2 F 0 VP 0

VP 0 VP

0 VP dG

1\VS

2 F 0 e2 i u 0

eiu 0 eiu

0 e2 iu 0G . ~26!

In the strong-field regimed!VP , the first term gives rise toa trapped state@12,14#. The last term gives rise to a multphoton resonance that is very weak whenVS!VP and thatthus can be treated as a small perturbation of the trapstate. This explains the existence of the trapped state wone of the fields is much smaller than the other one~leadingto D'S).

IV. CONCLUSION

In this paper we have explored the trapping mechanunder two laser fields in a strong regime. We have shothat a specific coherent superposition of the lower statetrapped during the dynamics for strong unequal field intsities with the suppression of quantum oscillations betwthe two states. This stabilization is robust with respect tofield parameters.

This suppression of quantum oscillations finds its mapplication in the coherent destruction of tunneling. In t

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context of tunneling, the possibility of trapped states had fibeen established by Grossmannet al. @13#. One can interpretthe linear combinationsuL&5@ u1&1u3&]/A252uFD

2(t)&and uR&5@ u1&2u3&]/A25uFD

1(t)& as wave packets localized in left and right wells, respectively, assuming that tbare statesu1& and u3& are the lowest eigenstates of a symmetric double-well potential. Grossmannet al. showed that,when driven by an external nonresonant monochromatic fiand for specificisolatedvalues of amplitude and frequencythe system exhibitsuL& anduR& as stationary states~up to anirrelevant phase!. Thus the tunneling is coherently suppressed for specific field parameters. This suppressionhowever, very sensitive to the field parameters. This stabzation has been technically explained within the Floquet fmalism by the occurrence of exact crossings of the quasergies for these specific field parameters@13,20#. Morerecently, Kilin et al. @12# and Paspalakis@14# have shown adifferent mechanism of stabilization. They predicted the spression of tunneling~i! in a robust way and~ii ! without everpopulating the intermediate upper levels, provided the upstate is properly chosen. This last point is crucial to avoidincoherent losses from the upper levels. For this purpose,needs to couple the two lower states involved in the tunning with one electronically excited state that has conquently to be neither symmetric nor antisymmetric with rspect to the tunnel coordinate@12#. This coupling can bealternatively realized by an additional laser that couplesper levels as shown in@21#. Our work in this context oftunneling shows that the use of a second laser does notprove significantly the stabilization~see Fig. 3!.

ACKNOWLEDGMENTS

R.U. thanks l’Universite´ de Bourgogne for the invitationand the Alexander von Humboldt Stiftung for financial asstance. We acknowledge support by INTAS 99-00019.

ev.

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in,

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