Volume 134, number3 PHYSICSLETTERSA 26 December1988

THE PERIODICALLY KICKED TWO-LEVEL ATOM

N.D. SEN GUPTATata InstituteofFundamental Research,Homi BhabhaRoad,Bombay400005,India

Received28 July 1988; revisedmanuscriptreceived12 October1988;acceptedfor publication12 October1988Communicatedby A.R. Bishop

The exactsolutionfor thetwo-level atomdrivenby aperiodicdelta-functionpulseis presented.

Recently, in this journalthereappeareda noteby d~PMuriel [1], in which he attemptedto presenta so- i — = (Q +a3w— aa1 ~ ~“c

5( I— nx)) W, (1)dt \.

lutionof the Liouville equationcorrespondingto theproblemof the two-levelatom driven by a periodic wheredelta-functionpulse.But the solution presentedin

2Q=w1+w2, W=m1—w2>O, (2)the referrednote is incomplete,as the elementsof

the densitymatrix at any instantt are expressedin andhW1, hw2 are the energiesof the two levels, hatermsof thoseat the instantsnr, wherer is the pe- is theproductof theatomictransitionfactorandthenod of the pulseddelta-functionandn are positive strengthof the driving externalfield. ~= 1 for steadyintegers. But what one needs, in practise, is an fields and~ = — 1 for alternatingfields.Fromeq. (1),expressionof the elementsin termsofthoseat some the discontinuityat t_— nr (n integer) is given byinitial instant.For this onehasto know the values

n~r+O)=exp(i2o1~t)Y’(nT—O),of the elementsat n; which is the major part of theproblemproposed.Notwithstanding,to obtain the wherewe usedthe relations[2—4]final expressiononefurtherhasto sumovera large

W(nr)=~[~P(nr+O)+W(nz—O)] (3)numberof similar terms,which in itself is a formi-dabletask.The objectof this note is to obtain the andsolutiondirectly in termsof initial values,in a con-

tanIz=~a. (4)cise form.

Sincethe densitymatrix is constructedfrom the Integratingeq. (1) between—0 and I,

wave function, it is betterto investigatethe Schrö-exp[i(Q+a3W)t] W(t)— ~?‘(—0)

dingerequation,which is easierto managethan then

Liouville equation.TheSchrodingerequationfor a = ia ~ exp[i (Q+a3w)pt] e~‘o~V’(pr),two-level systemunderthe actionof periodicdelta-function pulsesmaybe written, with the help of the

for n~<t<(n+l)r, (5)Pauli matricesn

0 l\ =ia>J~exp[i(Q+a3w)pt}e”a1~P(pi)~l 003=~~ —1) and oI=(l 0),

x(l—~c5~1)),fort=ni. (6)

as Expression(5) is alsovalid whenthe summationin

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Volume 134, number3 PHYSICSLETTERSA 26 December1988

eq. (1) extendsto ~. For finite N, the summation Finally for any arbitrary initial statein eq. (5) extendsto N for I>Nt. Writing eq. (6)for I=(n—l)r andsubtractingfrom it, oneobtains ~P(—0)=cos9u÷+sinOu_ (Oreal) (19)

W(nr)=e”~A~W( (n— 1 )r) (7) W(t)= [(A+iC) cos0+iB sinO]e’Te~1°’’u±

where + [(A—iC) sinO+iB sin0]e_~~mTe_iw2tu_(20)

A~(wr,~)=exp(i~”a1p) exp(—ia3wr) (nt<t< (n+ l)r). It maybe pointed outthat

Xexp(i~’~o1~1). (8) I !P(t) 12=1!P(—0) 2=1 for all t~ni (21)

For a steadyfield A~=4 (wr) (say) is independent andof n and

IW(nr)12=IW(0)12=cosl4 IW(—0)12 fort=n~,

W(t)=exp[—i(Q+a3w)(t—nt)] (22)

x (1 + ~iaa,)9~-’(nr) (9) andthe dependenceof ~P(t)on n is only through

cosn2and sinn2 (eqs. (15)—(20)).=exp(—iQt) exp[—io3w(t—nr)]For an alternating field (� = — 1),

~ (10)2nt)=e_

2i~~0tA~P(2(n_1)r), (23)Thus ~P(I) is expressedin termsof the initial state

—0) beforethe onsetof the driving field. A’~can wherebe easily found by noting that the eigenvaluesof ‘~ A~= ie1’~e~30~e_iOip0

aree~ande”~where=iA(wr+~it) (24)

cos2=cos2~.tcoSwi (11)and

andthecorrespondingorthonormaleigenvectorsE~andE_are ~P((2n+l)r)=e~TA(wr, ~=—1)!P(2nr). (25)

E+ = cosxu~— sin x U_ = 030! E_, (12) But for the factor i, theeigenvaluesandeigenvector

whereü+=(l, 0), ü_=(0, 1) and ofA0maybeobtainedfromthoseofAbyadding~it

___________ to wt. Henceone canproceedexactly as beforeto‘sin 2— sin wr expressW( t) in termsof the initial state !P( —0).

COSx \J 2 sin2 (13) Thestateat anyinstantis uniquelydeterminedby

the initial state,as it shouldbe. It seemsthat !P(t)Hence is a linearsuperpositionof the two stationarystates.

A” = e1~”E÷E.

4. + e-

1”5E ~ But this is only apparent.Thoughthecoefficientsdonotcontaintexplicitly theyimplicitly dependontime

=cosnA+isinn2(cos2Xa3—sin2Xo1). (14) becausen~r<t<(n+l)T. They are constantsonly

Thus duringtheinterval (nr+0, (n +1)r—0) andaredis-continuousat n~for all n. Of course,if the sum-

~t’(I) = exp( — iQt) exp[ — ia3 w(I— nr)] mation in eq. (1) is finite the coefficients are

constantsfor I> Nr andthe stateis simply a linearX(A+iCa3+iBo1)W(—0), (15)superpositionof the two stationarystates.The pe-

where riodiccollapseandapparentchaoticbehaviourofthe

A=cosn2 cos2~t+sinn2 sin 2j~sin2X, (16) systemas inferredin refs. [5—9]originatefrom thestronglyoscillatory terms cosn2 and sinnA in the

B= cosnA sin 2~t—sinn2 cos 2~zsin2X, (17) coefficients.

Before concluding,let us mention a few specialC=sinn2cos2~. (18)caseswhich mayhavesomephysicalsignificanceand

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Volume134, number3 PHYSICSLETTERSA 26 December1988

exhibit to someextentthe wide natureof variation the evolutionoperator.Henceone hasto introduceof the coefficientsA, B andC: an interactionparameterwhich is takenas time as

in ref. [101. Next, oneshouldmentionalsoanother(i) an=m7t, 2=±2~i,

similarclassofperiodicallykickedproblems.Casati,A= ±cos[2(n—l)~] , B= ±sin[2(n— l)1u], Chirikov, Israilev andFord [11] havestudiedthe

C=0. quantum pendulum under periodic perturbation.They havestudiednumericallyin detail the nature(+ or — accordingto m odd or even integer), ofevolutionof thesystem.Fromthetheoreticalstand

(ii) wr= (2m+1)it/2, 2=±x/2, point, it is veryinvolved sincethetimeindependentpart of the hamiltonianis a differential operatorof

A = 0, B = 0, C= ±1 . angle andthe periodically kicked interactionterm

Theevolutionof the systemmaybeperiodiconly dependsalso on the angle.whenü)T and~ (=tan~(~a))are suchthat2/it isa rationalnumber.

Finally, let uspointoutthatthereasonthata closedReferences

exactsolution oftheproblemcanbeexpressedis thatthesystemconsistsof two levels only. Notwithstand-ing the factthat the steadypart of thehamiltonian, [1] A. Muriel, Phys.Lett. A 128 (1988)367.

namelyQ+a3a,andthekicked (interacting)partof [2] C.K. Raju,J. Phys.A 15 (1982)381.

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mute,the ô-functionnatureof the interactingterms [5] J.H. Eberly,N.B. NarozhnyandJ.J. Sanchez-Mondragon,

asserts that the energy eigenvalues are always Phys.Rev. Lett. 44 (1980)1323.(Q±w)h. Therole of theinteractiontermis only to [6] N.B. Narozhny,J.J. Sanchez-Mondragonand J.H.Eberly,

inducefixeddiscontinuouschangesof theamplitude Phys.Rev. 23 (1981)236.

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general case (multi-level system), in which the processes(Springer,Berlin, 1984).

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5294.ther, if theyarenotcommuting,it is verydifficult to [11] G. Casati, B.V. Chirikov, F.M. Israilev and J. Ford, in:

expressthe evolutionoperatorin a closedform. In Stochasticbehaviourin classicalandquantumhamiltonianfact,solvingthe waveequationis a preludeto obtain systems,eds.0. CasatiandJ.Ford (Springer,Berlin, 1979).

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