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We onsider here the problem of population transfer within the rotational

framework as an optimization problem, subjet to the numerial modeling

for diatomi moleulespresentedearlier. The objetive to be metis dened

as the probability to populate a spei target rotational level, given the

initial groundstate:

wherepossibly

Jtarget

∈ {0,2,4,6,8, . . . , Nrot}

. Inouralulations,theyield subjet to maximization is simply

α

(g)

Jtarget(T)

2

, in terms of the notation

introdued earlier. Also,bydenition,

M

= 0

.

Weonsider

Nrot= 20

,where this expansionwasonrmed to giveon- verged resultsinthepresent alulations. Themoleule underinvestigation

hasa rotational onstant of

Brot

=Bg

=Be= 5cm

−1

.

Solving the dening dierential equations for the population transfer

problem(Eq.8.13)isobviouslyomputationallyexpensive. Inpratie,given

aneletrield,asingleevaluationoftheresultingwavepakethasthedura-

tionof approximately

5s

onasingle P4-HT

2.6GHz

proessor. We arethus interested in optimization proedures withas minimal funtion evaluations

aspossible.

8.2.1 Experimental Proedure

There are several dening parameters in the present alulations. Some of

them are ritial, as they pose diret onstraints on the quantum system

at hand, and pratially determine its ontrollability. In our model, suh

parameters are the peak Rabi frequeny, whih plays the equivalent role

of the laser intensity, as well as the pulse duration. Setting these two pa-

rameters denes the simulatedphysial system. Given thetarget rotational

level, itis thenpossible to aim at steering thesystem toward it. Thus, we

hooseto onsiderthe population transfer asa funtion of these two den-

ing parameters, where the fous will be on spei values that reet best

state-of-the-art laboratoryexperiments.

Fromthealgorithmiperspetive,wehoosetorestritouralulationsto

theDR2andtheCMAalgorithms,whihperformedbestontheTwo-Photon

Proess problems. Theyboth employsmall populations,and onsider rst-

order andseond-order information, respetively.

PreliminaryRuns Preliminaryalulationsrevealedalearpiture,whih

ouldhavebeenpreditedbyintuition 1

. Thesepreliminaryalulationswere

onsisted of

10

runsperalgorithm on

Jtarget

={0,2,4,6,8}

withthefollow- ingpeakRabifrequenies:

Ωge

={40,60,80, . . . ,160,180} ×1012s−1.

Given aRabi frequenyof

Ωge

= 160×10

12_s−1

,thequantum systemould

easily be steered into perfet ontrol for low

J

values

(J

={0,2,4})

. This task beame infeasible for higher

J

values with the given Rabi frequeny. However,whenthelatterwasinreased,e.g.,

Ωge= 180×10

12_s−1

,itbeame

1

AsmuhasintuitionexistsforQuantumMehanis;"Mybattingaverageonintuition

feasible. Hene, there is a trend of ontrollability as a funtion of thelaser

intensity,espeiallyforthehigherrotationallevels. Asfarasthealgorithmi

performanewasonerned, the DR2and theCMAperformedequally well

on the given systems. Most importantly, there was never a situation where

the DR2obtained ontrollability on a given system on whih theCMA did

not, nor vieversa.

We onsider the ase of a target rotational level of

J

= 4

as an inter- esting ase-study. This is due to the fat that it allows perfet ontrol at

Ωge

= 160×1012s−1

, but yet it is a hallenging task for the optimization routines. Also,theeetofdereasing the peakRabifrequenywhilelosing

ontrollability anbe observed relativelyeasily.

8.2.2 Numerial Observation:

J

= 0−→J

= 4

WeappliedtheDR2algorithmtotheoptimizationofthepopulationtransfer

problemfrom

J

= 0

to

J

= 4

. Theseoptimizationswereperformedforthree values ofthe peakRabifrequeny:

Ωge

=

80_×1012s−1,

120_×1012s−1,

160_×1012s−1

.

All alulations were arried out with

80

runs, limited to

10,000

fun- tion evaluations per run. These alulations obtained qualitatively dier-

ent results for the three intensities onsidered. For

Ωge

= 80×10

12_s−1

the optimizations were unable to aomplish the transfer from

J

= 0

to

J

= 4

with unit eieny. The best eieny obtained was

≈

32%

. For

Ωge

= 120×1012s−1

and for

Ωge

= 160×10

12_s−1

the transfer eieny

approahed

100%

inmost of thealulations.

Aimingatomparingtheresultsofindividualoptimization runs,wede-

neaorrelationoeientthatomparespulse-shapesattainedintworuns

i

and

j

,bymeans oftheir eld intensities:

ci,j

=

max∆t{PtIi(t)Ij(t+ ∆t)}

hq

P

tIi2(t)

q

P

tIj2(t)

i

(8.16)

where

Ii(t)

and

Ij(t)

are theeld intensities of the pulsesobtained inruns

i

and

j

, respetively. Taking the maximum as a funtion of

∆t

is due to the fatthatpulse-shapesattained bytheoptimization maybeshiftedwith

respettoeahother. Thesumsareoverthedisretetimesteps,asonduted

in the numerial alulation. Eq. 8.16 thus yields

ci,i

= 1

, and

ci,j

= 0

if pulses

i

and

j

do not overlap at all.

Case 1:

Ωge= 80×10

12_s−1

FigureA.4presentstheorrelationoeient

for the

80

optimization runsof the

Ωge

= 80×10

12_s−1

test-ase. The runs

population are highly orrelated. Upon examination of the atual alula-

tions,itisobservedthatallofthese solutionsareverylose toa singleFTL

pulse. Deviations from the FTL pulse do not only lead to a drop in the

orrelation oeient, but alsoin thepopulationtransferyield.

Case 2:

Ωge

= 120×10

12_s−1

In Figure A.5 the orrelation oeient

is plotted for the

80

optimization runs that were performed for the

Ωge

=

120_×1012_s−1

test-ase. Here,thelaserpulseenergywassuienttotransfer

populationfrom

J

= 0

to

J

= 4

withnear-uniteieny. Thebestsolutions, whihhaveapopulationtransfereienyof

99.982%

and

99.98%

,wereonly weakly orrelatedto eah other,and wereonlyweaklyorrelated tomost of

theothersolutions. Speially,therewereonly

9

solutionsamongthesetof

80

thatshareaorrelationoeientlarger than

0.95

withthebestsolution (indexedas

1

). Manyoftheremainingsolutionsarestronglyorrelated with the

3

rd

-best solution, whih hasa populationtransfer yieldof

99.975%

: As many as

41

solutions shared a orrelation oeient larger than

0.95

with thatsolution (indexed as

3

). Whilethethree good solutions

1

,

2

,and

3

are ratherdierentfromeahother, theyontainmost ofthedominantfeatures

of the identied optimized solutions.

Solutions1-3arepresentedinFigure8.1. Despitetheirdierenehara-

teristis,all threesolutions inFigure8.1aredominated bya seriesofpeaks

with a separation of

4.79×10

−13_s

. This orresponds to thebeating period

of a oherent superposition of

J

= 2

and

J

= 4

(

∆E

= 14B

). Additional good solutions likely exist, possibly ontinuously onneted on a ommon

level set, and further speialnumerial methods are needed to explore this

possibility,suh asthe D-MORPHalgorithm (Setion6.1.3).

Case3:

Ωge

= 160×10

12_s−1

FigureA.6presentstheorrelationoeient

for

80

optimization runs of the

Ωge

= 160×10

12_s−1

test-ase. While the

degreeofpopulationtransferisveryhighinalmostalltherunsatthisinten-

sity,the orrelationbetween thevarioussolutions is verylimited. Clearly,a

largenumberofsolutionsthattransferthepopulationwithuniteienyo-

exist,withvery little ommonality between them. Indeed, inspetion of the

atualpulseshapesobtainedintheserunsrevealshighlyompliatedpulses,

withfewregularfeatures,andanabseneofthepeakarisingfromoherene

between

J

= 2

and

J

= 4

inthe Fourier transform powerspetrum. 8.2.3 Intermediate Disussion

Uponinreasingtheintensityfrom

Ωge

= 80×10

12_s−1

to

Ωge

= 160×10

12_s−1

we nd that population transfer is aomplished with an ever inreasing

numberof distinguishablesolutions.

Figure 8.1: Comparison of the

3

best-performing pulse shapes that were obtained in

80

runsof the DR2for the population transferproblem of

J

=

0_−→J

= 4

at

Ωge

= 120×10

12_s−1

. All solutionsonsist oftrainsof pulses

with a spaing of

4.79×10

−13_s

, whih orresponds to the beating period

that ontrollable quantum systems with no onstraints plaed on the on-

trolsonly have extrema thatorrespond to perfet ontrol, or to no ontrol

at all; Additional analysis revealed the fundamental nature of ontrol level

sets(seeCorollary6.1.3)atthe absoluteextremaandatsub-optimalontrol

yields.

Astriking aspetof theresultsisthe evidenethatthe number

of independent solutions produed by an optimization seems to

ritially depend on the diulty of the problem. In the urrent

populationtransfer alulations we observed that at lowintensity,

wherereahing the target isahardproblem with less thanperfet

yield, the trials invariably onverge onto one and the same solu-

tion, whereas at higher intensity, where this represents an easier

problem, a wide variety of solutions are enountered.

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