Complete population transfer in a three-state quantum system by a train of pairs of coincident pulses

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by  Andon A. Rangelov

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. Dridi, S. Gu´erin, S; V. Hakobyan, H. R. Jauslin, and
Level Atoms (Wiley, New York,1975). H. Eleuch, Phys. Rev.A 80, 043408 (2009).
[2] B.W.Shore,TheTheory of Coherent AtomicExcitation [10] N. V.Vitanov, J. Phys. B 31, 709 (1998).
(Wiley,New York,1990). [11] E. A. Shapiro, V. Milner, C. Menzel-Jones, and M.
[3] C. P. Slichter, Principles of Magnetic Resonance Shapiro, Phys.Rev.Lett. 99, 033002 (2007);
(Springer,Berlin, 1990). [12] S. Zhdanovich, E. A. Shapiro, M. Shapiro, J. W. Hep-
[4] M. A. Nielsen and I. L. Chuang, Quantum Computation burn, and V. Milner, Phys. Rev. Lett. 100, 103004
andQuantumInformation (CambridgeUniversityPress, (2008); E. A. Shapiro, A. Peer, J. Ye, and M. Shapiro,
Cambridge, England, 1990). Phys. Rev.Lett. 101, 023601 (2008).
[5] N. V. Vitanov, T. Halfmann, B. W. Shore, and K. [13] M. H. Levitt, Prog. NMR Spectrosc. 18, 61 (1986); R.
Bergmann, Annu.Rev.Phys.Chem. 52, 763 (2001). Freeman, Spin Choreography (Spektrum, Oxford, 1997);
[6] U.Gaubatz,P.Rudecki,S.Schiemann,K.Bergmann, J. B. T. Torosov and N. V. Vitanov, Phys. Rev. A 83,
Chem. Phys.92, 5363 (1990). 053420 (2011); B. T. Torosov, S. Gu´erin, and N. V. Vi-
[7] N. V. Vitanov, M. Fleischhauer, B. W. Shore, and K. tanov, Phys. Rev.Lett. 106, 233001 (2011);
Bergmann, Adv.At.Mol. Opt.Phys. 46, 55 (2001). [14] G. T. Genov, B. T. Torosov, and N. V. Vitanov, Phys.
[8] G. S. Vasilev, A. Kuhn, and N. V. Vitanov, Phys. Rev. Rev. A 84, 063413 (2011).
A 80, 013417 (2009).