Laboratory Realization: Constraints

Therealizationofthequantumsysteminthelaboratoryposesonstraintson

the quantumdynamis, and maylead to adierent OCE searhlandsape,

landsape, with gradient-based steps and in logarithmi time omplexity,

may no longer be valid in OCE landsapes. Generally speaking, it is not

lear howdo QuantumControl landsapes appearinthe laboratory.

We disuss here brieyseveral aspets of laboratory experiments whih

arelikely to betranslated into onstraintsintheOCE landsape [140 ℄.

Theruialomponent oflaserpulseshapingproess isthephasemodu-

lation, whih istypially exposedto waveform distortion eets(fora om-

prehensivestudysee[141 ℄). Weoutlinehereseveralmodulationomponents.

Pixelation and Replia Pulses In pratie, the pulse shaping proess

is implemented by a so-alled Spatial Light Modulator (SLM), whih is

typially based on Liquid Crystal Display (LCD). This approah onsiders

individual pixels subjet to retangle-ativation-funtions, squ

(ν)

, ideally sharply-dened andwithno gapsbetween eahother. Thisis referredto as

the stairase approximation. Thetimemodulation ofthese step-funtionsis

attainedbymeans oftheir inverse Fourier transform,

F−1[

squ

(ν)]∼

sin

(τ)

where the width of sin

(τ) =

sin(τ)

τ

is inversely proportional to the pixel width. Expliitly, the resulting temporal eletri eld in this pixelization

an bedesribed asfollows:

e(t) =X

n

˜

e(t₋nτ)_·

sin

πt

τ

,

(6.34)

with

e(t)˜

as the desired eletri eld, and where

τ

=

1

∆ν

is the inverse frequeny spaing perpixel.

Pratially, step-funtion gaps between SLM eletrodes are responsible

for the onstrution of so-alled parasiti replia pulses inthetemporal do-

main,whih areloated atthe zerosof the sin envelopefuntion.

Pulse Break-Up A linear phase funtion results inthe timeshift of the

temporal pulse. This an easily be derived by a hange of variables, or

by the appliation of the so-alled Fourier Shift Theorem (see, e.g., [142℄).

The inuene ofthereplia pulsesbeomes moresubstantial when theyare

moved from the zeros of the envelope sin funtion, by breaking-up the

pulse energy into multiple parasiti replia pulses. This is equivalent to

the following statement: The steeper the linear phase, the more pronouned

beome the replia pulses, whih generally result in lower suboptimal yields

[140 ℄.

Phase Range: Wrapping Phases that dier in

2π

radians are mathe- matiallyequivalent. Thisperiodinatureofthephasein

[0,2π]

n

poses periodi boundary onditions on the modulator. Given

0

< ε

≤

2π

, the so-alled phase wrapping operator isimplementedasfollows:

φi

= 2π+ε

−→

φ˜i:=ε

φj

=−ε

−→

φ˜j

:= 2π−ε

(6.35)

or simplyas

φ˜i:=φi

mod

2π

.

From an optimization perspetive, this means that the searh spae is

pratially an

n

-dimensional hyperube spanning alength of

2π

ineah di- mension. Itislikelyto have impliationsontheoptimization routineinuse.

Intermsofonstraints,wrappedphasesmaybeexposedtosingularityeets

(

0−1

jumps), but it is not onsidered to be a signiant eet. Thus, we onsider it heremore as a mathematial feature of the searh spae, rather

than a onstraint.

Resolution Thenumberofpixels,

n

,determinestheontrolresolution, and poses a diret onstraint on the shaped-pulse in thetemporal domain:

Due to the reiproal nature of the Fourier transform with respet to fre-

quenyversustime,spetralresolutiondeterminestheupperboundfortem-

poralresolution. Forinstane, typial laboratoryrealizations urrently on-

sider

n= 128

pixelswithspetralresolutionof

0.25

nm/pixel, whih allowa shapedpulsewithmaximumtemporallengthof

8.5ps

atFWHMbandwidth of

10nm

.

Weherebysummarizethemainlaboratoryonstraintsinatypialquan-

tum systemrealization:

1. Temporal orspetralresolutionofthe eldLimitedspetralres-

olutionintherealizedshaperimplieslimitedpulsetemporalresolution.

State-of-the-art LCD pulse shapersontain

640

pixelsto betuned. 2. Limited eld uene, limited eld intensity Potential damage

to dierent experimental omponents restrits inpratie the applied

elduene andits intensity.

3. Limited spetral bandwidth or pulse duration State-of-the-art

ommerial lasers an produe nowadays pulses at the duration of

∼

20

f s

.

4. Proper basis The atual representation of the ontrol phase, e.g.,

pixel basis, polynomial expansion basis, et., poses by itself an addi-

tional onstraint onthelandsape.

6.3 Experimental Proedure

In thisstudy we areinterestedboth innumerial modeling of quantumsys-

tems,aswellasintheirreallaboratoryexperiments. Thenumerialmodeling

is typially driven by a known Hamiltonian, but designed in a laboratory-

oriented manner, as will be desribed shortly. Essentially, it is OCT om-

binedwith someOCE harateristis.

Inour alulations, we hoose to restritthis study mostlyto noise-free

simulations, as we are interested in the physis of the system, rather than

onduting an atual simulation of a real laboratory experiment. On this

note, weonsidertheabsene ofnoise inour alulations asablessing, asit

allowsforleaninterpretationofthephysisofthesystem. Inonepartiular

ase, we will arryout simulationswithnoise.

Generally speaking, onsidering the various quantum systems underin-

vestigationinthisstudy,thegoalthatwewouldliketoahieveinour exper-

imental work is three-fold,and maybe outlinedasfollows:

1. A preliminary part of our work on eah quantum system is devoted

to a large extent to an investigation of the performane of spei

derandomized EvolutionStrategies,aswellasparameterizations, with

respetto the given optimization task. Assuggested inSetion 1.4.4,

this wouldinlude theomma-strategy DES variants.

2. After having identied the routines whih perform best on our prob-

lems,furtherworkwouldtypiallyonentrateonthephysialinterpre-

tationoftheobtainedoptimalsolutions,whenappliabletothesystem

understudy. Inpartiular, wewill aim atlarifying whyertainpulse

strutures perform better than other trial solutions. Thiswill alsobe

aompaniedwithinvestigationofpulse-intensity,eldsalability,and

other deningfeatures.

3. Finally, we will be interested in applying misellaneous optimization

tehniques, at the level of deision making: multi-objetive optimiza-

tion, and theappliation ofnihing.

Next, we provide tehnial details onerning thetwo lasses of experi-

mental workonduted inthis study: numerial simulations andlaboratory

experiments.

6.3.1 Numerial Simulations

We present here the numerial modeling of our laser pulse shaping frame-

work,whihisinessenevalidforallthenumerialalulationsondutedin

thiswork,unlessspeiedotherwise. Theideaistosimulatetheexperimen-

As disussed earlier, in our alulations the ontrol is solely the phase

funtion

φ(ω)

. Itdenes thephaseat

n

frequenies

{ωi}

n

i=1

thatareequally distributedarossthespetrumof thepulse. These

n

values

{φ(ωi)}

n

i=1

are the deisionparameters tobeoptimally determined. Upon theiralibration

they are numerially interpolated into

n˜

= 2

14

points, using the spline()

proedure[143 ℄,forthealulationoftheeletrieldinEq.6.29. Thelatter

is implemented bymeans of theFFT() proedure [143 ℄.

The numerial resolution is naturally underposed to a onit with the

expetedoptimizationeieny. Inordertoahieveagoodtrade-obetween

the two,i.e., keeping both resolution and optimization eieny ashigh as

possible, thevalue of

n= 80

turned to be a good ompromise. The searh spae is therefore an

80

-dimensional hyperube spanning a length of

2π

in eah dimension.

Thespetralfuntion

A(ω)

istakentobeaGaussian,enteredat

800nm

, with a width hosen suh that the full-width-at-half-maximum (FWHM)

length of the Fourier transform limited (FTL) pulse (obtained by setting

φ(ω)_≡0

) is

∆τ

≈100f s

.

Mostof thesimulations were run withFORTRAN ode, aswritten and

provided by Prof. Mar Vrakking, of Amolf-FOM, Amsterdam 5

. This was

laterombinedwithaMATLABversionoftheoriginalode,asimplemented

bytheauthor. For thetwo-photon proessesreportedinChapter7 we used

a LabViewsimulator of PrinetonUniversity, oded byJonathanRoslund.

In document Niching in derandomized evolution strategies and its applications in quantum control (Page 129-133)