Complete population transfer in a three-state quantum system by a train of pairs of coincident pulses Andon A. Rangelov and Nikolay V. Vitanov Department of Physics, Sofia University, James Bourchier 5 blvd., 1164 Sofia, Bulgaria (Dated: January 6, 2012) A technique for complete population transfer between the two end states 1 and 3 of a three- | i | i statequantumsystemwithatrainofN pairsofresonantandcoincidentpumpandStokespulsesis introduced. A simple analytic formula is derived for theratios of thepulseamplitudes in each pair for which the maximum transient population P (t) of the middle state 2 is minimized, Pmax = sin2(π/4N). It is remarkable that, eventhough t2hepulses are on exact re|soinance, P (t) is da2mped 2 tonegligiblysmallvaluesevenforasmallnumberofpulsepairs. Thepopulationdynamicsresembles generalizedπ-pulsesforsmallN andstimulatedRamanadiabaticpassageforlargeN andtherefore 2 thistechniquecan beviewed as a bridge between these well-known techniques. 1 0 PACSnumbers: 32.80.Xx,33.80.Be,32.80.Rm,33.80.Rv 2 n a Introduction. Coherentexcitationofadiscretequan- states |1i and |3i of a three-state Λ-system with a train J tum system by an external resonant field represents of N pairs of resonant and coincident pump and Stokes 4 an important notion in quantum mechanics. Resonant pulses with negligibly small transient population in the pulses of specific pulse areas are widely used in a vari- middlestate|2i,whichvanishesasN−2 withthenumber ] h ety of fields, including coherent atomic excitation [1, 2], ofpulse pairsN. Wenote thatunlike theadiabaticsolu- p nuclearmagneticresonance[3],quantuminformation[4], tions, which are approximate,our technique is described - and others. Resonant excitation allows one to establish by an exact analytic solution. This technique formally t n a complete control over the quantum system, particu- resembles the techniques of piecewise adiabatic passage a larly in two- and three-state systems. Important exam- [11,12]andcompositepulses[13,14];thedifferenceswill u ples include complete population transfer between the be discussed toward the end. q two states in a two-state system and between the two Weshallfirstpresenttheexactanalyticsolutiontothe [ end states in a three-state Λ-system [2, 5]. Crucial con- three-state dynamics for a single pulse pair and then the 1 ditions for resonant excitation are exact pulse areas and exactsolutionforatrainofN pairsofpulses,inwhichwe v exact resonances between the frequencies of the external shalldemonstrateexplicitlythedynamicalsuppressionof 9 fields and the Bohr transition frequencies. Deviations the middle-state population. 2 0 fromexactresonancesorexactpulse areasleadtodevia- Single pulse pair. The probability amplitudes ck(t) 1 tionsofthetransitionprobabilityfromthedesiredvalue. (k = 1,2,3) of the three states that form the Λ-system . To this end, an alternative to resonant excitation is obey the Schr¨odinger equation, 1 0 provided by adiabatic passage techniques, which are ro- i~∂ c(t)=H(t)c(t), (1) 2 bust against such deviations. In three-state Λ-systems, t 1 thefamoustechniqueofstimulatedRamanadiabaticpas- where c(t) = [c (t),c (t),c (t)]T. The Hamiltonian in : sage (STIRAP) [5–7] allows complete population trans- 1 2 3 v the rotating-waveapproximation [1, 2] reads i fer between the two end states |1i and |3i in the adia- X batic limit without placing any transient population in r the middle state |2i, even though the two driving fields ~ 0 Ωp(t) 0 a H(t)= Ωp(t) −iΓ Ωs(t), (2) — pump and Stokes — can be on exact resonance with 2 0 Ω (t) 0 s their respectivetransitions,|1i↔|2iand|2i↔|3i. The   conditionsforSTIRAParethetwo-photonresonancebe- where Ω (t) and Ω (t) are the Rabi frequencies of the p s tween states |1i and |3i, the counterintuitive pulse order pump and Stokes pulses, respectively; each of them is (Stokesbeforepump),andadiabaticevolution. However, proportionaltotheelectric-fieldamplitudeoftherespec- adiabatic evolution requires large pulse areas, typically tive laser field and the corresponding transition dipole over10π for smoothpulse shapes, which may be hard to moment, Ω (t)=−d ·E (t) and Ω (t)=−d ·E (t). p 12 p s 32 s reachexperimentally. StrategiesforoptimizationofSTI- Γistherateofirreversiblelossfrommiddlestate|2i. For RAPwithminimalpulseareashavebeendeveloped[8,9], simplicitybothΩ (t)andΩ (t)willbeassumedrealand p s but these come on the expense of strict relations on the positivebecausetheirphasescanbe eliminatedbyredef- pulse shapes [8, 9] and require specific time-dependent inition of the probability amplitudes. More importantly, nonzero detunings [9]. we assume that the Rabi frequencies are pulse-shaped In the present paper we introduce a novel technique functions with the same time dependence f(t), but pos- that is both an alternative to the above techniques and sibly with different magnitudes, reducestothemintwooppositelimits. Thetechniqueen- ables complete population transfer between the two end Ω (t)=Ω0 f(t). (3) p,s p,s 2 Inthiscase—single-andtwo-photonresonancesandthe a transformation to the so-called bright-dark basis [10]. same time dependence of the pump and Stokes fields — The exact propagatorreads [10] the Schr¨odingerequation(1)issolvedexactlybymaking 1−2sin2θsin2 1A −isinθsin1A −sin2θsin2 1A 4 2 4 U(θ)= −isinθsin1A cos1A −icosθsin1A , (4) 2 2 2 −sin2θsin2 1A −icosθsin1A 1−2cos2θsin2 1A  4 2 4  where the root-mean-square (rms) pulse area A and the objective is to have P = P = 0 and P = 1 at the end 1 2 3 mixing angle θ are defined as of the pulse train, with as little transient population in the middle state |2i as possible. A= tf Ω2(t)+Ω2(t)dt, (5) Itis convenientto havean“anagram”pulse trainthat p s Zti q issymmetricwithrespecttotimereversal,i.e.,withmix- Ωp(t) Ap ing angles θN+1−k = π/2−θk, (k = 1,2,3...⌊N/2⌋). In tanθ = = , (6) ordertodeterminethevaluesofθ andA,weuseEqs.(4) Ω (t) A k s s and (8) and we first demand that after each pulse pair with A = tf Ω (t)dt being the pump and Stokes the population of state |2i vanishes; this gives immedi- p,s ti p,s ately rms pulse area A = 2π for each pulse pair, hence pulse areas. (RFor reasons that will become clear below the omission of A from the arguments of U in Eq. (4). we have omitted the area A from the arguments of U.) Next we require that the population is transferred to Due to condition (3) the angle θ is constant. state|3iintheend,P =P =0andP =1. Wefurther The propagator (4) allows us to find the exact ana- 1 2 3 require that the maximum of the transient population lytic solution for any initial condition; however, we re- P (t) excited by each pulse pair is the same. Among strictourattentionhereto asysteminitially instate|1i: 2 c(t )=(1,0,0)T. Thenthepopulationsattheendofthe the many solutions we pick the one that minimizes the i maxima of P (t). A simple algebra gives the angles interaction are 2 A 2 θ = (2k−1)π (k =1,2,3...N). (9) P1(tf)= 1−2sin2θsin2 , (7a) k 4N (cid:20) 4(cid:21) The maximum population of state |2i, which, as for A P2(tf)=sin2θsin2 , (7b) a single pulse pair above, occurs in the middle of each 2 pulse pair (at rms area π), is readily obtained, A P3(tf)=sin22θsin4 . (7c) π 4 Pmax =sin2 . (10) 2 4N (cid:16) (cid:17) Obviously, complete population transfer to state |3i is From here we conclude immediately that for large N achievedforθ =π/4,whichcorrespondstoA =A ,and p s the maximum population of the middle state vanishes rmspulse areaA=2π. Then,however,the intermediate as 1/N2. Obviously, for N ≧ 8 pairs, the transient pop- state |2i acquires a transient population which reaches a maximum value of Pmax = 1 at the intermediate time ulationinstate |2idoesnotexceed1%. Itis particularly 2 2 significantthatthissuppressionoccursonresonanceand when the accumulated rms pulse area A(t) reaches the it results from the destructive interference of the succes- mid-point value π. This scenario is illustrated in Fig. 1 siveinteractionsteps,ratherthanfromalargedetuning. (left frames). We note that for N ≫ 1 the total pulse area is very We shall show below that the application of a train large, which is the condition for adiabatic evolution on ofresonantpulsepairsallowsonetotransferthe popula- resonance in STIRAP. tionfromstate|1itostate|3icompletely,whilereducing We show in Fig. 1 the population evolution for sev- the transient population in state |2i to arbitrarily small eralpulse trains of different number of pulse pairs N. In value. all cases the population is transferred from state |1i to Trainofpulsepairs. AsequenceofN pairsofpulses, state |3i in the end in a stepwise manner. The transient each with rms pulse area A and mixing angles θ , pro- k population of the intermediate state |2i is damped as N duces the following total evolution matrix increases: from0.5forasinglepairofpulsesand0.15for U(N) =U(θN)U(θN−1)···U(θk)···U(θ1). (8) two pairsto below 1%for 8 pulse pairs. For small N the transition picture resembles (fractional) resonantexcita- When the system is initially in state |1i, the final popu- tion, whereas for large N the transition picture resem- lations are given by P = |U(N)|2, with n = 1,2,3. Our bles adiabatic passage. The lower frames demonstrate n n1 3 N=1 N=2 N=3 N=5 N=8 s 4 encie1/T ) 3 quof 2 bi Freunits 1 W S W P W S W P W S W P W S W P W S W P a( R 0 1.0 0.8 P1 P3 P1 P3 P1 P3 P1 P3 P1 P3 s n o 0.6 ati ul p 0.4 o P P 0.2 2 P2 P2 P2 P2 0 1.0 P P P P P 1 1 P 1 3 P 3 P 0.8 3 1 1 P P 3 ns 3 o 0.6 ati ul p 0.4 o P 0.2 P 2 P P P P 2 2 2 2 0 -5 0 5 -10 0 10 -10 0 10 -20-10 0 10 20 -30-15 0 15 30 Time ( units of T ) Time ( units of T ) Time ( units of T ) Time ( units of T ) Time ( units of T ) FIG. 1: Rabi frequencies (top frames) and populations (bottom frames) vs time for (from left to right) 1, 2, 3, 5 and 8 pairs of pulses. The pulse shapes are Gaussian, Ωp(t) = Ωsinθke−(t−τk)2/T2 and Ωs(t) = Ωcosθke−(t−τk)2/T2, with Ω = 2√π/T (corresponding to rms pulse area A=2π) and themixing angles θ (k=1,2,...,N) are given by Eq. (9). Middle frames: no k decay (Γ = 0). Complete (stepwise) population transfer 1 3 is achieved in all cases; however, the population P (t) of the 2 intermediate state is different. The maximum of P (t) is gi→ven by Eq. (10); it vanishes as 1/N2. Bottom frames: irreversible 2 loss from state 2 with a rate Γ=1/T. | i theeffectofirreversiblepopulationlossfromstate|2ifor appropriatelychosenamplitudes. InthelimitN ≫1,the a loss rate Γ = 1/T. As it can be expected from the present technique resembles STIRAP, because then the middle frames of lossless interaction,the damping of the successiveincrementsofthemixingangleθ becomevery k intermediate-state population by longer pulse trains re- small(nearlycontinuous),withtheStokesfielddominat- ducestheeffectofpopulationloss: thetarget-statepopu- ing over the pump field in the beginning and then the lationP decreasesto0.63forasinglepulsepairbutjust pump field dominating in the end, in exact analogy to 3 to 0.93 for a train of N =8 pulse pairs. This population the counterintuitive sequence Stokes-pump in STIRAP loss can be decreased even further by longer trains. [5–7]. However, the present technique achieves complete Wenotethatbecausethetechniqueusesfieldsonexact population transfer 1 → 3 also for small N, in a man- resonance the pulse shapes are unimportant. Although ner reminiscentofgeneralizedπ-pulses [2], while keeping the example in Fig. 1 uses Gaussian shapes, pulses of the middle-state population at very low values. In this rectangular or any other shape are equally suitable. manner,the presenttechnique canbe viewedasa bridge between generalized π-pulses and STIRAP. Discussion. We have demonstrated that complete population transfer in a three-state system driven by a Thepresenttechniquecanbeviewedalsoasanalterna- pairofexternalpulse-shapedlaserfieldsthatareonreso- tiveoftwoothertechniquesthatusepulsetrainsforcom- nanceandhavethesametimedependence,canbeaccom- plete population transfer. Piecewise adiabatic passage plished in such a way that all population is transferred (PAP) [11, 12] uses a train of a large number of pulses, fromthe initialstate|1itothe targetstate|3iwithmin- each of which produces a perturbatively small change in imaltransientpopulationinthe middle state|2i. This is thepopulations,whilethepresenttechniqueworksforan achievedwithatrainofN pairsofcoincidentpulseswith arbitrary number of pulse pairs and each pair may pro- 4 duce a large population change (for small N). In one of the relativephases, whereasin the presenttechnique the the implementations involving only two states [12], PAP controlparametersaretheamplituderatiosineachpulse demands phases that change quadratically from pulse to pair. Moreover, the objective in the composite pulse pulse,whichtranslateintoalinearchirpforalargenum- technique is the shape of the excitation profile while in ber of pulses; the present technique does not need such our technique the main objective, beside the complete quadratic phases but appropriate amplitude ratios. In population transfer 1 → 3, is the suppression of the the PAPimplementation withthree states[11], to which intermediate-state population. the present technique is more closely related, the pump Conclusion. The technique introduced in this pa- andStokesfields inSTIRAP areturnedabruptlyonand per allows complete population transfer between states off repeatedly; the amplitudes of the individual pulses |1i and |3i via an intermediate state |2i with a train of are determined such that they match the envelopes of N pairs of coincident pump and Stokes pulses, by plac- the pump and Stokes pulses in STIRAP. In the present ing only a negligible transient population in state |2i, technique the pulse amplitudes are determined from the which decreases as 1/N2 as the number of pulse pairs conditionstoachievecompletepopulationtransfertothe N increases. This technique resembles the technique of target state |3i and to minimize the population of the generalized π-pulses for small N and STIRAP for large middle state |2i. Indeed, the systematic suppression of N and therefore it can be viewed as a bridge between the transient population of state |2i with the increasing these two well-known techniques. It is remarkable that numberofpulsepairsN seeninFig.1isonlyobservedin the middle-state population P (t) is damped consider- the present technique. We also point out that the solu- 2 ably even for a small number of pulse pairs despite the tioninthepresentpaperisexactwhilePAPandSTIRAP fact that the pump and Stokes fields are on exact res- give approximate solutions in the adiabatic limit. onance with their transitions. All these features make The present technique is also reminiscent of the tech- this technique an interesting alternative of the existing nique of composite pulse sequences [13]. The latter uses techniques for coherent control of three-state quantum sequencesofpulsesintwo-statesystems[13],orsequences systems. of pulse pairs in three-state systems [14], with well- defined relative phases, which are determined from the This work is supported by the European network condition to produce a desired excitation profile. There- FASTQUAST and the Bulgarian NSF grants D002- fore the control parameters in the composite pulses are 90/08,DMU02-19/09 and IRC-COSIM. [1] L. Allen and J. H. Eberly, Optical Resonance and Two- [9] G. 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