Consideraciones legislativas en cuanto al cifrado de datos

When a moleule is exposed to a shaped, intense laser pulse the optimiza-

tion has to aomplish two things. First, the optimization has to reate a

wavepaketonsistingofalargenumberofrotationalstatesthatanserveto

align the moleule. Seond,theoptimization hasto preparethewavepaket

withtheorretphaserelationshipbetweentheomponentwavefuntions,so

thatduringitseld-freeevolutiontheseomponentswouldoherentlyadd-up

to generate an optimally aligned wavefuntion. While thereis no riterium

available that allows us to asertain whether the algorithm has optimized

the populationdistribution, itis possible to investigate the phaserelation-

ship ofthe omponent wavefuntionsinthe optimized solutions. Maximum

alignmentoursifatsomepointintimethe phasesofall omponentwave-

funtionsdierfrom eahother by 0(modulo

2π

). Expliitly,given awavefuntion,

ψ=X

j

a(_jt)_{· |}j_{i ·}exp

−iEjt

~

,

the oeients

a

(t)

j

are omplex numbers, and assuh an be expressed in their polar representation:

a(_jt)=r(_jt)_·expiϕ(_jt).

(9.9) Wethusquestionwhethergivenaertainpopulation-doestheoptimization

routine produe the optimal set of phases

ϕ

(t)

j

? In order to answer this question,asimpleoptimization proedurewasimplementedinthefollowing

manner: It aeptsthe

a

(t)

j

asinput, andaimsat optimizingthephases

ϕ

(t)

Figure 9.15: TOP: The distribution of the maximal and the best opti-

mized wavepakets over the rotational states. Stars represent the maximal

wavepaket inthe nite rotational basis(i.e., orrespondingto thehighest-

ranked eigenvetor of the observable matrix). Diamonds represent the

1

st

optimized setof solutions (CMA-Hermite/ DR2-Plain),and Squaresrepre-

sentthe

2

nd

optimizedsetofsolutions(CMA-Plain/DR2-Hermite);Cirles

represent alulations with doubled bandwidth and the same uene (

50fs

pulse with

Ωge

= 226

×10

12_s−1

), optimized by the DR2 subjet to plain

parameterization. Thegure learlyshows thatthelimitedeldbandwidth

uts othe rotational statesfor theoptimized solutions after

J

= 10

,when the original bandwidthis used,or after

J

= 12

whenthebandwidthis dou- bled. Furthermore,thisplotillustratesthedistintionbetweenthetwo

families of solutions for the original bandwidth (i.e., Diamonds ver-

sus Squares) arising from the dierent algorithmi approahes. BOTTOM:

Thealignment asafuntionoftheoverlapoftheoptimizedwavepakets

|Ψi

with the maximal eigenvetor

|Vi

. Note that the overlap for the original bandwidth never exeeds

0.8

in magnitude. Also note the three lusters

Figure9.16: Leftaxis: Normalizedangular probabilitydistributionfuntion

for the maximal ase

|ψmax(θ)|

2_{sin (θ)}

, and theoptimized ontrol funtion

|ψopt(θ)|2sin (θ)

. Right axis: The value of

cos

2_(θ)

. The onstraints pro-

hibit theevolutionaryalgorithmfromattaining theabsolutemaximalangu-

lar probability distribution funtion; However, theexpetation value of the

observable

cos2(θ)

opt

= 0.9621

when using theoriginal bandwidth orre- sponding to a

100fs

Fourier-limited pulse is within

0.025

of the maximum attainablevalue

cos2_(θ)

max

= 0.9863

. Whendoublingthebandwidth(i.e., basing the shaped laser pulse on a

50fs

Fourier-limited pulse)

cos2(θ)

opt

inreases to

0.975

,whihis only

0.0113

away fromthemaximumattainable value.

suh that the osine-squaredalignment is maximized. Pratially, it uses a

subroutine from the general alignment ode for the evaluation, and applies

the CMA algorithm for the tuning of the

10

relevant phases. Note that a single funtionevaluationhasthe duration of

≈0.5s

.

Weonsidered

50

dierentasesofhigh-qualitysolutionstothealignment problem(allsolutions haveosine-squared-alignment valuesintheregimeof

0.95

) -for eahtestase

100

independent optimizationswererun,aimingto tune the phases.

Theexperimentalresultsarelearandsharp. Theyarepresentedattwo

levels:

1. In all

100

runs for all

50

test-ases - the best solution has always synhronized phases. There are dierent phasevalues per run, but

itdoesnot make adierenefor theosine-squaredalignment,aslong

as the populated levels hold that same phase value. Expliitly, the

Sigma-RMSof the phases wasalulated:

2. The

50

test-ases, as originally obtained by the original optimization prior to this optimization proedure, held phases whih were not far

from beingsynhronized,

∆ϕDR2= 0.0566,

andindeed,theoptimizationsdidnotimprovetheosine-squaredalign-

ment dramatially: Alwayslessthan 1%improvement wasreorded.

We onsiderthisaverystrongresult-theevolutionaryoptimization routine

managed to takle the ne-tuning of the quantum ontrol problem, behind

theomplex transformationsand theso-alled Shrödingerblak-box.

To summarize, while we annot establish whether the optimization has

distributed the populationinthe best possible way, we do observe that the

algorithm hasproperlyphased-upallomponent wavefuntionswithrespet

to eah other. Thistype of oherent alignment of phases was also observed

to be optimal in the mehanisti analysis of another state-to-state ontrol

appliation [163 ℄.

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