light-matter interaction and quantum computing in rare-earth-ion-doped crystals kinos, adam

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Light-Matter Interaction and Quantum Computing in Rare-Earth-Ion-Doped Crystals

Kinos Adam

2018

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Download date 07 Jun 2020

AD

AM

KIN

OS

Light-M

atter Interaction and Quantum

Com

puting in Rare-E

arth-Ion-Doped C

rystals 2018

978

9177

5354

30

LUND UNIVERSITYFaculty of Engineering LTH

Department of Physics Division of Atomic Physics

ISBN 978-91-7753-543-0 (print)ISBN 978-91-7753-544-7 (pdf)

ISSN 0281-2762Lund Reports on Atomic Physics LRAP-543 (2018)

Light-Matter Interaction and Quantum Computing in Rare-Earth-Ion-Doped CrystalsADAM KINOS

DEPARTMENT OF PHYSICS | FACULTY OF ENGINEERING | LUND UNIVERSITY

Prin

ted

by M

edia

-Try

ck L

und

2018

N

ORD

IC S

WA

N E

CO

LABE

L 3

041

0903

This thesis written by Adam Kinos contains research on wave propagation in absorbing birefringent crystals slow light applications and quantum computing using rare-earth-ion-doped crystals as well as extensive light-matter interaction discussions surroun-ding the same topics

Light-matter interaction andquantum computing in

rare-earth-ion-doped crystals

Adam Kinos

Doctoral Thesis

2018

Akademisk avhandling som for avlaggande av teknologie doktorsexamen vid tekniska fakulteten vid Lunds Universitetkommer att offentligen forsvaras den 2 mars 2018 kl 0915 i Rydbergs sal pa Fysiska Institutionen Professorsgatan 1Lund

Fakultetsopponent Dr Charles W ThielMontana State University USA

Academic thesis which by due permission of the Faculty of Engineering at Lund University will be publicly defended on2nd of March 2018 at 0915 am in Rydbergrsquos hall at the Department of Physics Professorsgatan 1 Lund for the degreeof Doctor of Philosophy in Engineering

Faculty opponent Dr Charles W ThielMontana State University USA

Organization

LUND UNIVERSITY

Division of Atomic Physics Department of Physics PO Box 118 SE-221 00 Lund Sweden

Document name

DOCTORAL DISSERTATION

Date of issue January 2018

Sponsoring organization

Author

Adam Kinos

Title and subtitle


Abstract

Key words Rare-earth-ion-doped crystals Light-matter interaction Quantum computing Slow light

Classification system andor index terms (if any)

Supplementary bibliographical information

Language

English

ISSN and key title

ISSN 0281-2762 Lund Reports on Atomic Physics LRAP-543

ISBN

978-91-7753-543-0

Recipientrsquos notes Number of pages

219

Price

Security classification

I the undersigned being the copyright owner of the abstract of the above-mentioned dissertation hereby grant

to all reference sources permission to publish and disseminate the abstract of the above-mentioned dissertation

Signature Date January 22nd 2018

In this thesis crystals of yttrium orthosilicate (Y2SiO5) that are randomly doped with another rare-earth element such as

praseodymium (Pr) europium (Eu) or cerium (Ce) are investigated with lasers locked to ultra-stable cavities using the

Pound-Drever-Hall locking technique

Many of these rare-earth elements have long-lived 4f-4f transitions hundreds of microseconds to a few milliseconds with

even longer ground hyperfine lifetimes of up to several days The coherence properties are to the best of the authors

knowledge the longest achieved for any material currently with a record of six hours for the nuclear spin states of

Eu3+Y2SiO5 measured at cryogenic temperatures Furthermore due to natural trapping and differences in the local

environments each dopant ion experiences a slightly different crystal field and thus an inhomogeneity in the 4f -4f

transition exists between all ions in a crystal Since the homogeneous linewidth is in the order of kHz or below whereas

the inhomogeneous profile can be several GHz wide these materials have dense storing capabilities

This thesis explores how light interacts with such rare-earth-ion-doped crystals how the absorption and light polarization

varies during propagation how spectral features in the inhomogeneous absorption profile can be tailored to create

narrowband spectral filters how the speed of light is slowed down significantly in such narrow transmission windows

and how that can be used to either frequency shift incoming light control its group velocity or temporally compress

pulses

It also uses rare-earth-ions to research quantum computing the field of using quantum mechanical effects such as

superpositions and entanglements to outperform classical computers on certain specific problems This is done by

examining how two-color pulses can be used to rapidly induce coherence from an initially mixed state how qubit-qubit

interactions can be performed experimentally using ensemble qubits which opens the door to two-qubit experiments such

as the CNOT-gate and entanglements how a scalable quantum computer might be constructed using a single ion qubit

approach with a dedicated readout ion and buffer ion(s) to improve readout fidelity and how cerium is investigated as a

candidate for such a dedicated readout ion



Adam Kinos

Doctoral Thesis

2018

Light-matter interaction and quantum computing in rare-earth-ion-doped crystals

copy 2018 Adam KinosAll rights reservedPrinted in Sweden by Media-Tryck Lund 2018

Division of Atomic PhysicsDepartment of PhysicsFaculty of Engineering LTHLund UniversityPO Box 118SEndash221 00 LundSweden

httpwwwatomicphysicsluse


ISBN (print) 978-91-7753-543-0ISBN (pdf) 978-91-7753-544-7

Abstract

In this thesis crystals of yttrium orthosilicate (Y2SiO5) thatare randomly doped with another rare-earth element such aspraseodymium (Pr) europium (Eu) or cerium (Ce) are inves-tigated with lasers locked to ultra-stable cavities using the Pound-Drever-Hall locking technique

Many of these rare-earth elements have long-lived 4f-4f transi-tions hundreds of microseconds to a few milliseconds with evenlonger ground hyperfine lifetimes of up to several days The co-herence properties are to the best of the authorrsquos knowledge thelongest achieved for any material currently with a record of sixhours for the nuclear spin states of Eu3+Y2SiO5 measured at cryo-genic temperatures Furthermore due to natural trapping and dif-ferences in the local environments each dopant ion experiences aslightly different crystal field and thus an inhomogeneity in the4f-4f transition exists between all ions in a crystal Since the ho-mogeneous linewidth is in the order of kHz or below whereas theinhomogeneous profile can be several GHz wide these materialshave dense storing capabilities

This thesis explores how light interacts with such rare-earth-ion-doped crystals how the absorption and light polarizationvaries during propagation how spectral features in the inhomo-geneous absorption profile can be tailored to create narrowbandspectral filters how the speed of light is slowed down significantlyin such narrow transmission windows and how that can be usedto either frequency shift incoming light control its group velocityor temporally compress pulses

It also uses rare-earth-ions to research quantum computing thefield of using quantum mechanical effects such as superpositionsand entanglements to outperform classical computers on certainspecific problems This is done by examining how two-color pulsescan be used to rapidly induce coherence from an initially mixedstate how qubit-qubit interactions can be performed experimen-tally using ensemble qubits which opens the door to two-qubitexperiments such as the CNOT-gate and entanglements how ascalable quantum computer might be constructed using a single

iii

ion qubit approach with a dedicated readout ion and buffer ion(s)to improve readout fidelity and how cerium is investigated as acandidate for such a dedicated readout ion

iv

Popularvetenskapligsammanfattning

Ljus-materia vaxelverkan ar nagot som finns overallt ljus gener-eras i lampor absorberas eller reflekteras av material fokuseraspa nathinnorna med hjalp av linserna i vara ogon tapparna ochstavarna absorberar ljuset och signaler skickas via synnerven vi-dare till hjarnan sa att vi kan tolka varlden runtom oss i fargermorker och ljus

Valdigt kortfattat fungerar det enligt foljande det inkom-mande ljusets svangningar far atomernas elektroner att borjasvanga och skicka ut sitt eget ljus som tillsammans med detursprungliga ljuset likt tva vattenvagor kan slacka ut ellerforstarka varandra Samtidigt kan atomerna absorbera energinfran ljusvagen och exciteras

Denna avhandlingen handlar om hur ljus interagerar med ettspecifikt material namligen kristaller dopade med sallsynta jor-dartsmetaller som tex praseodymium eller europium Sadanaatomer ar intressanta att undersoka eftersom de har valdigtlanga livstider vilket betyder att de kan vara exciterade en rel-ativt (millisekunder) lang tid innan de ater faller tillbaka tillgrundtillstandet ibland genom att skicka ut nytt ljus

De dopade kristallerna undersoks i anknytning till kvantdatorersom i kontrast med en vanlig dator vars bitar endast kan vara nol-lor och ettor har kvantbitar som aven kan vara i tillstand som arbade noll och ett samtidigt sa kallade superpositioner Man kansaga att en kvantdator har tillgang till fler valmojligheter for varjekvantbit Dock nar man ska lasa ut vad en kvantbit ar far manalltid antingen noll eller ett som svar aven om kvantbiten innanman gjorde utlasningen var lite av bada samtidigt Detta betyderatt utlasningen av en vanlig bit och en kvantbit bada innehallerendast en enhet av information (aven kallat en bit av informa-tion) Vid forsta anblicken kan man darfor tro att en kvantdatorinte tillfor nagot som en klassisk dator inte redan har men tackvara en kvantdators formaga att anvanda kvantmekaniska fenomensasom superpositioner och sammanflatningar (engelska entangle-

v

ment) kan vissa problem losas mycket snabbare an pa en vanligdator man maste bara tanka till ordentligt angaende vilken infor-mation man vill fa ut i slutandan och hur denna beror pa kvantin-formationen under berakningen

I nulaget ar kvantdatorer i ett tidigt skede men utvecklingen avflera olika implementationer gar fort framat I denna avhandlingenundersoks som sagt kristaller dopade med sallsynta jordartsmet-aller dar tva olika energinivaer hos atomerna anvands som nolloroch ettor i kvantbitarna Det som undersoks ar hur man experi-mentellt kan fa tva kvantbitar var och en bestaende av miljontalsatomer att interagera For att dock kunna skala systemet tillflera kvantbitar maste man ga mot att anvanda enskilda atomertill varje kvantbit Svarigheten ar da istallet utlasningen av om enkvantbit ar i noll eller ett Detta har var grupp lange forsokt losagenom att anvanda cerium som en hangiven utlasningsatom ochaven om vi annu inte lyckats detektera en enskild ceriumatom sahar en permanent halbranningsprocess upptackts i materialet vistuderar nagot som dessvarre ytterligare forsvarar experimenten

Kristaller dopade med sallsynta jordartsmetaller ar ocksaspeciella eftersom man kan skraddarsy hur atomerna absorberarljus med lite olika frekvenser (farger) nagot som i vissa fall kanbibehallas i upp till dagar utan att forandras markbart Detta kananvandas till mycket och har undersoks vissa sadana strukturersom kan absorbera respektive transmittera ljus som skiljer sig yt-terst lite i frekvens sakta ner ljus sa att det endast ror sig meden brakdel av hastigheten som det normalt fardas med eller medhjalp av en elektrisk strom skifta frekvensen nagot pa inkommandeljus

vi

List of Publications

This thesis is based on the following papers which will be referredto in the text by their roman numerals Note that the thesis authorpreviously published under Adam N Nilsson

I Wave propagation in birefringent materials withoff-axis absorption or gainM Sabooni A N Nilsson G Kristensson and LRippePhys Rev A 93 013842 (2016)

II Development and characterization of highsuppression and high etendue narrowband spectralfiltersA Kinos Q Li L Rippe and S KrollAppl Opt Vol 55 No 36 10442 (2016)

III Slow-light-based optical frequency shifterQ Li Y Bao A Thuresson A N Nilsson L Rippeand S KrollPhys Rev A 93 043832 (2016)

IV Using electric fields for pulse compression andgroup-velocity controlQ Li A Kinos A Thuresson L Rippe and S KrollPhys Rev A 95 032104 (2017)

V Fast all-optical nuclear spin echo technique basedon EITA Walther A N Nilsson Q Li L Rippe and SKrollEur Phys J D 70 166 (2016)

vii


VI Towards CNOT operations using ensemble qubitsA Kinos L Rippe A Walther and S Kroll(2018) Manuscript in preparation

VII High-fidelity readout scheme for rare-earthsolid-state quantum computingA Walther L Rippe Y Yan J Karlsson D SerranoA N Nilsson S Bengtsson and S KrollPhys Rev A 92 022319 (2015)

VIII High-resolution transient and permanent spectralhole burning in Ce3+Y2SiO5 at liquid heliumtemperaturesJ Karlsson A N Nilsson D Serrano A Walther PGoldner A Ferrier L Rippe and S Kroll(Authors contributed equally)Phys Rev B 93 224304 (2016)

Other publications by the author not included in this thesis

Competitive binding-based optical DNA mappingfor fast identification of bacteria - multi-ligandtransfer matrix theory and experimentalapplications on Escherichia coli

A N Nilsson G Emilsson L K Nyberg C Noble L S StadlerJ Fritzsche E R B Moore J O Tegenfeldt T Ambjornsson andF Westerlund(Authors contributed equally)

Nucleic Acids Res 42 e118 (2014)

A fast and scalable kymograph alignment algorithmfor nanochannel-based optical DNA mappings

C Noble A N Nilsson C Freitag J P Beech J O Tegenfeldt andT Ambjornsson

PLOS One 10 e0121905 (2015)

Visualizing the entire DNA from a chromosome ina single frame

C Freitag C Noble J Fritzsche F Persson M Reiter-Schad A NNilsson A Graneli T Ambjornsson K U Mir and J O Tegenfeldt

Biomicrofluidics 9 044114 (2015)

viii


Bacteriophage strain typing by rapid singlemolecule analysis

A Grunwald M Dahan A Giesbertz A N Nilsson L K NybergEWeinhold T Ambjornsson F Westerlund and Y Ebenstein


Rapid identification of intact bacterial resistanceplasmids via optical mapping of single DNAmolecules

L K Nyberg S Quaderi G Emilsson N Karami E Lagerstedt VMuller C Noble S Hammarberg A N Nilsson F Sjoberg JFritzsche E Kristiansson L Sandegren T Ambjornsson and FWesterlund

Sci Rep 6 30410 (2016)

Super-resolution genome mapping in siliconnanochannels

J Jeffet A Kobo T Su A Grunwald O Green A N Nilsson EEisenberg T Ambjornsson F Westerlund E Weinhold D ShabatP K Purohit and Y Ebenstein

ACS Nano 10 (11) 9823-9830 (2016)

ix

Abbreviations

AOM Acousto-Optic Modulator

BP Band Pass filter

CCD Charge-Coupled Device

CNOT Controlled NOT

DM Dichroic Mirror

ECDL External Cavity Diode Laser

EIT Electromagnetically Induced Transparency

EOM Electro-Optic Modulator

FWHM Full Width at Half Maximum

OD Optical Diode

PBS Polarizing Beam Splitter

PDH Pound-Drever-Hall locking technique

PH Pinhole

PMT Photomultiplier Tube

SMF Single Mode Fiber

SPAD Single Photon counting Avalanche Diode

TC pulse Two-Color pulse

Ce Cerium

Eu Europium

Nd Neodymium

Pr Praseodymium

Y2O3 Yttrium Oxide

Y2SiO5 Yttrium Orthosilicate

xi

Contents

Abstract iii

Popularvetenskaplig sammanfattning v

1 Introduction 111 A personal perspective 212 A brief overview of the work performed 213 Outline of the thesis 3

2 Understanding light-matter interaction 521 Refractive index of a sparse material 6

211 Radiation induced by an accelerating charge 6212 Atom-swing analogy 9213 Phase relation between an incoming field and the in-

duced radiation 10214 Induced radiation from a plane of charges 12215 Refractive index and phase velocity 14

22 Dispersion group velocity and slow light 16221 Slow light 17222 Energy temporarily stored in the medium 18

23 Refractive index of a dense material 1924 Relation to phonons 22

3 Understanding the quantum mechanical nature of atoms 2331 Classical versus quantum picture 24

311 Analyzing the infinite well potential 2532 Light-matter interaction in quantum mechanics 27

321 Bloch sphere 28322 Connection between superpositions and light radiation 30323 Relation between the Bloch sphere and the wave de-

scription 3233 Decays 34

331 Coherence time 34332 Lifetime 35

34 DC Stark effect 37

4 Rare-earth-ion-doped crystals 3941 Rare-earth elements 39

411 4f-4f transitions 40412 Hyperfine energy level structures 43

42 Yttrium orthosilicate (Y2SiO5) host crystal 4343 Transition and static dipole moments 4444 An overview of praseodymium (Pr) and europium (Eu) prop-

erties in Y2SiO5 4545 Pr3+Y2SiO5 hyperfine and inhomogeneous absorption profiles 46

Contents

5 Light-matter interaction in rare-earth-ion-doped crystals 4751 Wave propagation in birefringent materials with off-axis ab-

sorption or gain (Paper I) 4852 Development and characterization of high suppression and

high etendue narrowband spectral filters (Paper II) 5153 Slow-light-based optical frequency shifter (Paper III) 5454 Using electric fields for pulse compression and group-velocity

control (Paper IV) 5655 Outlook of projects 59

6 A brief overview of quantum computation 6161 Pursuing quantum computing 6162 DiVincenzorsquos criteria 6263 Quantum Technologies Roadmap 63

7 Quantum computation in rare-earth-ion-doped crystals 6571 Qubit initialization 6672 Single qubit gate operations 66

721 One-color sechyp pulses 67722 Two-color sechyp pulses 68723 Fast all-optical nuclear spin echo technique based on

EIT (Paper V) 6973 Qubit-qubit interaction (Paper VI) 7274 Single ion readout scheme (Paper VII) 7575 Cerium (Ce) as a dedicated readout ion (Paper VIII) 7976 Investigating Eu3+Y2O3 nanocrystals 8077 Progress on DiVincenzorsquos criteria 8278 Outlook of projects 83

8 Conclusions 8581 A personal perspective 86

A Equipment 89A1 Main setup 89

A11 Dye laser setup 90A12 Dye laser locking 91A13 Experimental table 92

A2 Cryostat 93A3 Cerium (Ce) setup 94

The authorrsquos contribution to the papers 97

Acknowledgements 101

Contents

Papers

I Wave propagation in birefringent materials with off-axisabsorption or gain 115

II Development and characterization of high suppressionand high etendue narrowband spectral filters 129

III Slow-light-based optical frequency shifter 139

IV Using electric fields for pulse compression and group-velocity control 149

V Fast all-optical nuclear spin echo technique based on EIT 159

VI Towards CNOT operations using ensemble qubits 171

VII High-fidelity readout scheme for rare-earth solid-statequantum computing 183

VIII High-resolution transient and permanent spectral holeburning in Ce3+Y2SiO5 at liquid helium temperatures 191

Chapter 1

Introduction

The human striving for improvement be it in happiness powersports or manufacturing leads to an ever-increasing knowledgebase of society During the last few centuries we have gone throughan industrial revolution the atomic jet and space age and nowafter the digital revolution we are in the information age

As our knowledge increases it is important to remember whyon a fundamental level we believe nature behaves as it doesA comprehensive and thought-provoking overview is given in theFeynman Lectures of Physics volume 1 [1] volume 2 [2] and vol-ume 3 [3] which I use throughout my thesis as references for gen-eral physical phenomena

As equipment and techniques became more refined people wereable to probe into the large and small general relativity and quan-tum mechanics respectively The work presented in this thesis ismainly focused on a subset of the latter quantum computingparticularly using rare-earth-ion-doped crystals

This raises the eternal question of why Why should we investour time and resources in quantum computing One simple an-swer is curiosity As research progresses our understanding of theunderlying principles improves and for me that is reason enoughHowever a more fruitful answer might be the historical benefitsof pursuing new insights from the ballistic trajectories of New-tonian mechanics [4] to the general relativity-dependent GlobalPositioning System (GPS) [5] Similarly quantum mechanics isthe foundation on which modern computers with their transistorsand semiconductors are based In Chapter 6 I briefly provide morearguments as to why one should pursue quantum computing andlist several companies that are already investing time and moneyin solving the problem of building a quantum computer

Another question might also arise what is quantum comput-ing It is the practical engineering of a device capable of controllingits parts in a quantum mechanical way such that it includes su-

1

11 A personal perspective

perpositions and entanglements as well as inventing algorithmsthat use those quantum resources to achieve a (potential notproven) speed-up compared to classical computers Algorithmshave already been invented for specific problems but rememberthat day-to-day usage of a computer would not currently benefitfrom quantum mechanical effects

As mentioned earlier this thesis is primarily about engineeringsuch a quantum computer using rare-earth-ion-doped crystals butit also contains a fair bit of light-matter interaction especially inconnection with these materials


ldquoFall in love with some activity and do it Nobody ever figuresout what life is all about and it doesnrsquot matter Explore theworld Nearly everything is really interesting if you go into it deeplyenough Work as hard and as much as you want to on the thingsyou like to do the best Donrsquot think about what you want to be butwhat you want to do Keep up some kind of a minimum with otherthings so that society doesnrsquot stop you from doing anything at allrdquo- Richard Feynman (allegedly)

A few years ago I decided that quantum computing was a topicI wanted to explore further and so I started as a PhD student inthe quantum information group here at Lund University The mixbetween light-matter interaction theory and experiments intriguedme as it allowed for the opportunity to increase both my funda-mental and my application-based knowledge in a very interestingfield It has also given me time to teach undergraduates the ba-sics of waves and optics optical pumping high spectral resolutionspectroscopy light-matter interaction and quantum computingwhich has been a joyful learning experience for me and hopefullymy students

I have always enjoyed mathematics and my interest in physicshas grown to a desire to understand the fundamentals of naturebe it in quantum mechanics particle physics astronomy chem-istry or biology My work here has taught me about light-matterinteraction and quantum mechanics and it is my sincerest hopethat this thesis will convey some of that knowledge to future PhDstudents and remind me of everything I have learned during thistime

12 A brief overview of the work performed

The main goal of my project has been to improve the quantumcomputing progress in rare-earth-ion-doped crystals Some efforthas been put into generating high fidelity single qubit gates butthis has not yet resulted in any publication However qubit-qubit

2

Introduction

interactions have been studied for ensemble qubits in order toprovide useful experience in preparation for the single ion qubitscheme which has also been investigated through the unfortu-nately unsuccessful attempts to detect a single cerium ion to beused as a dedicated readout ion

I have not been able to resist the temptation to explore thelight-matter interaction topic through wave propagation simula-tions and slow light applications which was not my main aim butnevertheless became a large part of my PhD project

13 Outline of the thesis

Since understanding physics is my own personal driving force thisthesis starts off with the basic physical processes and slowly getsinto more application-based descriptions I have therefore chosento move most of the equipment description to the end of this thesisin Appendix A

I am a firm believer of the Feynman Technique [6] where learn-ing is an iteration between studying pretending or actually teach-ing the topic to a colleague or classroom in simple terms identi-fying knowledge gaps which is made easier by the teaching of thesubject and going back to studying and improving the simplifica-tions and analogues to take the complex subject into the familiarrealm in order to further increase knowledge and intuition aboutthe problem This thesis thus contains some sections that mightseem very easy and trivial but I wanted to include them for thesake of completeness

Chapter 2 investigates light-matter interaction from the electricfield and wave point of view whereas Chapter 3 discusses quantumphenomena from the perspective of atoms These two chaptersare very general and it is not until Chapter 4 that I introduce therare-earth-ion-doped crystals which are so central to this thesisby discussing their properties After this theoretical backgroundis given Chapter 5 presents the results of Papers I-IV

At this point the thesis shifts focus to quantum computingwith a brief overview in Chapter 6 before Chapter 7 presents ourefforts in the field and discusses Papers V-VIII

The thesis concludes in Chapter 8 where I try to compress theresults of my work during the past four years into a few pages

3

Chapter 2

Understanding light-matterinteraction

To understand and appreciate the work of Papers I-IV threethings are needed knowledge about atoms and rare-earth-ion-doped crystals which are discussed in Chapters 3 and 4 respec-tively and an intuitive understanding of light-matter interactionwhich is presented in this chapter Note that the discussions aresomewhat simplified as they do not fully consider all quantummechanical effects

Section 21 aims at giving the reader a qualitative picture ofhow light interacts with matter and why it normally slows downwhen propagating through anything but vacuum ie explainingthe refractive index of materials Starting in Section 211 an ac-celerating charge is studied with regard to the light it radiates outIn Section 212 an atom-swing analogy is presented to provide amore intuitive understanding of how the light field radiated byoscillating charges relates to an incoming field A mathematicaldescription of how light interacts with a sparse isotropic materialeg a gas in the linear excitation regime is given in Sections 213and 214 which closely follows the Feynman Lectures on Physics[1 chapter 30 31] but where I use the eminusiωt convention for the elec-tric field instead of eiωt The refractive index and phase velocityconcepts are explained in Section 215

Section 22 discusses dispersion the change of refractive indexas a function of the frequency of light group velocity the speedat which a pulse travels and slow light the concept of a group ve-locity that is much lower than the speed of light in vacuum Theeffects of strong dispersion and slow light are discussed in Sec-tion 221 The energy density of a pulse that propagates througha slow light material is distributed between the atoms and theelectric field and is investigated in Section 222

5

21 Refractive index of a sparse material

Figure 21 A charge shown as agreen circle has an electric fieldshown as red arrows associatedwith it which points outwards ifthe charge is positive (as in thefigure) and inwards if the chargeis negative The electric fieldstrength decays as 1r2 as seen inEquation (21)

Figure 22 A positive chargeshown in green is moving with aconstant velocity indicated by thedashed green arrow Its electricfield components (red lines)perpendicular to the movementdirection increase compared to thestationary charge seen inFigure 21 due to the Lorentztransformation of specialrelativity [2 chapter 26-3] Thistransformation also generatesmagnetic field lines shown inblue

The refractive index is once more derived in Section 23 thistime for dense matter instead of gases while still assuming anisotropic material and the linear excitation regime Multiple reso-nances and local field effects are also discussed in this section

Phonons vibrations in dense matter are qualitatively de-scribed in Section 24 using the framework presented in the previ-ous sections

While this chapter mostly discusses light-matter interactionfrom the lightrsquos point of view Chapter 3 goes beyond the lin-ear excitation regime and discusses what happens when the lightsignificantly excites an atom in the quantum picture


The main goal of this chapter is to lay the groundwork for thefollowing chapters but there is much more that can be understoodusing the knowledge presented here For example on a hot summerday you can sometimes see mirror-like puddles on the ground butas soon as you get closer they disappear and when looking abovea flame or candle the view seem distorted Effects such as theseare due to light traveling at different speeds in different materialsMore precisely in these two cases the speed depends on the airtemperature or density We call the ratio of the speed of light invacuum to the speed in matter the refractive index and the originof this is discussed in the following sections

211 Radiation induced by an accelerating charge

A stationary point charge has a radial electric field associated withit according to Coulombrsquos Law [2 chapter 4-2] see Figure 21 andthe equation below

E(r) =1

4πε0

q

r2er (21)

where ε0 is the vacuum permittivity q is the charge r is thedistance measured from the charge and er is a unit vector thatpoints radially outward

A charge moving with constant velocity has similar electricfield lines but the components perpendicular to the propagationdirection are increased due to the Lorentz transformation of specialrelativity [2 chapter 26-3] as seen in the red lines of Figure 22This transformation also creates a transverse magnetic field whichcan be seen in the blue lines but for now it is disregarded

When a charge is quickly accelerated from stationary to somevelocity it causes changes in the field lines but they propagateat the speed of light thus creating an outward moving spheri-cal shell beyond which the field lines still follow Coulombrsquos Law

6

Understanding light-matter interaction

from the original position and inside which the field lines are asin Figure 22 ie following the charge that now moves with con-stant velocity [7] Since the field cannot be discontinuous in thefree space surrounding the charge the lines within this shell lookapproximately as in Figure 23(a)

Even from this simple picture you can draw some remarkableconclusions First in the limit of an infinitesimal shell thicknessand thus a very small acceleration the electric field within thespherical shell is perpendicular to the propagation direction of theshell that radiates outwards as illustrated in the dashed box thatshows a more detailed view of the field within the shell This agreeswith the linear polarization you would expect from an acceleratingcharge in one dimension Second these transverse segments extendthe field lines by a factor r sin θ where θ is the polar angle thatis zero at the top of the figure This suggests that the long-rangeradial dependence is no longer the 1r2 from Equation (21) butinstead 1r such that the electric field induced by accelerating thecharge propagates much farther than the normal Coulomb fieldSince the intensity of a wave is proportional to the electric fieldsquared this shell contains a fixed amount of energy that is carriedaway from the charge (energy = ImiddotA prop E2middotA prop 1r2middotr2 = constantwhere I is the intensity and A is the area over which the intensity isspread ie the surface area of a sphere Third similar argumentscan be made for the magnetic field lines shown in Figure 22 andthe strengths of both the electric and magnetic fields induced areproportional to the magnitude of the acceleration since a largeracceleration shifts the field lines inside more than those beyond theshell In summary accelerating a charge a relatively small amountcreates a shell of transverse electromagnetic radiation proportionalto the retarded acceleration and decays as 1r such that it carriesa constant energy away from the charge [7]

7


(a)

(b)

120579

Figure 23 (a) A positive charge (green circle) has rapidly acceler-ated from stationary to some finite velocity (dashed green arrow) Thisacceleration forms a spherical shell denoted by the two black circlesthat expand outwards at the speed of light Beyond this shell the redelectric field lines originate from the original position whereas insideit the field lines move with the charge as in Figure 22 In the limitof an infinitesimal shell thickness the field lines within the shell areorthogonal to the radial direction and connect the internal and externalfield lines (b) The charge oscillates back and forth which generates asimilar oscillating behavior for the electric field8


Figure 24 An oscillating charge(green circle) generates a dipoleradiation pattern as seen in thered electric and blue magneticfield lines The electromagneticradiation decay as 1r exhibits asin θ behavior and is rotationallysymmetrical around φ theazimuthal angle The black linesdenote the spherical shell thatcontains the field lines andradiates outwards at the speed oflight

Before we discuss what makes the charge accelerate in the firstplace it is useful to consider a particular acceleration pattern iean oscillation where the particle is moving back and forth on a linein a sinusoidal manner The single shell is now replaced by severalshells stacked after one another as seen in Figure 23(b) Eachshell has fields that are proportional to the retarded accelerationSince the charge accelerates back and forth the fields also oscillateat the same frequency This is called dipole radiation Just as inthe single shell case the fields have an amplitude with a polar angledependence of sin θ and still exhibit rotational symmetry aroundthe azimuthal angle as seen in Figure 24

The results of this section are directly embedded in Maxwellrsquosequations [2 chapter 4-1] where Coulombrsquos Law is the static caseof Gaussrsquos Law for a point charge and where a moving charge iea current is related to a magnetic field Furthermore a changingelectric field induces a magnetic field and vice versa which corre-lates perfectly with our results of transverse electromagnetic fieldsthat propagate further than the normal Coulomb field

The following sections use the concepts presented here to de-scribe how light interacts with a single atom and a plane of atomsas well as why it seems to travel slower when propagating throughmaterials

212 Atom-swing analogy

When light interacts with a material it shakes the electrons inthe atoms that make up the material causing them to oscillatewith the frequency of the incoming field and send out their ownradiation as described in Section 211 Since the electrons are neg-atively charged and the nucleus of the atom is positive a restoringforce that tries to bring the electrons back to their original positionexists Energy loss in the form of eg heat or sound can also bepresent

This section goes through an analogy to the system describedabove A swing plays the part of the electron and a person steeringthe swing through the ropes acts as the light that drives it Thenucleus in this analogy is the stationary swing set that is too heavyto move and gravity is the restoring force trying to drive the swingback to its lowest point Air resistance and friction slow down theswing as it moves causing it to lose energy

As long as the material is sparse such as a gas and we arein the linear regime ie the electric field does not substantiallyalter the excitation of the electron through the interaction themathematical treatment of light that interacts with an atom isthe same as for the swing system [1 chapter 31-2] Thus it isenlightening to study how a swing behaves in different situationssince it is directly transferable to how light interacts with electrons

9

213 Phase relation between an incoming field and the induced radiation

Figure 25 A swing driven at alow frequency it is in phase (0degrees) with the hands

Figure 26 A swing driven atresonance frequency it is 90degrees after the hands

Figure 27 A swing driven at ahigh frequency it is opposite inphase (180 degrees) compared tothe hands

If a swing is pulled back and released it oscillates back andforth with one frequency that we call the resonance frequencyHowever the swing can oscillate at other frequencies if the personsteering it holds on to the ropes

When the swing moves back and forth very slowly in this man-ner ie with a frequency much lower than the resonance fre-quency it moves in phase with the person who steers it There isa 0 degree phase shift between the swing and the personrsquos handsas shown in Figure 25

If the swing is driven at its resonance frequency instead asshown in Figure 26 it is a quarter of an oscillation behind theperson ie 90 degrees after in phase

If you oscillate the swing at a frequency much higher than itsresonance it behaves as in Figure 27 ie being completely outof phase with the driving hands (180 degree phase shift)

The next section offers a mathematical treatment of how lightinteracts with an atom and causes it to send out its own radia-tion which is directly related to the swing described here Thephase relation of the induced electric field that originates from theelectron compared to the incoming field is the same as the phaserelations explained above for our more intuitive experiments withthe swing

213 Phase relation between an incoming fieldand the induced radiation

The system that we analyze in this section can be seen in Fig-ure 28 An incoming plane wave Ein(t r) = E0exe

minusiω(tminuszc)traveling in the z direction with a polarization along x and a fre-quency ω reaches an atom located at r = 0 with a single resonancefrequency ω0 A complex notation is used for the electric fieldwhere it is understood that the field consists only of the real partof Ein(t r) Similarly to a swing that oscillates much stronger ifdriven at its resonance frequency atoms have transitions ie cer-tain resonance frequencies at which the incoming field is able toexcite an atom to another energy state This is further discussedin Section 311

Newtonrsquos second law [1 chapter 9-1] can be used to relate the

acceleration x(t) = d2x(t)dt2 of the electron with the forces acting

on it Ftot(t) These forces are the restoring harmonic oscillatorforce proportional to x(t) the damping force proportional to x(t)and the force exerted by the light Fl(t) ie the Lorentz force [2chapter 13-1]

mex(t) = Ftot(t) = minusmeω20x(t)minusmeγx(t) + Fl(t) (22)

where me is the mass of the electron and τ = 1γ gt 0 is theexponential damping time The full Lorentz force is [2 chapter

10


119812in t 119851

119911

119909

120596 1205960

Figure 28 An atom located atr = 0 with resonance frequency ω0interacts with an incoming planewave with frequency ω that istraveling along the z-directionwith a polarization along x

13-1]FLorentz(t) = qe(Ein(t) + v(t)timesBin(t)) (23)

where qe lt 0 and v are the charge and velocity of the elec-tron respectively and Bin is the magnetic field associated withthe incoming electromagnetic wave Assuming a low electron ve-locity compared with the speed of light the magnetic term canbe neglected remembering that |Bin| = |Ein|c for a travelingwave Assuming the electron is located at z = 0 the force in thex-direction can be written as

Fl(t) = qeE0eminusiωt (24)

Solving Equation (22) for x(t) yields

x(t) = x0eminusiωt =

qeE0

me(ω20 minus ω2 minus iωγ)

eminusiωt (25)

Maxwellrsquos equations says that any accelerating charge sendsout its own field [2 chapter 18-4] as explained in Section 211In this case where the electron moves in an oscillatory pattern wecan write a simple expression for the induced radiation Erad(t r)at position r produced by an electron located at r = 0 as long aswe look far off in the z-direction ie |r| = r asymp z

Erad(t r) asymp minusqe4πε0c2r

exx(tminus rc) (26)

In reality the electric field produced has a polarization thatis orthogonal to the propagation direction Thus if r 6asymp z thepolarization would no longer be only along ex as explained inSection 211 and shown in Figure 24

Using the expression for the electron position given in Equa-tion (25) the field produced by the oscillating electron can bewritten as

Erad(t r) asymp qeω2

4πε0c2rx0exe

minusiω(tminusrc)

asymp q2eω

2

4πε0c2rme

1

ω20 minus ω2 minus iωγEin(t r)

(27)

where the second approximation uses r asymp z to write the inducedfield as the incoming field multiplied by two factors Since thefirst factor is just a positive number the middle part determinesthe phase relation of the induced radiation Erad(t r) and theoriginal field Ein(t r) Thus the induced field oscillates at thesame frequency ω as the driving field but with a phase relationdetermined by 1

ω20minusω2minusiωγ

Just as for the swing explained in Section 212 we can seethat if ω ω0 this factor is positive and real such that Erad(t r)

11

214 Induced radiation from a plane of charges

Figure 29 In an analogy to the induced electric field radiated by aplane of oscillating charges a house filled with people on swings thatoscillates in phase where each person shouts simultaneously as theyreach the most forward position is examined For a person standingright in front of the building the sound from the closest person on aswing traveling along the red line arrives earlier than the sound from aperson sitting farther away such as along the blue line Thus the soundexperienced by the person in front of the house is an elongated soundwith a center that is delayed compared to the sound arriving along thered line

oscillates in phase (0 degree phase shift) compared with Ein(t r)Similarly if ω = ω0 the fraction is positive and purely imaginary( 1minusi = i) which corresponds to being 90 degrees after in phase

Lastly if ω ω0 it is negative and real yielding a 180 degreephase shift between the two fields Note that γ is assumed to bemuch lower than ω and ω0

Now that we know how light interacts with a single atom thenext step in understanding refractive index is to continue onto athin plane of atoms which is the topic of the next section


To explain why light refracts and slows down when propagatingthrough matter the contribution from one electron to the field isnot enough Instead you have to evaluate the electric field contri-bution for a plane of electrons all emitting their own light due tothe acceleration induced by the incoming plane wave

Before continuing with the mathematical treatment of the elec-trons I will explain the effects using the analogy with the swings

Imagine a house with several floors filled with swings that alloscillate in phase as seen in Figure 29 Now each person sittingon a swing shouts as they reach their most forward position Since

12


119876

119875

120588119903

119911119889120588

120578

Figure 210 A plane of oscillatingcharges with a surface density of ηgenerates a field at P that can beevaluated by integrating overinfinitesimal rings of thickness dρat a distance ρ from the z axis

Re

Im

1205790 =120596119911

119888

radius =119888

120596

Figure 211 The integral inEquation (29) can be interpretedas the sum of infinitesimal arrowswith the same phase anglerelation starting at an angle ofθ0 = ωz

c For the integral in

question these arrows follow thedashed circle indefinitelyHowever in the physically morecorrect description the arrowsdecrease in length and thereforespiral inwards yielding thesolution denoted by the bluearrow

sound travels at a finite speed (as does light) the sound from theperson on the closest swing reaches a person standing in front ofthe house first see the red line in the figure A while later thesound from a person sitting further away arrives for example alongthe blue line in the figure The collection of sounds from everyoneon the swings is an elongated sound whose center is delayed com-pared to the sound from the person on the closest swing Fromthis analogy we can expect the electric field contribution from allelectrons to be delayed (later in phase) compared to the field fromthe closest electron

With this information in mind we can continue with the treat-ment of light that interacts with electrons in an infinite plane asin Figure 210 The electric field at point P from a charge locatedat Q is given in the first line of Equation (27) Since this dependsonly on the distance r between P and Q the total induced fieldcan be evaluated as the integral over infinitesimal rings of surfacearea 2πρdρ with the number of charges per unit area equal to η[1 chapter 30-7]

For symmetry reasons the only polarization that can benonzero is along the direction the electrons move in ie the x-direction given by the incoming electric field polarization

Eradtot(t r) asympint ρ=infin

ρ=0

qeω2

4πε0c2x0exe

minusiω(tminusrc)

rη2πρdρ (28)

Note that the expression is only approximately correct since wemade the assumption that r asymp z in Equation (26) before arrivingat Equation (27)

With the Pythagorean theorem r2 = ρ2 +z2 you can eliminatethe variable ρ from the integral (ρdρ = rdr) The new limits arefrom r = z to r =infin

Eradtot(t r) =qeω

2ηx0exeminusiωt

2ε0c2

int r=infin

r=z

eiωrcdr (29)

The solution to the integral is ciω (eiωinfinc minus eiωzc) the first

term of which is undefined mathematicallyReturning to the physics of the problem the integral is just the

sum of infinitesimal arrows all with a slight phase angle differenceof ∆θ = ω∆r

c For the integral above these arrows form a perfectcircle which is why it does not have a finite answer see the dashedcircle in Figure 211 However Equation (27) was only approx-imately true for distances r asymp z and this is no longer the casewhen ρ rarr infin Thus the electric field contribution from chargesfar out in the plane diminishes and eventually reaches zero as inFigure 24 for steep θ angles which results in a spiraling that tendstowards the center of the circle as the arrows decrease in lengthgiving the integral its value of minus c

iω eiωzc see the blue arrow in

Figure 211 [1 chapter 30-7]

13

215 Refractive index and phase velocity

|119812in|

|119812rad tot|

Figure 212 When adding theinduced field Eradtot to theincoming field Ein the resultingfield Eafter differs mostly in phasecompared to the incoming field

Propagation distance

119812in

119812rad tot

119812after

Figure 213 Compared to theincoming field Ein the inducedfield from a thin plane of chargesEradtot is 90 degrees later inphase and has much loweramplitude exaggerated here forvisibility The resulting field afterpropagating through the thinplane is therefore a small phasedelay as seen when comparingEafter with Ein

Finally the total electric field radiated by the oscillatingcharges in a plane of atoms driven by an incoming plane waveis

Eradtot(t r) =q2eη

2ε0cme

iω

ω20 minus ω2 minus iωγEin(t r) (210)

Once more we obtain a relation between the induced field andthe incoming field Two important notes can be made Firstthe electric field no longer decreases as a function of r Secondthe phase relations with the incoming field are the same as beforeexcept for an additional 90 degree after in phase due to the extrai in the numerator

The next section ties up the knowledge acquired so far in orderto finally explain the refractive index


Imagine a plane wave that propagates through air shaking theelectrons in the molecules that have resonances at ultraviolet andhigher frequencies amongst other For visible incoming light wetherefore have ω ω0 The field produced by the oscillatingcharges in a thin plane of air is 90 degrees (i) after in phase com-pared to the incoming field as explained in Section 214 and seenin Equation (210)

The total electric field after the plane is the sum of the twocontributions Eafter = Ein + Eradtot Currently we consider onlysparse materials like gases Thus η is small and as a result|Eradtot| |Ein| Thus the total field after the plane has roughlythe same magnitude as the incoming field but slightly rotated inphase as seen in Figure 212 The overall effect of passing througha thin slice of air compared to vacuum where no extra field isinduced is therefore a phase shift as seen in Figure 213 Thefield looks as though it arrives later By stacking several suchplanes after one another the light appears to travel slower whenpropagating through the material It travels at the phase velocityvp which is related to the refractive index by n = cvp

Before continuing to a discussion of the implications of lightslowing down in matter we should find an expression for the re-fractive index

Once more we examine the problem of a plane wave incom-ing perpendicularly towards a thin plane of thickness ∆z madeof a material with refractive index n According to our re-sults above this should lead to a phase shift or an equivalenttime delay Instead of taking time ∆zc to travel through thethickness of the plane it takes n∆zc ie an extra time of∆t = (n minus 1)∆zc Thus if the field before the plane was de-

14


scribed as Ein(t r) = E0exeminusiω(tminuszc) after the plane it is

Eafter(t r) = E0exeminusiω(tminus∆tminuszc)

= eiω(nminus1)∆zcE0exeminusiω(tminuszc)

asymp(

1 +iω(nminus 1)∆z

c

)E0exe

minusiω(tminuszc)

= Ein(t r) + Erad tot(t r)

(211)

where the approximation on the third line is made by assumingthat ∆z is small and therefore eiω(nminus1)∆zc asymp 1 + iω(nminus 1)∆zcThe field that causes the time delay is Erad tot in Equation (210)obtained in Section 214 which yields the fourth line of the equa-tion Since the surface density of charges η is the volume chargedensity N of the material multiplied by the thickness of the plane∆z we can after simple algebra write out an expression for therefractive index [1 chapter 31-2]

nsparse asymp 1 +1

2

q2eN

ε0me

1

ω20 minus ω2 minus iωγ (212)

The approximation sign and the subscript remind us that ourexpression for the refractive index is valid only for a sparse mediasuch as a gas The reason is that we have assumed that only theincoming electric field shakes the electrons whereas in reality thetotal field including the field induced by other charges can causeelectrons to oscillate In sparse materials the field induced byother charges is smaller than the incoming field which is why ourapproximation is valid for cases where n asymp 1 Later in Section 23 amore correct expression that also works for dense matter is derived

As in our example with air at the beginning of this section ifω ω0 we obtain a positive contribution to the refractive indexie a delay in time of the propagating field that is equivalent to areduced phase velocity vp = cn

If the incoming light is instead much closer to resonance ω asympω0 we can see from Equation (210) that the overall phase shiftof the induced field is 180 degrees ie out of phase such that itcancels a part of the incoming field The light energy is absorbedin the oscillation of the electrons just as a swing goes higher ifdriven at its resonance frequency by efficiently taking energy fromthe person pushing it In Equation (212) ω asymp ω0 yields a positiveimaginary contribution to the refractive index which if insertedin the second line of Equation (211) yields

Eafter(t r) = exp

[minus q2

eN

2ε0meγc∆z

]Ein(t r)

= exp[minusα

2∆z]Ein(t r)

(213)

where we have defined the absorption coefficient α with dimen-sion [mminus1] In conclusion if the incoming light is resonant with an

15

22 Dispersion group velocity and slow light

Figure 214 A schematic graph ofhow the real (blue) and imaginary(red) refractive index change as afunction of frequency over aresonance located at ω0 Notethat a positive imaginary partcorresponds to absorption

Figure 215 The upper graphshows the electric field oscillationswithin a pulse of finite durationThe pulse can also be visualizedwith monochromatic waves someof which are shown in the lowergraph When these componentsoscillate in phase they give rise tothe pulse whereas they canceleach other out when they oscillateout of phase

atomic transition the electric field amplitude decays exponentiallywhen traveling through the material Furthermore the energy lostfrom the field is gained by the atom as discussed further in Sec-tion 32

The refractive index can be written as a real and imaginarypart n = nRe+inIm A schematic graph of how these parts changeas a function of frequency can be seen in Figure 214 A strongabsorption positive imaginary refractive index is only obtainedclose to resonance

We have finally found the reasons why light slows down andsometimes is absorbed during propagation in a material Withoutgoing into detail light that passes between materials with differ-ent refractive indices bends or refracts and can be summed up bySnellrsquos Law [1 chapter 26-2]

n1 sin θ1 = n2 sin θ2 (214)

where light travels from refractive index n1 into n2 with anincidence angle of θ1 and refracts to an outgoing angle of θ2 bothmeasured from the normal to the interface

From Equation (212) we see that the refractive index dependson the charge density N Since the density of gases depends ontemperature so does the refractive index The distortion above acandle is light bending back and forth since the refractive indexchanges due to temperature changes in the air The mirages on aroad can be explained by light from the sky gradually bending dueto a temperature gradient from the very hot road to the slightlycooler air and eventually propagating up to the beholderrsquos eyes

Using this knowledge you can explain all kinds of effects Inthe next section we explore something exotic slow light


In Figure 214 you can see how the complex refractive index be-haves as a function of frequency when it crosses a resonance Thephase velocity thus changes as a function of frequency referred toas dispersion In particular when the refractive index increaseswith frequency it is called normal dispersion As explained abovelight with a specific frequency travels at its phase velocity vp(ω)

When thinking about a light pulse with a finite duration youcan imagine it as a distribution of single frequency light waves thatpropagate together The pulse is created through interference ofthese different frequencies which at some places is constructivegiving rise to a strong amplitude whereas it in other places isdestructive ie canceling each other out as in Figure 215

In a vacuum all frequency components move at the speed oflight and therefore travel together However in a material with

16


dispersion the waves shift relative to one another due to the differ-ences in phase velocities Thus the speed at which a pulse travelsie the speed at which the place where the individual componentsconstructively interfere moves can be very different from the phasevelocity It is called the group velocity vg(ω) The following sec-tions examine the slow light effect in which the group velocity ismuch lower than the phase velocity and the speed of light in vac-uum This effect occurs when light propagates through a mediumwith strong normal dispersion

221 Slow light

To explain the slow light effect I first present an analogy that usescombs The different frequency components of a light pulse areeach represented by a comb where the tooth separation is relatedto the frequency an example of which can be seen in Figure 216(a)

That a light pulse is built from several frequencies is illustratedby several combs stacked behind one another The location ofthe pulse is where you can see through all the combs see Fig-ure 216(b) where it is also indicated by a blue arrow

Two yellow teeth are used to track how far the individual combsmove ie the phase velocity whereas how fast the place where youcan see through all combs moves corresponds to the group velocityAt the beginning the yellow markers are in the same place as thepulse

In Figure 216(c) no dispersion is present and all combs moveexactly the same amount The phase and group velocities are thesame since you can see through the combs at the same place aswhere the yellow teeth are located

In the case of strong normal dispersion shown in Figure 216(d)the combs move slightly differently as shown by the small spreadof the yellow markers Note however that this difference is smallcompared to the absolute speed since all the yellow teeth almostline up with those in Figure 216(c) Finally you can see that thegroup velocity is much lower than the phase velocity as the partwhere you see through the combs have only traveled roughly halfthe distance as the individual combs

This is the explanation of slow light The individual frequencycomponents that make up a light pulse travel at slightly differentvelocities due to dispersion but the speed at which the construc-tive interference moves is greatly reduced Later in Sections 53and 54 and Papers III and IV slow light is used to reduce thegroup velocity of pulses to about vg = 15 kms which is 200 000times slower than the speed of light in vacuum c asymp 300 000 kmsand slower than the speed of sound in the material

The mathematical expression for the group velocity is

vg(ω) = cng(ω) ng(ω) = n(ω) + ωpartn(ω)

partω(215)

17

222 Energy temporarily stored in the medium

a

b

c No dispersion

d Dispersion

Pulse location

Pulse location

Pulse location

Figure 216 A slow light analogy using combs (a) Each comb has afrequency as represented by the distance between its teeth (b) A lightpulse is built by stacking combs with slightly different frequencies afterone another as visualized by being able to see through the combs alsoshown by the blue arrows in the figure The yellow teeth are used asmarkers to show how far each comb moves (c) In the case with nodispersion all combs move exactly the same amount which leads to agroup velocity equal to the phase velocity (d) In the case of strongnormal dispersion the combs move slightly differently which leads to agroup velocity that is much lower than the phase velocity

where ng is the group refractive index If a material has asteep normal dispersion curve (ω partnpartω n) the group velocity canbe much lower than the speed of light in vacuum as explainedabove


When light is slowed down by passing through a material someof its energy is temporarily stored in the atoms The more energythat is stored compared to the total energy of the pulse the slowerit travels

In Papers III and IV slow light is discussed and simulated forrare-earth-ion-doped crystals the topic of Chapter 4 Thus it isadvantageous to split the atoms of the material into two parts Thefirst part is called the host which includes all contributions to therefractive index from the crystal host atoms and is assumed not tocontribute to the dispersion of the material ie it is a constantaddition to the refractive index for all relevant frequencies Thesecond part consists of the dopant ions which this section refersto as resonant centers This part heavily influences the dispersionfor frequencies close to resonance As a result the energy densityof the system can be split into three parts Uhost Ures and Uvacfor the host resonant centers and energy density of the electro-

18


Figure 217 A schematic view ofa spectral hole (black) and thereal part of the refractive indexcurve (blue) that it generatesThere is a strong normaldispersion inside the transmissionwindow which slows down anylight propagating through it

magnetic field which is the same in the crystal as in a vacuumThe energy flow per cross section and time of a pulse traveling

in a vacuum is cUvac where c is the speed of light in vacuumWhen inside a material such as the crystal described above thespeed at which the pulse travels is reduced to the group velocityvg [8 9] However the energy flow into the crystal from a vacuummust be the same as the flow inside the material (light + host +resonant centers)

cUvac = vg(Uvac + Uhost + Ures)

vg =c

1 + Uhost+Ures

Uvac

(216)

For frequencies far from the absorption of the resonant centersUres = 0 and since we assume that the host does not have anydispersion the group velocity is equal to the phase velocity vg =vp = cn Thus the phase refractive index n is

n = 1 +Uhost

Uvac(217)

ie 1 + the energy density in the host compared to the energydensity of the light wave inside the crystal which is the same asfor the light in a vacuum

On the other hand a strong slow light effect can be achievedwhen the resonant centers make up a spectral hole ie a trans-mission window in an otherwise strongly absorbing region seeFigure 217 In such a case where the energy is stored mostlyin the resonant centers an approximate expression of the groupvelocity can be used [10ndash12]

vg asymp2πΓ

α(218)

where Γ is the frequency width of the transmission window andα is the absorption coefficient outside the hole

23 Refractive index of a dense material

The experiments performed in this thesis are based on light thatpropagates in solids not gases Thus it is useful to extend theframework in Section 21 to be valid for dense matter as wellNote however that the intuitive picture is still correct The onlyaddition needed is to let the fields induced by the atoms interactwith other atoms [2 chapter 32]

Still using the harmonic oscillator model from Section 213 forthe movement of an electron we arrive at Equation (25) Thistime however we do not directly relate it to an electric field as inEquation (26) Instead we calculate the induced dipole moment of

19


all atoms denoted by the polarization of the material P = Nqexwhere N is the number of atoms per unit volume

P(t r) =Nq2

e


Elocal(t r) = Nαpol(ω)ε0Elocal(t r)

(219)where αpol(ω) is the atomic polarizability (not to be confused

with the absorption coefficient introduced in Sections 215 and222) and Elocal is the local electric field felt by the atoms whichis related to the average field in the crystal Eavg by Elocal =Eavg + P

3ε0[2 chapter 11-4] For the sparse description used in

Section 21 Elocal = Eavg since the effect of atoms interactingwith the induced fields of other atoms was neglected

Writing the induced polarization as a function of Eavg yields

P(t r) =Nαpol(ω)

1minus (Nαpol(ω)3)ε0Eavg(t r) (220)

Furthermore real materials have more than one resonancefrequency with different damping factors γk and transitionstrengths fk which are numbers in the order of 1 that take intoconsideration the fact that some transitions are more difficult toexcite Thus we can add up the contribution from each transition

αpol(ω) =q2e

ε0me

sum

k

fkω2

0k minus ω2 minus iωγk(221)

In order to relate this to the refractive index you need to studyMaxwellrsquos equations which can be written as [2 chapter 18-1]

nabla middotE =ρ

ε0nabla middotB = 0

nablatimesE = minuspartB

partt

c2nablatimesB =

(j

ε0+partE

partt

)(222)

where ρ is the total charge density j is the total current and

nabla =(partpartx

partparty

partpartz

)is the nabla operator

The charge density can be split up into two parts ρ = ρpol +ρother ρpol describes the charges in the material that have beendisplaced due to the electric field and ρother represents all othercharges Similarly j = jpol + jother They can be related to thematerial polarization through ρpol = minusnabla middotP and jpol = partP

partt

20


When no other charges or currents except those induced in thematerial exist Maxwellrsquos equations can be rewritten as

nabla middotE = minusnabla middotPε0

nabla middotB = 0


partt

c2nablatimesB =part

partt

(P

ε0+ E

)(223)

Assuming that the average electric field is polarized along xand propagates in the z-direction with a wave vector k = ωvp =nωc ie Eavg(t r) = E0exe

minusi(ωtminuskz) and assuming an isotropicmaterial we can use Equation (220) where both Eavg and P haveonly an x component Solving Maxwellrsquos equations under theseconditions we arrive at the following expression that relates theelectric field to the polarization of the material

k2Eavg =ω2

c2Eavg +

ω2

ε0c2P

k2 =ω2

c2

(1 +

Nαpol(ω)

1minus (Nαpol(ω)3)

) (224)

The last expression can be related to the refractive indexthrough n = ckω

n2 = 1 +Nαpol(ω)

1minus (Nαpol(ω)3)(225)

where an expression for the atomic polarizability can be foundin Equation (221)

Note that for a sparse material where Nαpol(ω) 1 the secondterm in Equation (225) is small and n asymp 1 + 1

2Nαpol(ω) whichif we only include one resonance frequency with strength fk = 1is the same expression that we arrived at in Section 215 andEquation (212) for a sparse material

You should also be aware that the resonance frequencies damp-ing coefficients and transition strengths are generally different foran atom in a solid compared to a free atom since they are affectedby other nearby atoms

Because a solid can be made up of different atoms you needto sum over the different densities Nj of the atoms each withtheir own set of resonances resulting in a collection of αpol j(ω)Including this in Equation (225) and writing it in another formknown as the Clausius-Mossotti equation [2 chapter 32-3] youarrive at

3

(n2 minus 1

n2 + 2

)=sum

j

Njαpol j(ω) (226)

21

24 Relation to phonons

We have finally obtained an expression for the refractive indexthat works for a dense isotropic material in the linear excitationregime Even though we have only examined a simple model andthe real case is a bit different the overall idea and intuitive sensewill help when discussing Papers I-IV in Chapter 5 In a morecomplex model the transitions can also depend on the polariza-tion of light For example some polarizers are made of mate-rials for which electrons can easily oscillate and absorb light inone direction but not perpendicular to it Thus light polarizedalong the second direction is transmitted through the polarizerThis also means that the index of refraction can vary for differentpolarizations and propagation directions which is the case for abirefringent (or anisotropic) material


This chapter has covered how light interacts with atoms in a waythat makes the electrons oscillate but our model can also describehow infrared radiation interacts with phonons which essentiallyare just oscillations of atoms instead of electrons For a simplevibration the mathematical treatment is the same only that ω0k

now represents a resonance in the crystalline structureAs a side note it is not only your eyes that can see electro-

magnetic radiation your whole body is a sensor of heat radiationwhich is part of the electromagnetic spectrum with lower frequen-cies than visible light

22

Chapter 3

Understanding the quantummechanical nature of atoms

ldquo I think I can safely say that nobody understands quantummechanicsrdquo - Richard Feynman [13]

This quote should be interpreted in the way that nobody un-derstands quantum mechanics in terms of their common sense ofreality I write it here since quantum mechanics is a complex sub-ject where not even leading physicists agree on everything Someof the discussions I present in this chapter therefore only representmy own thoughts on the subject

The quantum mechanical picture of a particle is introduced inSection 31 where I also compare it with classical physics to high-light the similarities and differences A simple quantum systemthe infinite well is solved in Section 311 and the relationshipbetween position and momentum in quantum mechanics is alsodiscussed

Section 32 returns to the light-matter interaction topic thistime focusing on the atom A useful depiction of a two levelquantum system is described in Section 321 Following this Sec-tion 322 connects the superposition of an atom to the oscillationof a charge and the induced field it creates Finally Section 323links the description of this chapter to the results of Chapter 2These sections are especially important for the quantum compu-tation results discussed in Chapter 7

Two important time scales for an atom are described in Sec-tion 33 where Section 331 describes phase decay and coherencetime while Section 332 explains the decay of excitation and life-time

Finally Section 34 describes the DC Stark effect which is theshift of energy levels due to a static electric field and is used inboth Chapters 5 and 7

23

31 Classical versus quantum picture

Figure 31 Shows the three lowestenergy states E1 rarr E3 for theinfinite well discussed inSection 311 The wave functionsΨn(x) are shown in blue and theprobability densities |Ψn(x)|2 inblack


To describe quantum mechanics I first go through an example ofclassical physics so that the similarities and differences can be seenmore easily

Imagine a football in the physical system of a field with twogoals At a specific time the ball can be located at any positionr with a momentum p = mv where m and v are the ballrsquos massand velocity respectively In this non-relativistic example of clas-sical physics Newtonrsquos second law [1 chapter 9-1] applies and theacceleration is determined by the forces acting on the ball through

F = ma (31)

where a is the acceleration and F is the total resultant forceGiven that the physical system is already determined for ex-

ample the field is located on Earth with a gravitational constantof g = 981 ms2 normal air pressure temperature and a certainwind speed the forces are explicitly determined by the system it-self but depend on where the ball is positioned and how it travelsie r and p Thus r p a and F cannot be chosen independentlyin classical physics since F determines a

Furthermore not all ball positions are allowed For examplesince the physical system of the football field including two goalswere taken as the starting point the ball cannot be placed at thealready occupied position of a goalpost

In classical physics the football is at one specific location rbut in quantum mechanics we learn that this view is not alwayspossible for elementary particles (for a deeper discussion of wave-particle duality see reference [3 chapter 1 2]) Instead the bestyou can do is specify a probability of where the particle may befound We describe this through a complex wave function (proba-bility amplitude) Ψ(t r) which is related to the actual probabilityof where the particle may be found through P (t r) = |Ψ(t r)|2see Figure 31 for an example

Instead of following Newtonrsquos Law the wave function evolvesaccording to the Schrodinger equation [14 p 97-98]

i~partΨ(t r)

partt= H(t r)Ψ(t r) (32)

where ~ is the reduced Planck constant and H(t r) is theHamiltonian that is related to the energy of the system TheHamiltonian takes the place of the forces in the classical descrip-tion in the sense that it is determined by the physical system It isalso a Hermitian operator such that its eigenvalues are real As aresult for a static problem where the Hamiltonian does not changewith time ie H(t r) = H0(r) the wave function must dependon time in the form of a complex exponential function

Ψ(t r) = Ψn(r)eminusiωnt (33)

24

Understanding the quantum mechanical nature of atoms

0 L 119909

119881 = infin119881 = 0

119881 = infin

Figure 32 An infinite well wherethe potential is infinite outsidethe area 0 rarr L and zero inside

When plugged into Equation (32) we arrive at the time-independent Schrodinger equation [14 p 98-100]

~ωnΨn(r) = EnΨn(r) = H0(r)Ψn(r) (34)

This is an eigenvalue problem For a given Hamiltonian itresults in a set of eigenvalues or energies En = ~ωn with cor-responding wave functions Ψn(r) that evolve with time in phasespace at frequency ωn according to Equation (33)

Regardless of whether the Hamiltonian is time-dependent ornot the wave function evolves deterministically according to theSchrodinger equation Equation (32) Furthermore when per-forming a measurement of the particlersquos position Bornrsquos rule statesthat the result is given with a probability |Ψ(t r)|2 [15] Measure-ments are therefore completely random in quantum mechanics butthe evolution of a particlersquos probability function is not

The Copenhagen interpretation of quantum mechanics statesthat the wave function randomly collapses to one of the possiblepositions when a measurement is performed [16]

Analogously to the football that cannot be located at the goal-posts the physical system described in the Hamiltonian does notallow the wave function to be arbitrarily chosen This is seen ex-plicitly in the next section where the Hamiltonian of an infinitewell is examined

311 Analyzing the infinite well potential

This section solves the one-dimensional time-independentSchrodinger equation for an electron inside the infinite well po-tential V (x) seen in Figure 32 The potential energy outside thearea 0 lt x lt L is infinite whereas it is zero if x is inside the well

The Hamiltonian of the system is

H0 =minus~2

2m

part2

partx2+ V (x) (35)

where the first term is related to the non-relativistic kineticenergy of the electron EK = 1

2mv2x = 1

2p2xm where px is the classical

momentum related to the quantum operator px = minusi~ partpartx

Since the particle cannot have infinite energy the wave functionmust be zero outside and at the boundary of the well Possiblesolutions to the time-independent Schrodinger equation are

Ψn(x) = A sin(πnLx)

0 lt x lt L (36)

Ψn(x) = 0 x le 0 x ge L

where A =radic

2L is a normalization constant to ensure thatthe probability of finding the electron somewhere inside the wellis 1 and n is a positive integer 1 2 3

25


When inserting this into Equation (34) you arrive at

En =~2

2m

π2n2

L2(37)

In Figure 31 the first few solutions for n = 1 rarr 3 are shownAs can be seen the physical system characterized by the Hamil-tonian H0 limits the wave function to certain states Ψn(x) thatare related to the probability of finding the electron at a positionx through P (x) = |Ψn(x)|2 These states have specific energiesEn Note that in our current simple model if the electron is in aneigenstate it stays there indefinitely

The splittings between these energy levels divided by ~ are theresonance frequencies discussed in Chapter 2 where the strengthof the interaction fk in Section 23 is set by the transition dipolemoment discussed later in connection with Equation (318)

In quantum mechanics the position and momentum wave func-tions are related through a Fourier transform [14 p 53-55] ie byknowing the wave function in space Ψn(x) you also know the wavefunction in momentum Φn(px) where |Φn(px)|2 is the probabilityof the electron having a certain momentum px

Φn(px) =1radic2π~

intΨn(x)eminus

i~pxxdx (38)

Since time and frequency are also related by a Fourier trans-form the description of how a pulse with finite duration is createdby a distribution of frequencies described in Section 22 is validhere as well The wave function in space Ψn(x) is created by adistribution of the spatial frequencies or momenta Φn(px) Thusto make the position more local the distribution in momentumspace must become wider or vice versa This leads to the Heisen-berg uncertainty principle [14 p 33-35 45-46]

∆x∆px ge~2

(39)

∆t∆E ge ~2

(310)

where ∆ denotes the spread or uncertainty of the positionmomentum time or energy In other words the Heisenberg un-certainty principle is a mathematical consequence of the particlebeing described as a wave that exhibits Fourier relations betweenposition and momentum or alternatively time and energy

In conclusion from the classical point of view r v and F caninitially be chosen independently whereas a depends on F(rv) perNewtonrsquos second Law and is therefore already determined and can-not be chosen arbitrarily In quantum mechanics you can chooseonly the initial Ψn(x) and H independently ie the position prob-ability and the physical system since the momentum wave func-tion Φn(px) is related to the position through Equation (38)

26


32 Light-matter interaction in quantummechanics

Let us now return to light-matter interaction but this time studyit from the perspective of matter

Consider an electron that surrounds an atom and for the sakeof simplicity neglect all states except the ground and first excitedstate with energies Eg and Ee and wave functions |g〉 = Ψg(x)and |e〉 = Ψe(x) where Dirac notation is used [14 p 10-23]

When light disturbs the atom an interaction term HI(t) mustbe added to the Hamiltonian The electron oscillates due to thefield and can be described as a dipole qer where r is the displace-ment with respect to the electronrsquos center of mass The interactionterm should therefore correspond to the energy of a dipole in anelectric field E(t) = 1

2E0eminusiωt + cc where cc denotes the com-

plex conjugate of what was written before The electric field isnow described in its real instead of complex form as in the previ-ous chapter where it was understood that only the real componentyielded the actual field The reason is that quantum mechanics isalready a complex theory Note however that E0 is still complexin order to describe all possible incoming fields ie combinationsof cos(ωt) and sin(ωt) The interaction term of the Hamiltonian is

HI(t) = minus1

2qer middotE0e

minusiωt + cc (311)

This interaction couples the ground state |g〉 with the excitedstate |e〉 such that the wave function can transition between orinto a superposition of the two states

|Ψ〉 = α|e〉+ β|g〉|α|2 + |β|2 = 1

(312)

where α and β are complex coefficients and the second expres-sion normalizes the wave function to 1 Alternatively it can bewritten as a function of two angles θ and φ which will be usefulin Section 321

|Ψ〉 = cosθ

2|e〉+ eiφ sin

θ

2|g〉 (313)

The wave function can also be described as a density matrix[14 p 180-185]

ρ = |Ψ〉〈Ψ| (314)

In matrix form where |e〉 =

(10

)and |g〉 =

(01

) this becomes

ρ =

(αβ

)(αlowast βlowast

)=

(|α|2 αβlowast

αlowastβ |β|2)

=

(ρee ρegρge ρgg

)(315)

27

321 Bloch sphere

w

z

v

R

|e

uy

x

Figure 33 The blue arrowrepresents the Bloch vectorR = (u v w) shown here in thepositive octant of the Blochsphere together with the definitionof θ and φ from Equation (313)|e〉 and |g〉 are located at z = 1and z = minus1 respectively

This can be rewritten to form three new real variables u andv that relate to the phase of the wave function and w that corre-sponds to the excitation

u = ρeg + ρge

v = i(ρeg minus ρge)w = ρee minus ρgg

(316)

The evolution of these variables under the interaction Hamil-tonian HI(t) can be described by the Bloch equations [17 18]

u = minus u

T2minus δv + ΩImw

v = δuminus v

T2+ ΩRew

w = minusΩImuminus ΩRev minusw + 1

T1

(317)

where T1 and T2 are the life and coherence times that determinehow fast the atom decays as discussed further in Sections 321 and33 δ is the detuning between the transition (resonance) frequencyω0 = (Ee minus Eg)~ and the light frequency ω ie δ = ω0 minus ω

ΩRe and ΩIm are the real and imaginary parts of the Rabifrequency Ω which describes how rapidly the transition is drivenie how fast it oscillates between the two states It is related tothe transition dipole moment and electric field through

Ω =microeg middotE0

~(318)

where the transition dipole moment microeg = 〈e|qer|g〉 is theoverlap of the two states with respect to the dipole operator qerWhether the Rabi frequency is real imaginary or complex is setby the complex electric field amplitude E0 and can therefore easilybe controlled by experimentalists

Now that the mathematical description is complete the nextthree subsections go through an intuitive way of describing theevolution of a two level atom interacting with light the connectionbetween a superposition and the oscillation of a charge as well ashow to relate this view to what was described in Chapter 2

321 Bloch sphere

A two-level superposition state can according to Equation (313)be written as a function of two angles and therefore be picturedas lying on a sphere with radius 1 referred to as the Bloch spherethe positive octant of which is shown in Figure 33 The ground|g〉 and excited |e〉 states are placed at the bottom and top ofthe sphere respectively and any state can be depicted by a Blochvector R that points from the center towards the surface of the

28


R2

|g

R1

W1

W2

z |e

yx

Figure 34 The evolution of theBloch vector R for starting state|g〉 is shown for the resonant casein blue with subscript 1 and inred for a detuned case withsubscript 2 The state vectorsrotate around the driving vectorsW

sphere Conveniently the variables u v and w introduced inEquation (316) are the components of the Bloch vector along theaxes x y and z ie R = (u v w)

The lifetime T1 in Equation (317) determines how quicklythe atom relaxes toward the ground state at w = minus1 and thecoherence time T2 sets how long phase knowledge described inu and v can be kept Both of these effects can reduce the lengthof the Bloch vector |R| = R to be less than 1 which describesincomplete knowledge about the state

When the life and coherence times are long compared to an in-teraction the T1 and T2 terms can be neglected in Equation (317)If the interacting electric field also has a constant frequency theevolution of the Bloch vector is a rotation using the right-handrule around the vector W = (minusΩReΩIm δ) since the Bloch equa-tions can then be written as

R = W timesR (319)

See Figure 34 for two examples where the initial state is |g〉Ω is purely imaginary and for the blue line δ = 0 whereas for thered line δ gt 0 The angular velocity at which the state vectorrevolves around the Bloch sphere is given by the generalized Rabifrequency ΩG =

radic|Ω|2 + δ2 Since all decays are neglected the

Bloch vectors trace out perfect circles as time goes onResonant interaction with light therefore causes a two level

atom to absorb energy and be put into first a superposition andeventually the excited state If the interaction continues the atomstarts to deexcite towards the ground state once more but now onthe opposite side of the Bloch sphere

Since the z component w describes the excitation (energy)of the atom and the evolution is driven at a constant angularspeed given by ΩG the atom does not absorb the same amount ofenergy in each time interval For the resonant case the absorptionper time interval starts low and increases to a maximum whenthe Bloch vector reaches the equatorial plane before once moredecreasing until it reaches the top pole If the atom is driven evenfarther the excitation decreases such that it gives off energy to theelectric field ie stimulated emission

Even though Chapter 2 only discussed absorption in the linearregime ie very close to the bottom pole it is also valid for thelinear regime of gain at the top pole by changing the sign of thedecay parameter γ to negative instead of positive The phasesdescribed in Section 213 therefore switch signs and when addingthe 90 degree phase shift due to the sum of all induced fields froma plane of electrons as described in Section 214 the total phasechanges from 90 + 90 = 180 degree for the resonance case whichcaused absorption to minus90 + 90 = 0 degree such that the inducedelectric field constructively add to the incoming field as describedabove

29

322 Connection between superpositions and light radiation

322 Connection between superpositions and lightradiation

According to Equation (33) each eigenstate of the Hamiltonianevolves with a phase factor eminusiωnt where ωn = En~ is relatedto the energy of the state For the superposition seen in Equa-tion (312) the two terms therefore evolve differently as a functionof time Any global phase factor of the wave function can beneglected since taking the absolute square to arrive at the proba-bility density makes it 1 However as discussed below the relativephase matters

The time evolution of such a superposition can be written as

|Ψ(t)〉 = α|e〉+ βeiω0t|g〉 (320)

where ω0 = ωe minus ωg and |α|2 + |β|2 = 1 to keep the statenormalized

The wave function Ψ(x t) and probability density |Ψ(x t)|2are shown in Figure 35 for four different superpositions of the twolowest infinite well states described by Equation (36) for n = 1and 2 at three different times In case 1 α = 0 β = 1 case 2α =

radic18 β =

radic78 case 3 α = 1

radic2 β = 1

radic2 and case 4

α = 1 β = 0 As can be seen the probability density does notchange with time in case 1 or 4 where the system is in a pure statebut in case 2 and 3 the probability of where to find the electronoscillates back and forth Using the knowledge of Chapter 2 weexpect the electron to induce its own field that depending onphase relations can destructively or constructively interfere withthe incoming electric field and absorb or release energy to thefield This oscillation is highest in case 3 where α = β = 1

radic2

which again shows that the induced electric field and absorptionare strongest for states at the equator

30


Figure 35 Shows the evolution of four different superposition caseswritten in the form of Equation (320) at three different times Thewave functionrsquos real and imaginary parts are shown in blue and redrespectively and the probability density is shown in black Case 1α = 0 β = 1 case 2 α =

radic18 β =


radic2

β = 1radic

2 and case 4 α = 1 β = 0

31

323 Relation between the Bloch sphere and the wave description

323 Relation between the Bloch sphere and thewave description

In this subsection the Bloch sphere introduced in this chapter isconnected with the wave description presented in Chapter 2 Forthese equations to be directly applicable to rare-earth-ion-dopedcrystals which are discussed in Chapter 4 I assume that in ad-dition to the interaction with the atoms discussed here the lightinteracts with background atoms such as the host crystal atomswhich generates a background refractive index of n that is alsoincorporated into the wave vector k = nωc

Once more the incoming electric field is defined as Ein(t r) =12E0e

minusi(ωtminuskz) + cc ie the same convention as in the previouschapter but now written in its real form Alternatively it can bewritten as

Ein(t r) = ERe0 cos (ωtminus kz) + EIm

0 sin (ωtminus kz)E0 = ERe

0 + iEIm0

Ein(t r) =~ex|microeg|

(ΩRe cos (ωtminus kz) + ΩIm sin (ωtminus kz))(321)

where it in the last line has been related to the complex Rabifrequency Ω = ΩRe + iΩIm through Equation (318) where Iassume that the field polarization direction ex is aligned withthe transition dipole moment

In order to see how the atom affects the light the Bloch equa-tions shown in Equation (317) are expanded into Maxwell-Blochequations which include the following additions [19 20]

(part

partz+n

c

part

partt

)ΩRe =

α0

2π

int +infin

minusinfing(δ)vdδ

(part

partz+n

c

part

partt

)ΩIm =

α0

2π

int +infin

minusinfing(δ)udδ

(322)

where n is the background refractive index and g(δ) is the ab-sorption distribution as a function of the detuning δ = ω0 minus ωnormalized such that it equals one where α0 the absorption coeffi-cient is measured This back action from the atoms to the light isbased on a macroscopic polarization and should qualitatively yieldthe same results as in Section 215 ie it includes the additional90 degree phase shift originating from summing over a plane ofoscillating atoms as described in Section 214 Note that u v wΩRe and ΩIm now also varies with the spatial variable z and notjust with time

The ncpartpartt term can be neglected in our experiments since it

contributes only when changes are fast ie on a timescale of nLc

where L is the crystal length

32


|g

R

(a)

W

z |e

yx

|g

(b)

R

W

z |e

yx

|g

R

(c)

W

z |e

yx

Figure 36 Driving of the Blochvector R around the drivingvector W using a positiveimaginary Rabi frequencyΩIm gt 0 for three differentdetunings (a) δ gt 0 and|δ| gt |ΩIm| (b) δ = 0 and (c)δ lt 0 and |δ| gt |ΩIm| See themain text for further description

In order to more easily examine a small change in the Rabi fre-quency or electric field as it propagates through the material andobtain the same phase relations between the incoming and inducedfields as in Chapter 2 the equations are rewritten as follows

∆ΩRe =α0

2πv middot∆δ middot∆z

∆ΩIm =α0

2πu middot∆δ middot∆z

(323)

where the integration over all detunings is assumed to give vmiddot∆δand u middot ∆δ respectively and ∆ is used to denote a small changeor interval for the parameter in question

In Figure 36 three examples are given where the incomingRabi frequency is always purely imaginary ΩIm gt 0 but thedetuning is varied In Figure 36(a) the detuning is positiveand |δ| gt |ΩIm| but in order to show the effect it is not mademuch larger Using the right-hand rule to rotate the Bloch vectorR = (u v w) around the driving vector W = (minusΩReΩIm δ) theBloch vector first gains a negative u component then a negativev component and finally a positive u component before it returnsto the ground state From Equation (323) this corresponds toRabi frequency changes of ∆ΩIm lt 0 ∆ΩRe lt 0 and ∆ΩIm gt 0respectively Since the incoming Rabi frequency was ΩIm gt 0the first and last parts correspond to absorption and gain respec-tively A negative real Rabi frequency is a factor of i 90 degreeafter in phase compared to the incoming field ΩIm gt 0 Thusthe middle part corresponds to a delayed light field In summarylight is first absorbed then delayed and finally re-emitted as gainThe rotational speed of the Bloch vector is set by the magnitudeof the driving vector |W| For large detunings it processes on atimescale of 1|δ| Since the absorption and gain parts cancel eachother out the overall effect is a delay of the incoming field ie anincrease in the refractive index just as obtained in Section 215

The detuning is zero in Figure 36(b) ie showing the reso-nant interaction case The Bloch vector gains a positive and thennegative u component corresponding to absorption and gain re-spectively according to similar logic as discussed in the previousparagraph This agrees with a positive imaginary part of the re-fractive index for absorption using γ gt 0 or a negative part forgain when γ lt 0

In Figure 36(c) the detuning is negative which results inBloch vector changes of negative u positive v and positive uA v gt 0 corresponds to ∆ΩRe gt 0 ie minus90 degree (minusi) earlier inphase or a reduction of the refractive index which is also obtainedusing Equations (212) or (225)

33

33 Decays

R2

|g

z |e

y

R1

x

Figure 37 When dephasing ispresent at random intervals andmagnitudes it alters the phaseangle φ of the Bloch vectorlighter-colored segments Thusthe evolution of the resonant blueline and detuned red line casesno longer trace perfect circles asin Figure 34 In this figure it isassumed that each dephasingevent occurs on a time scale muchshorter than 1 over the Rabifrequency

33 Decays

The interaction that we added in Section 32 HI(t) was used todescribe a field controlled by the experimentalists but any field(both electric and magnetic) in the presence of the atom addssimilar terms to the Hamiltonian as do phonon interactions Thusit is not surprising that the atom may be disturbed and change itsphase or excitation without the knowledge of the experimentalistsThese decays are described by their exponential time dependencethrough the coherence time T2 and lifetime T1 Much work goesinto isolating systems such that these interactions are kept to aminimum [21ndash23]

331 Coherence time

If an interaction between an atom and its environment causes onlyphase changes it shortens the coherence time T2 of the atomSuch a decoherence process results in a difference between thephase of the atom and eg a laser controlled by the experimen-talists As a result the path in the resonant case blue lines doesnot evolve as in Figure 34 but as in Figure 37 where the de-phasing events lighter-colored segments are seemingly random inboth direction and size and occur on a time scale much shorterthan 1 over the Rabi frequency This lack of knowledge leads to aperceived shortening of the Bloch vector as the final state becomesmore uncertain each time a random dephasing event occurs

This decoherence also allows an atom to more easily absorblight that is slightly detuned red lines since instead of keepingits rotation close to the bottom part of the Bloch sphere as inFigure 34 it can reach a higher excitation due to these phasedisturbances see Figure 37 Thus incoming light needs only tobe sufficiently close to the resonance frequency of the atom in orderto be absorbed The frequency width of where the atom absorbsreferred to as the homogeneous linewidth Γhom is inversely relatedto the coherence time

Γhom =1

πT2(324)

To understand another effect of decoherence picture a laserthat partly excites N atoms situated in a crystal Since the laserbeam travels at a finite speed its phase evolves during propaga-tion and the first atom has a different phase compared to the lastatom of the crystal However the field induced by the first atomthat travels in the same direction as the laser reaches each newatom in phase with its oscillations since the phase of this inducedfield evolves at the same rate during propagation as the phase ofthe initial laser field As a result all atoms oscillate such thattheir fields add constructively in the forward direction creating

34


a b

Figure 38 Oscillating atoms green moving upwards and orange down-wards induce fields that in (a) add coherently in the forward directionas the atoms oscillate in phase whereas in (b) the intensities are spreadout over all directions due to the random phases of the atoms

a formidable intensity scaling as N2 see Figure 38(a) In otherdirections the resulting field is much lower due to the phase differ-ences of the atoms However if decoherence processes target eachatom randomly the constructive interference in the forward direc-tion quickly dies off The atoms are still excited but with randomphase relations Thus the induced field radiates in all directionsand its intensity is proportional to N ie the net result is asthough we were adding intensities instead of fields for incoherentprocesses see Figure 38(b)

Sometimes you can compensate for the interactions that causethe decoherence For example if the dephasing occurs graduallyover some time tde and the experimentalist can alter the directionof the phase shift on a time scale talt tde the overall dephasingcan be made almost negligible as the phase first evolves in eg thepositive direction but is then switched by the experimentalist aftera time talt to evolve in the negative direction thereby cancelingout the previously gained positive phase Such interactions cantherefore be disregarded and a longer coherence time can oncemore be obtained This method of increasing the coherence timeis called dynamic decoupling [24 25]

In conclusion coherence time is a measure of how long two ormore systems can remain in phase and can sometimes be compen-sated for by clever tricks that utilize our knowledge of the processesthat cause the decoherence [26]

332 Lifetime

Whereas coherence time T2 measures the decay of phase lifetimeT1 describes the decay of excitation T1 interactions are similar tothe ones described in the previous section but change θ instead ofφ in Equation (313) and Figure 33 and can only move the Bloch

35

332 Lifetime

vector towards the ground stateThe lifetime also limits the longest possible coherence time by

T2 le 2T1 since the state eventually decays to the ground statewhere no phase exists A factor of 2 is needed since lifetime mea-sures decay of excitation ie energy whereas coherence time mea-sures decay of phase ie amplitude and energy prop amplitude2

Even an isolated atom decays from excited states This is calledspontaneous emission and gives rise to the finite lifetime of excitedlevels The exact reason for the decay is still being discussed Thegeneral consensus is that it relates to interactions with vacuumfield fluctuations andor self-reaction (also known as the radiationreaction field) ie interaction with its own induced field How-ever the degree to which these effects come into play is debatedIn reference [27] Milonni offers an overview of the discussion andemphasizes the fluctuation-dissipation connection between the twoeffects [28] one cannot exist without the other These effects can-cel each other out for the ground state thus rendering it stable[27] Puri later argued that the vacuum field fluctuations could bemathematically removed and therefore made no real contributionto the spontaneous emission [29] but Milonni quickly refuted himand once more emphasized the importance of the connection be-tween the two [30] Very recently Pollnau has suggested that foreach frequency there exist two counter-propagating modes eachcarrying a zero-point energy of 12 vacuum photon that projecttheir vacuum energy onto each other since the modes are not or-thogonal but anti-parallel and vacuum fluctuations can thereforealone quantitatively describe spontaneous emission [31]

Aware of this debate I will anyway try my best to explain thephysical reason for spontaneous emission with the precaution thatthe rest of the section may be open to question First I presenta semiclassical hypothesis for spontaneous emission Second Idiscuss statistical properties of the process

Consider a two level atom in the excited eigenstate at time t0Due to the vacuum field fluctuations the Hamiltonian and there-fore the eigenstates changes such that at time t0 +∆t the atom isin a tiny superposition between the new ground and excited eigen-states As explained in Section 322 this superposition radiateslight According to our present understanding this should start offslowly but as more and more energy is radiated away the groundstate probability increases and the Bloch vector turns towards theequator where the oscillation and radiation are strongest Theprocess slows down again as the ground state population becomeslarger than that of the excited state which results in an exponen-tial tail of the decay process This dissipation of energy due tothe oscillation of the charge can be seen as self-reaction [27] Inreality the decay is exponential from the beginning (disregardingsmall- and long-time deviations [32ndash34]) which this semiclassicalmodel does not capture but the vacuum fluctuations might be

36


responsible for this initial increase of the decayThe decay process is seemingly irreversible ie the probability

that the atom reabsorbs the spontaneously emitted photon is neg-ligible (disregarding quantum revival [35] for now as we consideran isolated atom) This may be seen as a statistical property sincethe photon can be created in an infinite number of modes eachcorresponding to sending it out in different directions or polariza-tions whereas the atom can only be excited or not Thus it isstatistically much more likely for an atom to decay from the ex-cited state and never reabsorb the emitted photon resulting in anapparent spontaneous decay [36] However as hinted above theprocess can be made unitary and reversible if you keep track of thevacuum modes see cavity quantum electrodynamic [37] and quan-tum revival [35] and spontaneous emission is thus stimulated butfrom an unknown source such as the vacuum field fluctuations

This also indicates that incident light covering only a few modesis much more likely to scatter from one mode to another comparedto being absorbed In order to achieve significant absorption iea large percentage of the incoming photons are absorbed manymodes must be covered by the incoming light for example byfocusing on the atom or with induced fields generated by otheratoms that oscillate together as described in Section 214 Thisis understandable since as the atom oscillates it radiates out ina dipole pattern see Section 211 which should at least partlycancel out the incoming light if any significant absorption is to beachieved Thus the larger the fraction of the dipole pattern modesthat are covered by the incoming light the higher the absorptioncan be

34 DC Stark effect

If the external electric field is static instead of oscillating as de-scribed previously in Section 32 the energy levels of an atomicsystem shift due to the additional term in the Hamiltonian Thisis called the (DC) Stark effect and occurs since a static field has adefinite positive and negative side and the electron wave functionsthat are shifted towards the positive side have lower energy thanwave functions that are shifted towards the negative side Fur-thermore electrons in outer shells are more loosely bound to thenucleus and are therefore generally more affected by the externalfield since they can move more easily

Because each energy level has its own wave function the energysplitting between levels generally changes when an external fieldis applied which is used in Papers III and IV as described inSections 53 and 54

37

Chapter 4

Rare-earth-ion-doped crystals

This chapter provides an overview of the properties of rare-earthelements doped into a crystal For more detailed information seeeg Sun [38] or Macfarlane et al [39]

In the growth process used for the crystals in this thesis theCzochralski method the desired rare-earth element is added to amelt of pure crystal ingredients and the crystal is slowly grownfrom this melt The goal is to minimize defects such as oxy-genation centers vacancies imperfect crystal structure and otheratomic impurities while attaining the desired doping concentra-tion

This chapter begins with Section 41 which describes the rare-earth elements then Sections 411 and 412 discusses the benefitsof 4f-4f transitions and the hyperfine energy level structure ofPr3+Y2SiO5 and Eu3+Y2SiO5 respectively

The host crystal Y2SiO5 in this thesis is discussed in Sec-tion 42 Following this Section 43 discusses transition and staticdipole moments after which some properties of Pr3+Y2SiO5 andEu3+Y2SiO5 are provided in Section 44

Finally the absorption profile of a rare-earth-ion-doped crystalis discussed in Section 45 using Pr3+Y2SiO5 as an example

41 Rare-earth elements

The rare-earth elements are the 15 lanthanides scandium andyttrium shown in the periodic table in Figure 41 This thesisinvolves mostly praseodymium (Pr) europium (Eu) and cerium(Ce) doped in yttrium orthosilicate (Y2SiO5) Thus the discus-sions in this chapter are based on their properties

39

411 4f-4f transitions

21Sc

39Y

57La 58Ce 59Pr 60Nd 61Pm 62Sm 63Eu 64Gd 65Tb 66Dy 67Ho 68Er 69Tm 70Yb 71Lu

Figure 41 Placement of the rare-earth elements scandium yttriumand the 15 elements in the lanthanides series in the periodic table


The electron configurations of the lanthanides are based on thenoble gas Xenon which has filled 5s and 5p states but an empty4f shell For triply ionized (trivalent) lanthanides which is thepredominant oxidation state for the majority of them the elementshave 0-14 filled 4f states starting from Lanthanum (La) and endingin Lutetium (Lu)

The filled 5s and 5p states are spatially extended farther outthan the 4f electrons see Figure 42 and therefore interact morewith the surroundings causing the chemical properties of the lan-thanides to be roughly the same This extension also causes themto act as a Faraday cage that protects the inner-lying 4f electronsfrom surrounding disturbances

For a free atom dipole transitions require a change of parity inthe final state compared to the initial one However when dopedinto a crystal which is the main topic of this thesis the atommay sit in a site that is no longer inversion symmetric Thus par-ity is not a very good quantum number and forbidden transitionscan become weakly allowed This is the case for 4f-4f transitionsin rare-earth elements doped into non-centrosymmetric crystalsSince these transitions are weak and shielded by the outer lyingstates the life and coherence times can be long but require cryo-genic temperatures of around 4 K to eliminate phonon interactions

Phonons follow Bose-Einstein statistics such that the probabil-ity of finding phonons in a given state with a certain frequency ωpis

P (ωp) =1

exp(

~ωp

kBT

)minus 1

(41)

where kB is the Boltzmann constant and T is the temperaturein Kelvin Phonons generally scatter and cause decoherence butthe probability of this process scales as T 7 [41ndash43] for tempera-

40


Figure 42 Radial distribution of 4f 5s 5p and 6s electron orbitals inGadolinium (Gd) The 4f orbital is shielded from the environment bythe energetically favorable outer-lying orbitals The figure is reprintedwith permission from [40]

tures T lt 0084TD where TD is the Debye temperature which isequal to 580 K for Y2SiO5 [44] Hence at low temperatures (le 4K) phonons no longer significantly disturb the properties of the4f states

Phonons can also cause non-radiative decay that reduces thelifetime of states but these processes are considered unlikely if thetransition require 5 or more phonons For Y2SiO5 the maximumphonon energy is about 12 THz and so transitions above 60 THzare mostly unaffected [44 45]

Figure 43 shows an overview of calculated 4fn energy levels oftrivalent rare-earth ions doped into Lanthanum trifluoride (LaF3)Even if the host material is different from Y2SiO5 the placementsof the energy levels are similar

41


50

48

46

44

42

40

38

36

34

32

30

28

26

24

22

20

18

16

14

12

10

8

6

4

2

0

Ener

gy [103

cm-1

]

1500

1440

1380

1320

1260

1200

1140

1080

1020

960

900

840

780

720

660

600

540

480

420

360

300

240

180

120

60

0

Frequ

ency [TH

z]200

300

400

500

600

700

8009001000

2000

300040005000

Wavelen

gth [n

m]

Figure 43 Calculated 4fn energy levels of most trivalent rare-earth el-ements doped into LaF3 The levels also have similar positions in otherhost crystals such as Y2SiO5 since the electrostatic interaction of theelectron and nucleus as well as the spin-orbital interaction mostly de-termine the energies The first figure which shows the 4fn energy levelsin this way referred to as a Dieke diagram was originally printed inreference [46] for Lanthanum trichloride (LaCl3) but has been reprintedseveral times since I slightly modified this version from reference [47]and reprinted it here with permission

42


plusmn12g

plusmn32g

plusmn52g

plusmn12e

plusmn32e

plusmn52e

606 nm

495 THz

102 MHz

173 MHz

46 MHz

48 MHz

Figure 44 Energy structure ofPr3+Y2SiO5 (site 1) [48 49]The arrows show the possibleoptical transitions also seen inFigure 47 whose relativeoscillator strengths are printed inTable 43

10265degab

c

1041 nm

79deg

C2 (b)

D1

D2

Figure 45 The monoclinicY2SiO5 crystal is shown withcrystal axes a b and c as well asprincipal axes D1 D2 and C2(often denoted by b instead sincethey are parallel)

412 Hyperfine energy level structures

The transition used for Pr3+Y2SiO5 throughout this thesis is the3H4 rarr 1D2 located at 606 nm (495 THz) see Figure 43 Itshyperfine energy level structure can be seen in Figure 44

The first-order hyperfine interaction in Pr3+Y2SiO5 (alsoEu3+Y2SiO5) vanishes since the orbital angular momentum isquenched [39] A second-order interaction does however exist anddepends on the square of the nuclear spin causing all energylevels to be doubly degenerate at zero magnetic field The lev-els in Figure 44 are denoted by the hyperfine quantum numbersmf = plusmn12g plusmn32g plusmn52g for the ground states and similarly forthe excited ones However as mentioned in the previous sectionthe ion interacts with the crystal and these quantum numbers arenot ldquogoodrdquo Thus caution is warranted when eg applying rulesfor transitions

The transition of interest in Eu3+Y2SiO5 is 7F0rarr 5D0 locatedat 580 nm (517 THz) Eu has a similar energy level structure as Prie three ground and excited hyperfine states but with slightlylarger energy splittings ranging from around 30 MHz to almost300 MHz [38] Furthermore it has two isotopes 151Eu and 153Euwhich differ in energy splittings

Finally any dopant is doped into both sites of yttrium by re-placing one of the two possible Y in Y2SiO5 and have transitionsat two slightly different wavelengths For Eu this means a total offour different combinations of isotope and site locations in Y2SiO5but the complexity can be reduced by doping with isotopically pureEu

Ce3+ does not have any optical 4f-4f transition available in-stead the 4f-5d zero-phonon transition at 371 nm (site 1) is usedthroughout this thesis The excited state lifetime is only 40 ns[50] and the transition can be cycled many times with continuousexcitation making Ce a good candidate for single ion detectionThis is discussed further in Sections 74 and 75

42 Yttrium orthosilicate (Y2SiO5) host crystal

Y2SiO5 crystals are monoclinic with eight molecules per cell seeFigure 45 where a b and c are the crystal axes It is birefrin-gent and therefore has three principal axes with slightly differentrefractive indices These are called D1 D2 and C2 (often denotedwith b instead since they are parallel) with indices nD1

= 17881nD2 = 1809 and nC2 = nb = 17851 (for 606 nm) [51] TheY2SiO5 crystal is frequently used as a host due to its low magneticmoments silicon and oxygen have no nuclear spin and yttrium hasspin 12 but with a low magnetic moment of minus0137microN where microNis the nuclear magneton [41] Low magnetic moments are good

43

43 Transition and static dipole moments

Δ1206412static

124deg

Δ1206414static

Δ1206411static

Δ1206413static

C2 (b)

Figure 46 The four possiblestatic dipole moments ∆microstatic

i in

Pr3+Y2SiO5 They all form a124 degree angle with the b axis[55] and lie in a plane that istilted 35 degrees from the D2 axis[56 57]

since they minimize the effect of spin flips that otherwise disturbthe coherent properties of the dopants


A strong interaction with the environment generally result in shortlife and coherence times However a transition with a strong inter-action ie a large oscillator strength (proportional to the squareof the transition dipole moment [52]) allows for easier control sincethe ions interact more with the laser light The oscillator strengthof Pr (site 1) is about 20minus 60 times as strong as Eu which lowersthe requirements for available laser power in experiments [53 54]

When Pr is used as a dopant in Y2SiO5 its transition dipolemoment microeg have a direction that lies mostly in the D1-D2 planewith a 746plusmn19 degree tilt from the D1 axis [38] In order to excitethe ions incoming light must have a polarization component alongthis direction Fortunately due to the phase retardation causedby the different refractive indices of the host crystal absorptionof light can be achieved regardless of polarization in the D1-D2

plane since the polarization changes during propagation throughthe crystal The strongest absorption is however achieved if theincoming polarization is aligned along the D2 axis see Paper I

Furthermore the static dipole moment of the ground and ex-cited states are different due to differences in their wave functionssee Section 34 such that an external electric field can Stark shiftthe resonance frequency of the ions by ∆ω

∆ω =(microstatic

e minus microstaticg ) middotE

~=

∆microstatic middotE~

(42)

where microstaticg and microstatic

e are the static dipole moments of theground and excited states respectively and E is the external elec-tric field at the location of the ion This effect is used in Papers IIIand IV where absorption structures are altered by shifting theresonance frequencies of absorbing ions using an external electricfield In Pr3+Y2SiO5 four different possible directions exist forthe static dipole moment difference ∆microstatic which all make a124 degree angle with the b axis [55] see Figure 46

Since the static dipole moment of an ion is state dependentit alters the surrounding electric field if the ion is excited whichin turn DC Stark shifts other nearby ions This leads to instanta-neous spectral diffusion where the excitation of some ions causesdephasing or detuning of nearby ions Usually this is problematicand is avoided by going to ever lower excitation populations How-ever as discussed in Sections 73 and 74 it can also be used tomake an ion control other nearby ions

44


Property Pr (site 1) Pr (site 2) Refλvac 605977 nm 607934 nm [48]ν 494726 THz 493133 THz [48]T1 optical 164 micros 222 micros [48]T2 optical 152 micros (B = 77 mT) 377 micros (B = 77 mT) [48]T1 hyperfine asymp 100 s asymp 100 s [58]T2 hyperfine 42 s (B = 80 mT) - [26]Oscillator strength 77 middot 10minus7 45 middot 10minus8 [53]microeg 25 middot 10minus32 Cmiddotm 63 middot 10minus33 Cmiddotm [53]∆microstatic 24 middot 10minus31 Cmiddotm 20 middot 10minus31 Cmiddotm [53]

Table 41 Properties of Pr3+Y2SiO5 in order vacuum wave-length frequency optical lifetime optical coherence time hyper-fine lifetime hyperfine coherence time oscillator strength transi-tion dipole moment and the difference between the static dipolemoment of the excited and ground states Magnetic field strengthsare noted when used and the coherence times are achieved by im-plementing techniques in order to preserve phase information

Property Eu (site 1) Eu (site 2) Refλvac 579879 nm 580049 nm [54]ν 516991 THz 516840 THz [54]T1 optical 19 ms 16 ms [54]T2 optical 26 ms (B = 10 mT) 19 ms (B = 10 mT) [54]T1 hyperfine gt 20 days gt 20 days [41]T2 hyperfine 6 hours (B = 135 T) - [23]Oscillator strength 12 middot 10minus8 31 middot 10minus8 [54]microeg 66 middot 10minus33 Cmiddotm - [53]∆microstatic 76 middot 10minus32 Cmiddotm - [53]

Table 42 Properties of Eu3+Y2SiO5 see Table 41 caption fordescription

44 An overview of praseodymium (Pr) andeuropium (Eu) properties in Y2SiO5

An overview of life and coherence times as well as oscillatorstrengths transition dipole moments and differences of staticdipole moments between excited and ground states is providedin Tables 41 and 42 for Pr and Eu respectively For a broaderoverview of material properties see reference [38]

The coherence times in the tables (especially the hyperfine) arereduced if necessary precautions are not taken such as a magneticfield with correct magnitude and direction as well as dynamicdecoupling schemes as discussed in Section 331 Additionally asmentioned in Section 411 phonons rapidly destroy the coherenceof the excited state as the temperature goes above 4 K

45

45 Pr3+Y2SiO5 hyperfine and inhomogeneous absorption profiles

Frequency [MHz]

046

94102

148196

275321

369

Ab

sorp

tio

n

~kHz

Figure 47 Schematic view of theabsorption spectrum for a Pr iondoped into Y2SiO5 correspondingto the transitions seen inFigure 44 The peak heightscorrespond to their relativeoscillator strengths as written inTable 43

Frequency

Ab

sorp

tio

n

~GHz

Crystal

Figure 48 Due to therandomness of the doping processeach dopant ion experiencesslightly different electricalsurroundings as shownschematically in the crystal insetfigure where the green symbolsrepresent dopant ions and thegray spheres represent host atomsThe transitions from ground toexcited states therefore vary inthe GHz range [38] leading to awide absorption profile if thecontributions of all ions in thecrystal are taken into account

Levels plusmn12e plusmn32e plusmn52eplusmn12g 056 038 006plusmn32g 039 060 001plusmn52g 005 002 093

Table 43 Relative oscillator strengths for Pr3+Y2SiO5 based onexperimental results in reference [49] but normalized to 1

45 Pr3+Y2SiO5 hyperfine and inhomogeneousabsorption profiles

In Pr3+Y2SiO5 the energy splittings of the hyperfine groundstates are in the order of tens of MHz which is much lower thanthe thermal fluctuations (even at 4 K) Thus ions at equilibriumare equally likely to be in any of the three ground states Anion in the plusmn12g can absorb light to any of the excited states andtherefore has an absorption profile as seen in the blue peaks of Fig-ure 47 These transition lines are in the order of kHz wide for Prand even narrower for Eu see Tables 41 42 and Equation (324)By repopulating the Pr ion to any of the other two ground statesits absorption features shift 102 MHz or 102 + 173 MHz = 275MHz respectively as seen in the red and green peaks

The nine possible transitions between the ground and excitedstates have different oscillator strengths as depicted by the peakheights in Figure 47 The relative strengths can also be seen inTable 43

Furthermore the local crystal field surrounding a particular iondetermines its exact excitation frequency through the DC Starkeffect see Section 34 which generally varies between ions in theGHz range [38] but depends on the doping concentration [41] Thecombined absorption of all dopants in a crystal thereby createsan inhomogeneous broadening that is roughly this wide and canschematically be seen as the black line in Figure 48 As a resultdifferent ions in the crystal can absorb light at a particular fre-quency on different transitions thus forming nine groups of ionsat each frequency corresponding to the nine different transitions

Ions in the crystal can therefore be selected in groups depend-ing on their excitation wavelength Since the individual transitionsare in the order of kHz or less whereas the inhomogeneous broad-ening is GHz this results in up to millions of addressable frequencychannels This number is important for determining the capabilityof storing or processing information

Using clever spectral hole burning techniques you can relocateions between their hyperfine ground states in order to tailor theabsorption profile in various ways [49] either to create sharp nar-rowband absorption filters as in Section 52 or more complicatedstructures to initialize and control qubits as in Chapter 71

46

Chapter 5

Light-matter interaction inrare-earth-ion-doped crystals

The previous three chapters have laid the groundwork for light-matter interaction atoms and properties of rare-earth-ion-dopedcrystals so that this chapter can focus on the main results ofPapers I-IV all of which study effects in Pr3+Y2SiO5 crystals

To start off with Section 51 (Paper I) theoretically and intu-itively discusses how the electric field polarization and absorptionchanges during wave propagation in birefringent crystals with anoff-axis transition dipole moment

Section 52 (Paper II) takes a more experimental approachas the absorption of spectral holes is analyzed with the goal ofachieving a narrowband spectral filter

Such spectral holes have strong slow light effects and by usingthe DC Stark effect you can shift the entire transmission windowin frequency space The next topic covered in Section 53 (Pa-per III) uses these effects to frequency shift optical light pulsesby changing an external voltage

When the spectral hole is prepared slightly differently an ex-ternal voltage can instead be used to change the frequency widthof the transmission window and thereby alter the group velocitythat an incoming pulse experiences This group velocity controlleris discussed in Section 54 (Paper IV) where it is also used totemporally compress a probe pulse

These last two devices described in Sections 53 and 54 cantheoretically be prepared such that they are able to operate regard-less of the incidence angle and polarization of the incoming lightmaking them ideal for scattered light applications Furthermorethey are controlled by only an external voltage and are therebyadvantageous in weak light situations

Finally Section 55 provides an outlook for these projects

47

51 Wave propagation in birefringent materials with off-axis absorption or gain (Paper I)

51 Wave propagation in birefringent materialswith off-axis absorption or gain (Paper I)

Research on rare-earth-ion-doped crystals is a diverse field withmany applications eg quantum memories and computing [2159 60] spectral filtering [61] and as laser gain media [62] Thusit is useful to understand the basics of wave propagation in suchbirefringent absorbing materials which is exactly what Paper Ianalyses

In a birefringent material such as the Y2SiO5 crystal the re-fractive index depends on both the propagation direction and theincoming polarization Three principal axes referred to as D1 D2and b exist for Y2SiO5 and have refractive indices of nD1

= 17881nD2 = 1809 and nb = 17851 respectively for 606 nm [51]

If incoming light propagates along one of these principal axes ina pure crystal the polarization is preserved However if the lighthas a polarization which is in between these axes they acquire aphase retardation with respect to each other due to the differencein optical path length

If the crystal is doped with some resonant center such as Pranother orientation is important the transition dipole momentdirection microeg since the dopants absorb light polarized in thisdirection For Pr the transition dipole moment is tilted 746plusmn 19degrees from the D1 axis towards the D2 axis with almost nocomponent along b [38] see Figure 51(a)

To intuitively understand what happens when light propagatesthrough a birefringent crystal with off-axis absorption you canexamine the two effects of absorption and phase retardation oneat a time This is explained below for the case of a Pr3+Y2SiO5

crystalAn incoming wave with electric field Ein propagates along b

with its polarization along D2 as shown in Figure 51(a) It po-larizes the material along microeg which can be seen as an inducedelectric field EP that is out of phase with the incoming polar-ization The two fields interfere and the resultant field Eres inFigure 51(b) is tilted away from the transition dipole momentdirection

To examine the effects of birefringence it is easier to split theresultant field into its components along D1 and D2 see Fig-ure 51(c) Due to the difference in refractive index the phasesof these polarization components evolve differently as the wavepropagates through the crystal see Figure 51(d)-(e)

From the effects explained in Figure 51 it is clear that theincoming polarization is not generally maintained during propaga-tion even if it is initially polarized along a principal axis Howevertwo steady state polarizations exist but only the solution with thelowest absorption (or highest gain) is stable Generally speaking

48

Light-matter interaction in rare-earth-ion-doped crystals

1198631

1198632

120641119890119892119916in

(a) Input polarization

(d) Propagation

1198631

1198632

(b) Absorption

119916in

119916P

119916res

1198631

1198632

(c) Projection on 1198631 1198632

119916res

1198631

1198632

(e) Propagation

746 plusmn 19deg

1198632

1198631

Figure 51 An intuitive explanation of absorption in (a)-(b) and prop-agation through a birefringent crystal in (c)-(e) (a) An incoming elec-tric field Ein polarized along D2 is shown together with the transitiondipole moment microeg (b) The material is polarized along the transitiondipole moment direction which can be seen as an electric field vectorEP that is out of phase with the incoming field This causes a decreaseand rotation of the incoming field resulting in Eres (c) The field is de-composed into components along D1 and D2 (d) Due to the differencein refractive index for the principal axes the electric field polarizationchanges from linear to elliptical (e) Further propagation leads oncemore to linear polarization this time with a different polarization an-gle

they are elliptically polarized and differ only by a 90 degree rota-tion in the D1 D2 plane (assuming the transition dipole momentlies in this plane) If the incoming polarization matches one ofthese steady state solutions it does not change during propaga-tion similar to the principal axis polarizations in the pure crystalcase

The ellipticity and direction of these polarizations depend onthe ratio seen below

R =n2D2minus n2

D1

χabs(51)

where χabs is the electric susceptibility connected to the Prabsorption

49


Ele

ctric

fiel

d al

ong

(a)

D2

Ele

ctric

fiel

d al

ong

(c)

D2

Ele

ctric

fiel

d al

ong

(b)

D2

Electric field alongD1

Figure 52 The stable (green) andunstable (red) polarization steadystate solutions are shown fordifferent values of R given inEquation (51) (a) R asymp 85 (ieR 1) which is the case forPr3+Y2SiO5 In (b) R asymp 1 and(c) R 1

Assuming nD2asymp nD1

asymp nbg where nbg is a background refrac-tive index you can write n2

D2minus n2

D1= (nD2

+ nD1)(nD2

minus nD1) asymp

2nbg∆n Thus R is approximately the ratio of the rate of accu-mulating phase retardation ∆n and the absorption χabs

If R 1 phase retardation overshadows absorption whichmakes all polarizations that are symmetric along the crystal axesequivalent since they evolve into each other due to the effectsexplained in Figure 51(c)-(e) As a result the polarization steadystates lie along D1 and D2 This is the case for Pr3+Y2SiO5where R asymp 85 (χabs = [882plusmn 08] middot 10minus4 based on αD1 = 36plusmn 05cmminus1 and αD2

= 47plusmn5 cmminus1 [38]) as can be seen in Figure 52(a)If the ratio is at the other extreme R 1 absorption dom-

inates and any phase retardation can be ignored which leads toone steady state along the transition dipole moment with high ab-sorption and another along the orthogonal direction with very lowabsorption see Figure 52(c)

If R asymp 1 both effects are important and the steady state solu-tions are more elliptical lying between the transition dipole mo-ment and the principal axes as seen in Figure 52(b)

As mentioned previously only the steady state solution withthe lowest absorption (highest gain) is stable When light prop-agates through a crystal with an incoming polarization that dif-fers from these solutions the polarization changes and eventuallyreaches the stable steady state solution In any practical situa-tion irregularities and impurities cause the polarization to shifttoward the stable solution even if the initial polarization is per-fectly aligned with the unstable steady state solution

Based on the explanation in Figure 51(a)-(b) an electric fieldcomponent along D2 induces a small component along D1 whenit is absorbed along a tilted transition dipole moment and viceversa During propagation both the D1 and D2 components areabsorbed generally with different absorption coefficients For thesteady state solutions however the ratio of the two polarizationcomponents are such that the difference between the absorptioncoefficients are compensated for by the induced field and the po-larization remains the same during propagation

Three important conclusions can be drawn from these resultsFirst the maximum absorption direction depends on the ratio Rand might not be along the transition dipole moment but alongthe principal axis D2 as in the case of Pr3+Y2SiO5 describedabove

Second since the polarization of incoming light shifts duringpropagation the absorption coefficient α also tends to vary withdistance For example in Pr3+Y2SiO5 the stable steady statelies close to the D1 axis since it has the lowest absorption andeven if the initial polarization is perfectly aligned with D2 thelight experiences only the high absorption coefficient αD2

untilit has propagated approximately 54 mm after which the stable

50


Figure 53 A schematic view of anarrowband spectral filter is seenin black The goal is to have hightransmission for the pulse insidethe spectral hole the green linewhereas the pulse detuned by 125MHz relative to the center of thehole the blue line shouldexperience strong absorption

steady state is reached and the field starts to decay with the lowabsorption coefficient αD1

Third even if the incoming polarization is perfectly aligned

with D2 it almost immediately creates a D1 component due tothe effects explained in Figure 51(a)-(b) Thus there is little rea-son to have an incoming polarization purity that is better thanthe ratio of this created D1 component compared to the initial

D2 For Pr3+Y2SiO5 this ratio is in the order ofID1

ID2= 10minus5

where Ix denotes the initial intensity along direction x and cor-responds to a polarization angle precision of approximately 017degrees In other words a polarizer with a suppression of 105 isenough to achieve the longest propagation distance with maximumabsorption

For information about how the simulations were performed seePaper I

52 Development and characterization of highsuppression and high etendue narrowbandspectral filters (Paper II)

Narrowband rare-earth spectral filters based on hole burning havea wide range of applications including quantum memories [56]self-filtering of laser frequencies [63] and ultrasound optical to-mography (UOT) [64] Paper II address the critical issues in con-structing such high suppression filters and targets mainly the ap-plication of UOT which is briefly described in the paper and moreinformation can be obtained from references [61 64]

The filter is illustrated in Figure 53 where the black curveshows the absorption coefficient of the filter as a function of fre-quency ie the spectral hole The green and blue lines representthe frequency distributions of two pulses located 0 MHz and 125MHz detuned from the center of the hole respectively The goalof the spectral filter is for the pulse inside the spectral hole tohave high transmission and low distortion whereas the pulse lo-cated outside the hole should have high absorption Since the twopulses are separated by only a few MHz the requirements for thehole edge sharpness and homogeneity throughout the crystal arestringent

The setup used for the experiments is shown in Figure 54 Thelight comes from a stabilized Coherent 699-21 ring dye laser andcan be frequency tuned by an acousto-optic modulator in a doublepass configuration A beam splitter reflects a small portion of thelight to a reference detector PD1 (Thorlabs PDB150A) but themain part is transmitted through a lambda-half plate and a polar-izer before it reaches the crystal that is situated in the cryostatA second polarizer is used to probe the polarization of the trans-mitted light after which two different detectors PD2 (Thorlabs

51

52 Development and characterization of high suppression and high etendue narrowband spectral filters

(Paper II)

Figure 55 The graphs show theexperimental transmission insideand outside the spectral filter for3 micros full width at half maximum(FWHM) probe pulses averaged25 times The D2 and bcomponent of transmitted lightinside the spectral filter are shownin (a) and (b) respectively (c)Transmitted D2 light whenprobing the spectral filter in thehigh absorption area 125 MHzdetuned from the center of thehole (d) Same as (c) but showsthe b component of thetransmitted light instead Theblack lines show the time gatingused for calculating thesuppression values such as thoseshown in Figures 56 and 57

Double pass AOM

Polarizer Polarizer

λ2-plate Beamsplitter PD3

PD2

Stablized laser light

Cryostat

Shutter

119887

1198632

1198631

Schematic setup Top view

PD1

Pr3+ Y2SiO5

Figure 54 Overview of the experimental setup see the main text formore information

PDB150A) and PD3 (Hamamatsu PMT R943-02) are used to de-tect it PD2 is used to analyze the strong probe pulse transmittedthrough the spectral hole whereas PD3 is used to detect the weaklight transmitted outside the hole For more detailed informationabout the setup see Appendix A

A 10 mm thick 005 doped Pr3+Y2SiO5 crystal with prin-cipal axis orientation as seen in Figure 54 was used for theseexperiments In order to increase the lifetime of the spectral holea 10 mT magnetic field was applied across the D2 axis

Typical results for the filter when propagating insideoutsidethe spectral hole can be seen in Figure 55 All pulses were sentin at time 0 with a D2 polarization and propagated along the D1

crystal axisIn Figure 55(a) the transmission inside the spectral hole is

shown when the polarizer located after the crystal was alignedwith D2 Due to the large dispersion in the hole the light slowsdown and exits after roughly 13 micros (see slow light description inSection 221) All graphs in Figure 55 are normalized to the samescale where the maximum transmission inside the spectral holeseen in Figure 55(a) is 1

The transmitted b component shown in Figure 55(b) is muchlower and its magnitude is relatively consistent with the suppres-sion of the polarizer 105 considering that the light propagatesthrough the cryostat windows and is partly reflected at the crys-tal surface both of which may reduce the extinction ratio of thepolarization that reaches the crystal Moreover a slight misalign-ment of the crystal principal axes with the polarizers can explainwhy the delayed b component which is assumed to be due to theoutgoing cryostat window and the polarizer after the cryostat islarger than the non-delayed component

The D2 and b components of the transmitted light when prop-agating outside the spectral hole can be seen in Figures 55(c)and (d) respectively The b component is not delayed since theabsorption coefficient is almost zero along that direction and the

52


Figure 56 The suppression in dBas a function of the probe FWHMduration The suppression iscalculated by comparing the D2transmission for inside andoutside probes time gatedbetween 11 micros and 15 micros shownas black lines in the examplegiven in Figure 55 The two datasets correspond to experimentsperformed in quick successionusing the same spectral hole Themarkers show the mean of 25measurements and the error barsshow one standard error of themean

Figure 57 The suppression in dBas a function of the outside probedetuning For more informationabout the data treatment see thecaption of Figure 56

magnitude is in the same order as seen in Figure 55(b) indicatingthat it comes from an impure polarization that reaches the crystalas discussed above

Fortunately a polarizer that is aligned with D2 and is locatedafter the crystal can attenuate the b component of the transmittedlight Thus this b component does not reduce the potential of thespectral filter (remember that the light transmitted through thespectral hole is oriented almost exclusively along D2 such that thepolarizer after the crystal does not affect it) Alternatively you canuse two stacked crystals with their axes rotated 90 degrees relativeto one another in order to achieve high absorption regardless ofincoming polarization [65]

The results of Figure 55(c) show that the D2 light has twoparts one that is slowed down similarly to the transmission insidethe spectral hole shown in Figure 55(a) and one part that is notThe slow part is assumed to be light leaking through the spectralhole due to an overlap of its frequency components with the spec-tral hole This indicates that the hole edges are not as good asillustrated in Figure 53 or that the frequency distribution of thepulse is not as narrow Using time gating the light that is notdelayed can be neglected when evaluating the spectral filter sup-pression which is calculated by comparing the total transmissionbetween the two black bars located at 11 micros and 15 micros for thecases of being inside or outside the spectral hole

After a spectral hole was created four sets of 25 probe pulseseach were sent into the crystal the first and third inside the holeand the second and fourth outside When calculating the suppres-sion two values were obtained by comparing the first two sets andlast two sets respectively This was done in order to evaluate anydeterioration of the hole and thus the suppression as it was used

In Paper II the application discussed is UOT where the pulseduration and frequency separation are not yet fixed Thereforethe suppression of the filter was examined as a function of the in-coming pulse full width at half maximum (FWHM) duration asseen in Figure 56 where the overlap of the two suppression curvescalculated as described in the previous paragraphs indicates thatthe hole does not deteriorate Since the frequency width and thepulse duration are inversely proportional to each other a shortpulse has poorer suppression due to an increased frequency over-lap between the spectral hole and the incoming pulse However ashorter pulse is preferred in UOT such that a compromise of 3 microslong pulses was chosen In these experiments the pulse propagat-ing outside the spectral hole was detuned 1 MHz compared to thecenter

The suppression was also measured as a function of detuningfor a fixed FWHM duration of 3 micros as seen in Figure 57 Youcan see that the suppression is almost constant as long as thedetuning is larger than 125 MHz where a suppression of 534plusmn07

53

53 Slow-light-based optical frequency shifter (Paper III)

Figure 58 Schematic view of theabsorption structure The blueareas represent ions that shifttoward higher frequencies if apositive voltage is applied acrossthe crystal The red ions shift inthe opposite direction

dB is achieved Note also that a suppression of 632 plusmn 06 dBwas obtained if the detuning was increased to 102 MHz whichagrees well with previous results at that detuning [56] Such largefrequency shifts are not suitable however for the application ofUOT

The main advantage of these filters is their insensitivity to theincoming beam direction as simulated and discussed further inPaper II This is especially important for the application of UOTwhere a simple Fabry-Perot cavity which can act as a high sup-pression narrowband filter does not cover enough spatial modesto be useful

To summarize in Paper II a narrowband spectral filter wascreated and analyzed It achieved a relative suppression of 534plusmn07 dB for two 3 micros long pulses separated only by 125 MHz

53 Slow-light-based optical frequency shifter(Paper III)

The DC Stark effect explained in Section 34 can be used to alterthe absorption profile of a material as examined in Papers IIIand IV This section covers Paper III which uses the absorptionprofile shown in Figure 58 to frequency shift incoming probe pulsesby controlling an external electric voltage See the article for adetailed description of the structure preparation

The DC Stark effect causes the blue and red ions to shift theirresonant frequencies in opposite directions If a positive voltage isapplied the blue ions shift to higher frequencies and vice versa forthe red ions Thus the spectral hole in the blue absorption shiftstoward higher frequencies while keeping the same width as long asthe shift is small enough that the red absorption does not overlapwith the hole Experimental results of the transmission windowmovement can be seen in Figure 59(a) for five different voltagesettings

When a light pulse propagates through the spectral hole mostof its energy is stored in the nearby ions as described in Sec-tion 222 This leads to a greatly reduced group velocity comparedto the speed of light in vacuum making it possible to store the en-tire light pulse inside the crystal In these experiments a 1 micros pulsepropagated through a 10 mm thick 005 doped Pr3+Y2SiO5 crys-tal

If an external DC electric field is applied when the pulse isstill inside the crystal the ions that store the energy of the pulseshift their resonance frequencies as described above and so doesthe light they re-emit Denoting the frequency of the incominglight pulse as 0 MHz on a relative scale the experimental resultsof the outgoing pulse frequencies can be seen in the solid lines ofFigure 59(b)

54


Figure 511 Frequency shift andrelative efficiency as a function ofapplied voltage across the crystal

0

1

2

3

Ab

sorp

tio

n

L (a) - 15 V- 5 V0 V5 V15 V

-4 -3 -2 -1 0 1 2 3 4Frequency [MHz]

FF

T In

ten

sity

(b)

Figure 59 (a) Experimental αL (absorption coefficient multiplied bycrystal length) as a function of frequency shown for five different volt-ages applied across the crystal (b) Using heterodyne detection theFourier transform intensity of the transmitted probe light is examinedThe solid and dashed lines show the experimental and Maxwell-Blochsimulation results respectively As can be seen the probe frequencyshifts together with the spectral hole

The dashed lines in the same figure are the results of a Maxwell-Bloch simulation and coincide well with the experimental resultsFurthermore the simulations can show the energy distribution ofthe ions during propagation as in Figure 510(a)-(b) right beforeand after applying the external electric voltage respectively Thespectral window is approximately 1 MHz wide and the energy isstored mostly in the ions just outside the hole More detailed in-formation about how the simulations were performed can be foundin Paper III

Finally experimental results of the frequency shift and relativeefficiency are shown together as a function of the applied volt-age in Figure 511 The linear behavior of the frequency shift isexpected since the DC Stark effect linearly shifts the transitionfrequencies Furthermore the Stark coefficient in the simulationsthat yielded the best overlap with the experimental results was1167 kHz(Vcm) which agrees well with the literature value ofasymp 112 kHz(Vcm) [55 57]

The efficiency drop for large shifts might have several expla-nations For example a misalignment of the electric field to thesymmetry axis of the static dipole moment see Figure 46 in Sec-tion 43 or inhomogeneities in the applied electric field can causeions to move differently and reduce the coherent reemission of lightAlternatively some red ions in Figure 58 move closer to the spec-tral hole thereby increasing the overall absorption

55

54 Using electric fields for pulse compression and group-velocity control (Paper IV)

Figure 510 Simulation results of the ion excitation as a function fre-quency and propagation distance when a 1 micros probe pulse propagatesthrough a spectral hole initially centered around 0 MHz with a widthof a little over 1 MHz Red describes the maximum excitation in thematerial whereas dark blue represents zero excitation Figure (a) isjust before an instantaneous frequency shift of the ions is applied and(b) is right after the shift

In conclusion Paper III uses the strong slow light effect of anarrowband transmission window and the DC Stark effect to fre-quency shift optical light using a voltage source as a controllerIn experiments up to plusmn4 MHz shifts were achieved with efficien-cies above 85 The technique can potentially be improved to belimited only by the breakdown voltage of the crystal (at least 1MVcm [66]) which would lead to frequency shifts in the order of100 GHz

Furthermore it is argued that these frequency shifters can beconstructed to have a solid acceptance angle close to 2π and workfor any incoming polarization and thus also work for scatteredlight These arguments are based on related works [61 65]

These devices should be particularly suitable in weak light situ-ations due to the fact that the shifters are controlled by an externalvoltage source and not any additional light pulses

54 Using electric fields for pulse compressionand group-velocity control (Paper IV)

In this section which describes the results of Paper IV the DCStark effect is once more examined but now for the absorption

56


Figure 512 A schematic view ofthe absorption profile and thecorresponding refractive indexcurve when no external electricvoltage is applied The dispersionis relatively low and the groupvelocity is around 30 kms Blueions shift toward higherfrequencies if a positive voltage isapplied and vice versa for redions

Figure 513 The 16 MHz wideabsorption structure inFigure 512 changes into the 1MHz hole seen above when anexternal voltage of 40 V is appliedacross the crystal creating astrong dispersion for the narrowspectral hole with a groupvelocity of around 5 kms

Figure 514 Shows the group velocity and relative efficiency as a func-tion of the applied voltage The inset shows a zoomed in view of thehighest voltages ie the narrowest holes (gray area)

profile seen in Figure 512 and used to alter the group velocity ortemporarily compress probe pulses

Similarly as described in Section 53 blue ions shift to higherfrequencies when a positive external voltage is applied whereasred ions move to lower frequencies If 40 V is applied across thecrystal the absorption profile changes to reduce the width of thespectral hole from 16 MHz to 1 MHz as seen in Figure 513

The narrower hole has a much steeper dispersion curve whichcan be seen when comparing the refractive index curves the solidgreen lines in Figures 512 and 513 leading to a lower groupvelocity See Section 221 for more details

By choosing an external voltage and thus a spectral hole widthbefore sending a probe pulse into the crystal you can choose thegroup velocity with which it should propagate through the trans-mission window see Equation (218)

In Paper IV a 6 mmtimes10 mmtimes10 mm (crystal axes btimesD1timesD2)Y2SiO5 crystal doped with 005 Pr was used to control the groupvelocity of pulses in the manner described above

The experimental results are shown in Figure 514 where theblue line represents the group velocity as a function of the appliedvoltage which relates to the spectral hole width where lower volt-ages mean a wider hole The red curve is the relative efficiency

57


Figure 515 The blue line shows areference of the input pulsecentered at 0 micros If it propagatesthrough a 16 MHz wide spectralhole it is delayed only slightly asseen in the green line For a 1MHz hole the group velocity ismuch lower and the delay islonger see the purple line If thehole is initially narrow but quicklywidens when the pulse iscompletely inside the crystal thepulse is compressed as seen in thered line

ie the transmission of the pulse normalized such that the max-imum is 1 The data shown is averaged 150 times and the barscorrespond to one standard deviation

The device can alter the group velocity of the probe from 30kms to 15 kms corresponding to voltages of 0 V and 422 Vrespectively while having a relative efficiency above 80 Theslowest speed recorded was almost 1 kms but with a relative effi-ciency of around 50

The reasons for the efficiency drop are the same as those dis-cussed in Section 53 inhomogeneities in the applied electric fieldmisalignment of the field with the symmetry axis of the staticdipole moment and increased absorption of the light as the holenarrows

In Paper IV it is argued that the tunability of the controllercan be extended from the current factor of 20 to around 600 iffor example Er3+Y2SiO5 is used where the maximum width of aspectral hole is 575 MHz (based on the hyperfine splittings from[67]) instead of the 18 MHz for Pr3+Y2SiO5

The current device can also be used for temporal pulse shapingif the external voltage is changed when the pulse is still propagatinginside the crystal

To demonstrate this a 16 MHz wide spectral hole was createdas in Figure 512 and a 1 micros long pulse was sent into the crystal attime t = 0 a reference of which is shown in the blue line of Fig-ure 515 When no voltage was applied the transmission windowremained wide and the pulse arrived at the delayed time τ1 = 350ns as shown in the green trace This corresponds to a high groupvelocity of vg1 = Lτ1 asymp 29 kms where L = 10 mm is the crystallength in the propagation direction

When a constant voltage of 40 V was used instead the hole wasapproximately 1 MHz wide and the pulse became further delayedas shown in the purple line It arrived at time τ2 = 154 micros cor-responding to a group velocity of vg2 = Lτ2 asymp 65 kms At thislow group velocity the 1 micros light pulse was strongly compressedfrom a FWHM length of about 300 m in a vacuum to 65 mm inthe crystal such that almost the entire pulse was accommodatedinside the crystal at one time

In a final experiment the external voltage was rapidly switchedoff from 40 V to 0 V just as the pulse was about to exit the crystalThe first part of the pulse still went through the entire crystalwith group velocity vg2 whereas the last part initially traveled atspeed vg2 but traveled the remaining distance at speed vg1 vg2after the external field changed and altered the absorption profileThus the last part of the pulse spent a much shorter time insidethe crystal and the entire pulse was compressed as shown in thered trace of Figure 515

Although these initial results show that pulse shaping is possi-ble the compression was only a factor of about 2 and the efficiency

58


calculated as the area under the red curve relative to the area of thegreen trace in Figure 515 was only 66 In Paper IV howeverMaxwell-Bloch simulations were performed and compared to theexperimental results showing that larger compressions and higherefficiencies should be possible

As a recap of Paper IV the group velocity of a pulse was tunedby a factor of 20 while keeping the efficiency above 80 initial re-sults of pulse reshaping were performed showing that pulse com-pression is possible by altering the external voltage over the crystalas the pulse propagates inside it these devices can be used advan-tageously in weak light situations since they do not require anyother light fields and are controlled solely by the external voltageand if two crossed crystals are used they should theoretically havea solid acceptance angle close to 2π [61] and work for all incomingpolarizations [65]

55 Outlook of projects

This last section briefly reviews the current or potential follow-upsto the projects discussed in this chapter

The wave propagation model presented and used in Papers Iand II is not currently being expanded any further However theability to model light propagation in materials remains of interestto our group in the ultrasound optical tomography (UOT) projectwhere the focus is on absorbing and scattering materials [64]

The Pr3+Y2SiO5 spectral filter shown in Paper II will be usedin preliminary experiments for UOT when measuring on tissuephantoms More effort is currently being devoted to find other ma-terials for narrowband spectral filtering in the tissue optical win-dow also for the application of UOT These filters should prefer-ably have similar or better properties than the one discussed inSection 52 but at a wavelength more suitable for penetratinginto the human body

The UOT project may one day use the frequency shifter schemepresented in Paper III when taking the next step of phase conju-gating and amplifying the scattered light signal in an attempt torefocus it back into the human body and achieve a lasing effect toimprove the penetration depth and the signal-to-background ratio

The group-velocity controller project Paper IV is not cur-rently being continued or improved upon but our group has anongoing general interest in slow light applications

59

Chapter 6

A brief overview of quantumcomputation

This chapter shifts focus toward quantum computing by provid-ing an overview of the field and is followed by Chapter 7 whichpresents the progress achieved by our group in the rare-earth quan-tum computing field especially projects where I have been in-volved

In Section 61 a justification for pursuing quantum computersis offered Section 62 describes the DiVincenzo criteria which arecommonly accepted requirements that a fully functional quantumcomputer needs to meet

Lastly the Quantum Technologies Roadmap as a part of theEuropean Commission Quantum Technology Flapship is discussedin Section 63

61 Pursuing quantum computing

In a speech at a conference in May 1981 at MIT Richard Feyn-man presented the idea that simulating the evolution of a quantumsystem on a classical computer in an efficient way appeared impos-sible and urged the world to build a quantum computer ldquo natureisnrsquot classical dammit and if you want to make a simulation ofnature yoursquod better make it quantum mechanical and by golly itrsquosa wonderful problem because it doesnrsquot look so easyrdquo [68] Theconjecture that any local quantum system can be simulated on aquantum computer was proven by Seth Lloyd in 1996 [69]

As always scientific progress is not due to one person aloneand other scientists made important contributions before Feyn-manrsquos famous speech in 1981 R P Poplavskii showed in 1975 theinfeasibility of using a classical computer to simulate quantum sys-tems [70] and Y Manin proposed in 1980 the idea of a quantum

61

62 DiVincenzorsquos criteria

computer [71] A relatively thorough and expanding timeline ofthe quantum computation field is available on Wikipedia [72]

Nevertheless the field of quantum computing had been initi-ated With the prospect of efficiently simulating quantum systemsand discoveries of Shorrsquos algorithm which provides exponentialspeedup compared to classical ones when factoring large integersand might theoretically break many present-day encryptions [73]Groverrsquos algorithm which provides a quadratic speedup for thecommon problem of brute force searching a database [74] andthe quantum annealing algorithm which was shown by HidetoshiNishimori et al to outperform classical simulated annealing [75]the field of quantum computing became a worldwide interestwhich includes leading corporations like IBM [76] Google [77]Microsoft [78] and Intel [79] not to mention new businesses forthe sole purpose of quantum computing such as D-wave [80]

The benefits of such quantum simulators and algorithms in ad-dition to the prospect of even more discoveries both fundamentaland application-based and an overall gain in human knowledgepropel the field forward and make it a worthwhile investment forscientists and companies alike


Back in 1996 [81] DiVincenzo put forth the idea of minimal re-quirements for creating a quantum computer In 2000 he pub-lished such a list which contained five thoroughly evaluated re-quirements later referred to as the DiVincenzo criteria [82] Thislist appears below as presented in reference [82] where the last twopoints are requirements for quantum communication and includedhere for purposes of completeness

(i) A scalable physical system with well characterized qubits

(ii) The ability to initialize the state of the qubits to a simplefiducial state such as |000〉

(iii) Long relevant decoherence times much longer than the gateoperation time

(iv) A ldquouniversalrdquo set of quantum gates

(v) A qubit-specific measurement capability

(vi) The ability to interconvert stationary and flying qubits

(vii) The ability to faithfully transmit flying qubits between spec-ified locations

These criteria will be discussed more in Section 77 with regardto the progress of rare-earth quantum computing by our group

62

A brief overview of quantum computation

63 Quantum Technologies Roadmap

This section discusses the Quantum Technologies Roadmap [83]provided by the European Commission in its Quantum Technol-ogy Flagship [84] which is a e1 billion investment in research andapplications in the field The document lists six topics Quan-tum Computation Quantum Communication Quantum Simula-tion Quantum Information Theory Quantum Metrology Sensingand Imaging and Quantum Control The focus of this section isthe computation part

Below is a general list of future directions as they appear in theQuantum Technologies Roadmap (2015 edition of the Europeanroadmap for Quantum Information Processing and Communica-tion) [83]

(i) Further development of all current technologies to under-stand their limitations and find ways around them

(ii) Assessment of the capabilities of different technologies forbeing scaled up

(iii) Optimization of the performance of error correcting codesby both increasing the error threshold and decreasing theoverhead of required qubits

(iv) Investigation of new ways of performing quantum compu-tation in particular based on self-correcting codes (as theyappear in topological systems)

(v) Development of new quantum algorithms and search forproblems where quantum computers will be required

(vi) Development of quantum complexity theory and its applica-tion to many body physics

(vii) Building interfaces between quantum computers and com-munication systems

(viii) Development of quantum-proof cryptography to achieveforward-in-time security against possible future decryption(by quantum computers) of encrypted stored data

As can be seen the list is comprehensive in its goals but pur-posely vague about the specific experimental platform The firstpoint even specifically requests further investigation into all cur-rent technologies

Reference [83] explains some experimental approaches withstate-of-the-art (2015) updates on progress and challenges as wellas short medium and long-term goals The topics discussed aretrapped ions quantum computing with linear optics supercon-ducting circuits and electronic semiconductor qubits as well as

63


impurity spins in solids and single molecular clusters which in-clude rare-earth-ion-doped crystals Anyone who wants an experi-mental overview of the quantum computation field would thereforedo well to read this roadmap or an updated version of it

64

|0rang

|1rang

|auxrang

|119890rang

606 nm

495 THz

102 MHz

173 MHz

46 MHz

48 MHz

1205960 Ω0 1206010

1205961 Ω1 1206011

Figure 71 Hyperfine states forPr3+Y2SiO5 using the quantumcomputation notations for thelevels The arrows indicate twopossible electromagnetic fieldswith frequencies ω0 and ω1 andcomplex Rabi frequencies Ω0eiφ0

and Ω1eiφ1 respectively

Chapter 7

Quantum computation inrare-earth-ion-doped crystals

This chapter describes the quantum computation progressachieved by our group at Lund University particularly the projectsI have been involved with

First Section 71 describes how an ensemble qubit is initializedin Pr3+Y2SiO5 the material that is used for most discussionsthroughout this chapter and whose relevant energy levels can beseen in Figure 71 now employing quantum computation notationsfor the states

Single qubit gates are discussed in Section 72 where Sec-tion 721 presents complex hyperbolic secant pulses (sechyp pulsesfor short) using a single colorfrequency and Section 722 goesthrough two-color pulses Finally Section 723 (Paper V) useselectromagnetically induced transparency (EIT) ie two-colorpulses in order to perform a spin echo sequence

The pulses described so far can be used to perform any singlequbit operation Thus the next step is two-qubit gates which arediscussed in Section 73 (Paper VI) still using ensemble qubits ina Pr3+Y2SiO5 crystal

Due to poor scaling properties of ensemble qubits howevera high-fidelity single ion scheme is presented in Section 74 (Pa-per VII) where a dedicated readout ion is also doped into thecrystal together with the qubit ion but with a much lower con-centration Two possible readout ions are discussed neodymium(Nd) in a micro-cavity [85 86] and weakly doped cerium (Ce) thatis detected using a confocal microscope setup [87] Although bothmethods are still under investigation some results of the ceriumapproach can be seen in Section 75 (Paper VIII)

The confocal microscope setup has also been used to investi-gate Eu3+Y2O3 nanocrystals a paper on which is currently under

65

71 Qubit initialization

Figure 72 Absorbing ions areremoved from a frequency regionin preparation for the qubitinitialization

Figure 73 The qubit has beenprepared in the |0〉 state 0 MHzcorresponds to the |0〉 rarr |e〉transition shown in Figure 71

Figure 74 The qubit has beentransfered to the |1〉 state by usingtwo consecutive sechyp pulses

preparation and is briefly discussed in Section 76In Section 77 the progress of the single ion quantum com-

puting scheme is evaluated based on the DiVincenzo criteria pre-sented in Section 62 Lastly Section 78 discusses the outlook ofthe quantum computing projects of our group


The inhomogeneous broadening in rare-earth-ion-doped crystalsrenders it impossible to target only a specific transition such as|0〉 rarr |e〉 with a laser pulse without preparing the sample firstThe reason is that at each specific light frequency nine groups ofions are resonant all at different transitions due to the DC Starkshift induced by the spatially varying crystal field see Section 45

The first step in initializing a qubit is to create a wide regionwhere no absorbing ions exist This is done by repeatedly sendingin light pulses that scan the targeted area exciting and opticallypumping ions away from the region [39 88] The resulting spectralpit can be seen in Figure 72 and is the largest possible region thatcan be completely emptied in Pr3+Y2SiO5 (set by the hyperfinesplittings [49])

A narrow absorption peak sim170 kHz wide is then createdby burning back some ions into the pit This is done in a carefulmanner and after cleaning up unwanted absorption the remainingpeaks inside the pit all correspond to ions in the |0〉 state andrepresent our ensemble qubit The absorption of the ions to thethree excited states can be seen in Figure 73 corresponding to theblue spectrum in Figure 47 The peak heights are proportional tothe relative oscillator strengths of the different transitions as seenin Table 43

For the sake of completeness the absorption spectrum whenthe qubit is in state |1〉 is shown in Figure 74 corresponding tothe red part of Figure 47 Only the transitions to the two lowestexcited states can be seen since the last transition lies outside thespectral pit The qubit has been transferred from |0〉 via |e〉 to |1〉using two sechyp pulses that are described in Section 721

72 Single qubit gate operations

After initializing the qubit the next step is to perform simple gateoperations Section 721 uses one-color pulses ie pulses withonly one frequency at a time to perform transfers between anyground and excited state Two-color pulses which can be used todo arbitrary single qubit operations in a robust manner are thendescribed in Section 722 Finally an all-optical nuclear spin echotechnique is discussed in Section 723 (Paper V)

66

Quantum computation in rare-earth-ion-doped crystals

Figure 75 The absolute value ofthe Rabi frequency and theinstantaneous light frequency of asechyp pulse are shown in theupper and lower graphsrespectively

|0

R0

Re

z |e

yx

Figure 76 The Bloch vectorevolution for the sechyp pulseshown in Figure 75 The initialstate is R0 = (0 0minus1) ie |0〉and the final state Re is close to|e〉

721 One-color sechyp pulses

Since square pulses are very sensitive to Rabi frequency fluctua-tions ie variations in laser intensity sechyp pulses are often usedinstead In such pulses the intensity and frequency of the lightchange as a function of time [89ndash91] and can be described by thecomplex Rabi frequency Ω(t)

Ω(t) = Ω0sech(β(tminus t0))1minusimicro (71)

where Ω0 is the maximum Rabi frequency t0 is the center of thepulse and β and micro are parameters related to the FWHM durationand frequency width

tFWHM asymp176

β(72)

νwidth =microβ

π(73)

An example of how the absolute Rabi frequency and the instan-taneous light frequency change as a function of time for a sechyppulse can be seen in Figure 75 For a qubit initially in |0〉 such asechyp pulse resonant with the |0〉 rarr |e〉 transition ie the bluearrow in Figure 71 causes the Bloch vector to evolve as seen inFigure 76 The result is a transfer to the |e〉 state

The benefits of using a sechyp pulse is its stability toward Rabifrequency fluctuations as well as the fact that it works well forslightly detuned ions while still not exciting ions far from the res-onance [91] This is important because our qubit is an ensembleof ions with frequencies within a sim170 kHz region

In Pr3+Y2SiO5 the maximum transfer efficiency reached fortwo such sechyp pulses first exciting and then deexciting is 96[60] which is close to being lifetime limited ie the decay fromthe excited state during the duration of the pulses 8 micros in total isconsistent with the efficiency loss in the experiments For higherefficiency transfers one option is to use much shorter pulses butthey require stronger fields and off-resonantly excite ions whichdestroy the qubit either by introducing unwanted absorption inthe pit or exciting the qubit on another transition

An alternative for obtaining higher fidelities is to examinematerials with longer lifetimes andor larger hyperfine splittingsEu3+Y2SiO5 fulfills both of these conditions (see Sections 44 and412) and fidelities of 9996 may be possible see Paper VII

Currently our group have measured the relative oscillatorstrengths in 151Eu3+Y2SiO5 site 2 see Table 71 but due to alack of available Rabi frequency only one transition could be drivenwith a potentially high transfer efficiency In order to estimate thisefficiency the qubit was cycled back and forth between |12g〉 and|32g〉 via |52e〉 where only the |12g〉 rarr |52e〉 transition was

67

722 Two-color sechyp pulses

|1

|B

|D

z |0

y

x

Figure 77 Shows the Blochsphere for the ground hyperfinelevels |0〉 top and |1〉 bottom aswell as the bright |B〉 and dark|D〉 states which lie at oppositeends of the equator and are set bythe angle φ


Table 71 Relative oscillator strengths for 151Eu3+Y2SiO5 site 2The uncertainties are estimated to plusmn003 and can be compared tothe values measured for 153Eu3+Y2SiO5 site 1 in reference [92]

driven with the good pulse Unfortunately the uncertainties inthe measurements eg spontaneous decay from |52e〉 the read-out pulse affecting the population and that two different pulseswere used to transfer from |12g〉 and |32g〉 respectively madethe analysis inconclusive about the efficiencies of the two differ-ent pulses [93] Thus more effort is needed in order to reach theexpected results


A one-color sechyp pulse is good for pole to pole transfers on theBloch sphere but is not designed to do arbitrary single qubit gatessuch as creating a superposition or performing a NOT operationFortunately two-color pulses with center frequencies ω0 and ω1 asseen in Figure 71 where both components change as a sechyp asshown in Figure 75 and have the same Rabi frequency Ω0 = Ω1but different phases φ0 and φ1 are able to perform any arbitrarysingle qubit gate in a robust manner [91] Note that this sectionassumes two-color pulses that change as a sechyp but the entiremethod is valid even when using simple two-color square pulses

The two frequency components of the light interfere and de-pending on their phase relation φ = φ1 minus φ0 two ground statesuperpositions bright |B〉 and dark |D〉 are useful to consider

|B〉 =1radic2

(|0〉+ eminusiφ|1〉

)

|D〉 =1radic2

(|0〉 minus eminusiφ|1〉

) (74)

The bright state interacts with the light whereas the dark statedoes not due to destructive interference of the two excitation pathsthe ion can take ie from |0〉 or |1〉 to |e〉 Using a Bloch spherewhere |0〉 and |1〉 are on the top and bottom respectively the twostates |B〉 and |D〉 lie at opposite ends of the equator and makeup an alternative basis see Figure 77

A two-color sechyp pulse abbreviated to TC pulse transfersany population in |B〉 to the excited state |e〉 By sending inanother TC pulse but with an overall phase shift of π minus θ ie

68


|1

R0

R

|D

z |0

y

x

Figure 78 The pair of TC pulsesdescribed in the main text andEquation (75) rotates the initialBloch vector R0 an angle θaround the Dark state |D〉 whichis determined by the angle φ

φnew0 = φ0 + π minus θ and φnew

1 = φ1 + π minus θ the population returnsto |B〉 once more but with an added phase eiθ

In summary two TC pulses sent in after one another performthe following operation

|Ψinit〉 = α|B〉+ β|D〉First TC pulserarr α|e〉+ β|D〉

Second TC pulserarr eiθα|B〉+ β|D〉(75)

This is equivalent to a rotation with an angle θ around |D〉 ifone neglects any global phase factor see Figure 78 Since anyunitary operation on a single qubit can be written as a set ofrotations [94 p 174-177] these TC pulses can be used to performany arbitrary single qubit gate

In practice two additional TC (compensation) pulses areneeded after the pair described above where φcomp = φ + π andθcomp = 0 These two pulses excite the former |D〉 state (sinceφcomp is opposite the original φ on the Bloch sphere) and bringsit back without any overall phase shift (since θcomp = 0) Theyare needed to compensate for the phase spread obtained when thequbit peak is in the excited state which occurs due to the slightdifference in transition frequency of the various ions in the sim170kHz wide absorption peak For example an ion with slightly higherfrequency evolves its phase faster than an ion resonant with thelaser and therefore gains an additional phase when excited If onlythe first two TC pulses are used simply the |B〉 part of the ionis excited and gains the extra phase spread By using these com-pensation pulses the former |D〉 part is also excited and thereforegains the same phase spread which ensures that all ions indepen-dent of resonant frequency gain the desired phase shift of θ for the|B〉 state when all four pulses are considered

723 Fast all-optical nuclear spin echo techniquebased on EIT (Paper V)

This section contains an overview of Paper V in which the bright-dark state description and TC pulses otherwise known as electro-magnetically induced transparency (EIT) pulses are used in orderto perform an all-optical Raman spin echo sequence on the hyper-fine ground states |0〉 and |1〉 using the two transitions shown inFigure 71

Assuming no state preparation the relevant absorbing ions arein a completely mixed state which can be written using the densitymatrix formalism see Section 32

ρmixed =1

2(ρ0 + ρ1) =

1

2(ρB + ρD) (76)

where ρx = |x〉〈x|

69

723 Fast all-optical nuclear spin echo technique based on EIT (Paper V)

Time

1205872 120587 read

Spin echo sequence

|119890⟩

1

|0⟩

|119890⟩

1

|0⟩

|119890⟩

1

|0⟩

ൗ120591 2 ൗ120591 2

Figure 79 An all-optical Raman spin echo sequence on the hyperfineground states see main text for more details

A single TC square pulse brings |B〉 to |e〉 ie ρB rarr ρe Theremaining part of the state is ρD which is a superposition of thehyperfine ground states since |D〉 = 1radic

2


) In other

words starting from a completely mixed state a few microsec-onds long TC pulse creates a superposition between the hyperfineground states

Many similar techniques rely on optical pumping and thereforetake much longer many lifetimes of the excited state (T1 asymp 164 microsfor Pr3+Y2SiO5 [48]) in order to create a superposition Howeverthe downside of the TC pulse method is that only half of thestate is in a superposition whereas the other half is in the excitedstate but fortunately for some experiments this is not a crucialdisadvantage

For example the TC pulse can be used as the first step in aspin echo sequence which is generally called a π2 pulse as seen inFigure 79 In our case the TC pulse is a π pulse on the transition|B〉 rarr |e〉 but we will nevertheless call it a π2 pulse and thinkabout it in the |0〉-|1〉 basis as seen in Figure 710(a) where oncemore half of the wave function is in |e〉 after the pulse is completed

A general spin echo sequence is used to measure the coher-ence time of an inhomogeneous system which in our case is theground hyperfine inhomogeneity of 30 minus 80 kHz [58] The firstpulse π2 pulse generates a superposition After a waiting timeof τ2 during which the ions quickly dephase φdephase due todifferences in their hyperfine splittings a π pulse is used to inversethe phase obtained so far φinverse = π minus φdephase After anotherwaiting time of τ2 the ions rephase and emit a strong signal seeSection 331 but now on the opposite side of the Bloch sphereφrephase = φinverse+φdephase = π regardless of their hyperfine inho-

70


y

|1

x1

z |0(a)

y

|1

x2

z |0(b)

y

|1

1x2

z |0(c)

Figure 710 Bloch simulationresults of the π2 and π TCpulses ie the first two steps ofthe spin echo sequence inFigure 79 are shown in (a) and(b) respectively Instead ofstarting in a fully mixed state thesimulations show the two pathsions take if they start in |0〉 bluelines or |1〉 red lines Note thatin both (a)-(b) the blue and redarrows which indicate the Blochvector after the pulse(s) overlap(c) For a qubit initially in |0〉experimental quantum statetomographies were performedafter the π2 pulse shown ingreen denoted by 1 and shouldmatch the results in (a) and afterboth a π2 and π pulse shown inpurple denoted by 2 and shouldmatch (b)

Figure 711 The readout strength of the spin echo sequence seen inFigure 79 as a function of waiting time τ and external magnetic fieldstrength The decay with time is due to the limited coherence of thehyperfine ground states The coherence time curve is at a maximumfor an external field of about 50 microT which is of similar strength as theEarthrsquos local (Lund University Sweden) magnetic field [95]

mogeneity The coherence time of the system is measured by vary-ing the waiting time τ and analyzing the readout strength whichdecreases for longer waiting times due to increased decoherence ofthe system because of more interactions with the environment

For our sequence seen in Figure 79 the π pulse is anotherTC pulse but with twice the Rabi frequency and a different phaseon the |1〉 rarr |e〉 transition φnew

1 = φ1 + π2 performing therotation seen in Figure 710(b) The rephasing of the hyperfineground states is detected by continuously exciting only the |0〉 rarr|e〉 transition see the last pulse in Figure 79 which generates abeating signal at 102 MHz between the incoming laser light andthe rephased coherence between |1〉 and the part excited from |0〉to |e〉

In Paper V simulations of this spin echo sequence were madeusing the optical Bloch equations as seen in Figure 710(a)-(b)A qubit was also prepared experimentally as described in Sec-tion 71 and the pulse sequence was tested and analyzed usingquantum state tomography the results of which can be seen inFigure 710(c)

Furthermore the technique was used to optimize an externalmagnetic field in order to achieve a longer hyperfine coherencetime This experiment was performed without any qubit initializa-tion ie on an initially completely mixed state A coherence wascreated within microseconds using the fast π2 pulse which can becompared with other methods that normally require preparationthrough optical pumping thereby taking several optical lifetimesie milliseconds A full coherence time measurement curve con-sisting of 30 spin echo measurements with different waiting timesτ was performed and analyzed with an update frequency of 1 Hz

71

73 Qubit-qubit interaction (Paper VI)

Figure 712 (a) Hyperfine groundstate coherence time and (b)readout signal strength of the spinecho sequence both as a functionof temperature The red data usesonly the spin echo sequence as inFigure 79 whereas the blue dataalso initially burns a 2 MHz widehole centered around the |0〉 rarr |e〉transition At around 8 K thishole burning procedure stopsworking due to the increasedoptical linewidth and the twocurves overlap

making it possible to optimize the magnetic field strength in realtime The results of this optimization along the most sensitivemagnetic field direction can be seen in Figure 711 where a com-pensation field of sim50 microT corresponding to the local (Lund Uni-versity Sweden) vertical component of the Earthrsquos magnetic field[95] yielded the best result

The ground state hyperfine coherence time was measured asa function of temperature that reached all the way up to 11 Kbefore the readout signal became too weak to yield a reliable mea-surement value see Figure 712 The difficulties inherent to suchexperiments lie in the rapid decrease of the optical coherence timewith increasing temperature T optical

2 prop 1T 7 [41ndash43] which haslimited previous measurements up to 6 K [96] Thus these new ex-periments work for optical linewidths (116)7 = 70 times broaderthan previous ones

The results presented in Figure 712 are based only on the spinecho sequence for the red lines but use an additional preparatoryhole burning procedure that empties a 2 MHz wide region aroundthe |0〉 rarr |e〉 transition for the blue lines Hole burning yieldedbetter results due presumably to the decrease in instantaneousspectral diffusion that affects the nuclear spin states At around 8K it is no longer possible to burn a spectral hole and the two curvesoverlap This might indicate that the decrease in coherence timearound 6 minus 8 K is due wholly to increased optical excitation andnot a real decrease in the hyperfine coherence time See Paper Vfor a more in-depth discussion of these results

In conclusion a fast spin echo sequence was implemented andused to adjust a magnetic field in real time as well as measuringthe hyperfine coherence time up to a temperature of 11 K


The next step after performing arbitrary single qubit operationsinvolves two-qubit gates The idea investigated in this section andPaper VI is based on the dipole blockade effect ie the use of theDC Stark effect and the state dependent static dipole momentexplained in Section 34 and 43 respectively to control the targetqubit based on the state of the control qubit as illustrated inFigure 713

Unfortunately the dipole blockade effect works only for targetions situated close to a control ion since the frequency shift scaleswith 1r3 where r is the distance between the ions [97] and themagnitude of the shift needs to be large enough to move the ionout of resonance with the laser Thus the preparation scheme ofour two qubits is more complex than described in Section 71

In the first step of emptying frequency regions of absorptionas shown in Figure 72 you have to burn the two pits interleaved

72


Figure 714 When sequentiallyemptying two frequency regions ofabsorbing ions the first region ispartially refilled due tooff-resonant excitation during theoptical pumping of the secondregion This refilled absorption isshown in the blue squares as afunction of the frequencyseparation of the two regions Theerror bars which lie inside thesquares show the standard errorof the mean A Lorentzian fit isshown in the red line and anestimation of the refilledabsorption when using aninterleaved burning scheme isshown in the black circle

|0rang

|1rang

|119890rang

1 3

(a)

|0rang

|1rang

|119890rang

2 2

Control Target

|0rang

|1rang

|119890rangΔ120584

2 2

|0rang

|1rang

|119890rang

1 3

(b)

Control Target

Figure 713 The pulse sequence for a two-qubit interaction controlledNOT (CNOT) gate between a control and target qubit is shown with asimplified energy level structure The target is initially in state |0〉 (a)The control is in |1〉 The first transfer pulse denoted by 1 is ineffectiveand the control remains in the ground state The desired single qubitoperation denoted by 2 and shown here as a NOT gate is performed onthe target qubit The third pulse 3 is also ineffective since the controlis not in |e〉 (b) Control is initially in |0〉 Pulse 1 excites the controlwhose new static dipole moment illustrated by the dashed red linesaround the control qubit shifts the resonance frequency of the targetby an amount ∆ν The two-color pulse pulse 2 is no longer resonantand does nothing Finally the control will be brought back to |0〉 bypulse 3

and not one after the other since off-resonant excitation other-wise reintroduces absorption in the first region during the opticalpumping of the second one The magnitude of this refilled absorp-tion in the first region was studied as a function of the frequencyseparation between the pits as seen in Figure 714 A Lorentzianfit agrees well with the experimental results Thus off-resonantexcitation appears to be the reason for the refilling As mentionedearlier this effect can be minimized by alternating the burningbetween the two regions which results in a very low refilled ab-sorption This is estimated for the 90 MHz separation used in allthe following experiments by assuming that all 700 burn pulsesused to create one empty region contributes equally to the refilledabsorption shown in Figure 714 and that only the last burn pulsecontributes when the interleaved burning scheme is used This re-sults in a refilled absorption of 007700 αL = 10minus4 αL as shownby the black circle in Figure 714

After the absorption peaks have been created for both qubitsinitializing them to state |0〉 the scheme shown in Figure 715 is

73


|0rang

|1rang

|119890rang

1 3

Control

|auxrang

|0rang

|1rang

|32119890rang

2

Target

|auxrang

4

|12119890rang

|52119890rang

4648

102

173

[MHz]

Figure 715 The main part of thepreparation scheme to removeunwanted target ions is shown 1)The control is excited 2)Unwanted target ions are excitedto |52e〉 whereas wanted targetions are not excited due to thedipole blockade effect shown inFigure 713(b) 3) The control isdeexcited back to |0〉 4) Theunwanted target ionsspontaneously decay for exampleto |aux〉 where they are removedfrom the target qubit SeePaper VI for more detailedinformation about the fullpreparation scheme

used to remove unwanted target ions ie those not located closeenough to a control ion for the dipole blockade effect to work Themain part of the scheme consists of four steps

1 The control is excited on the |0〉 rarr |e〉 transition

2 Unwanted target ions located far from any control ion are ex-cited to |52e〉 whereas target ions situated sufficiently closeto a control ion are left in |0〉 due to the dipole blockade effect

3 The control is deexcited from |e〉 to |0〉

4 Waiting the unwanted target ions spontaneously decaypreferably to |aux〉

Using this scheme each target ion located close enough to acontrol ion has a resonance frequency shift after step 1 is neverexcited by step 2 and remains part of the target qubit placed in |0〉However undesired target ions are excited in step 2 and therebycan decay to |aux〉 where they are no longer part of the qubitSince this approach is probabilistic it needs to be repeated severaltimes in order to remove as much as possible of the unwantedbackground ions The scheme is presented with more details inPaper VI which includes repumping of ions from state |1〉 forboth the target and control qubit and finally repeating the wholeprocedure once more after the control and target roles of the qubitshave been switched

In order to estimate the absorption that originates from targetions close enough to a control ion (referred to as the signal) thetarget absorption is measured in two cases First after performingsteps 1-4 once Second after only performing steps 2 and 4 wherestep 2 excites all target ions with equal probability since the con-trol ions were never excited Thus after performing the schemeonce the remaining absorption in the target peak is lower in thelatter case compared to the former case where target ions situatedclose enough to a control ion are not removed The difference inabsorption of these two measurements correspond to the absorp-tion from target ions situated close enough to a control ion Moreinformation about the data treatment can be found in Paper VI

This difference in absorption ie the signal was studied as afunction of the control qubit ion density as seen in Figure 716In these experiments the control qubit peak was approximately1 MHz wide (FWHM) whereas the target peak was about 140kHz in order to increase the likelihood that a target ion was closeenough to a control ion in order to be shifted out of resonance withthe laser pulses that have a frequency width similar to the peakwidth The signal scales linearly with the control qubit ion densityas indicated by the linear fit shown in the red line The signal alsoscales linearly with the target qubit ion density Thus the signal

74


Figure 716 The target signalabsorption as a function of thecontrol ion density is shown in theblue squares where the error barsdisplay the error of the mean Alinear fit is shown in the red line

Figure 717 The signal ie theabsorption originating from targetions situated sufficiently close to acontrol ion and background iethe absorption from unwantedtarget ions can be seen in theblue and red lines respectively asa function of the number ofrepetitions of the preparationscheme presented in Figure 715The black line shows theuncertainty or noise in theevaluation of a zero signal

ie the number of target ions situated close enough to a controlion scales quadratically with the total ion density per frequencychannel For a given material the absorption coefficient α istherefore a good parameter to maximize by eg being in the centerof the inhomogeneous profile or minimizing crystal defects that canotherwise cause a broadening of the inhomogeneous linewidth andthereby a reduction in the maximum absorption coefficient

In Figure 717 the signal and background absorptions areseen as a function of the number of repetitions of the prepara-tion scheme The background decreases exponentially at a fasterrate than the signal such that the signal-to-background ratio canbe increased However the uncertainty or noise in the evalua-tion of a zero signal as shown in the black line indicates that thepreparation scheme cannot be repeated many more times beforethe signal-to-noise ratio decreases below 1 For these experimentsboth the control and the target qubits were 1 MHz wide sincethe transfer pulses can be made shorter in time when using widerpeaks which reduce the duration spent in the excited state for thecontrol ions and thus reduce lifetime decay as well

The interleaved burning scheme the quadratic scaling of thenumber of target ions located close enough to a control ion as afunction of the ion density per frequency channel and the demon-stration that the signal-to-background ratio can be increased byrepeating the preparation scheme are the current results in Pa-per VI but some final experiments with a CNOT are plannedbefore the manuscript can be finalized and submitted for peer re-view and publication

Unfortunately the ensemble qubit approach is not scalable inthe long run as the number of ions in each qubit decreases ex-ponentially as more qubits are added Calling the probability ofone ion being close enough to an ion in another qubit p the prob-ability of finding n qubits that interact in a chain ie qubit 1interacts with qubit 2 which in turn interacts with qubit 3 etcis pnminus1 Fortunately if single ions are used instead as individualqubits you can find long chains of interacting qubits The problemthen lies in reading out the state of single ions a topic discussedin the next section

74 Single ion readout scheme (Paper VII)

To circumvent the poor scaling of ensemble qubits this section andPaper VII investigate a single ion scheme where a dedicated read-out ion of a different element than the qubit ion is used to performstate selective detection of single ion qubits Whereas the qubitelement can be heavily doped into the crystal to make it easier tofind qubit chains as seen in Figure 718 the readout ion shouldbe sparsely doped such that single ion detection is possible The

75


q7

q6

q5 q3

q4

q1

q2

Buffer

Readout

Figure 718 A qubit chainconsisting of seven qubits onebuffer (also of the qubit kind)and one readout The black linesindicate possible two-qubitinteractions The procedure forreading out qubit 1 is shown inFigure 719 Any other qubit isread out in the same way after thequantum state of the desired qubithas been switched with qubit 1

|0rang

|1rang

|119890rang

1 3

Qubit 1

|0rang

|1rang

|119890rang

2

Buffer

Δ120584

4

Readout

Figure 719 The procedure for reading out the quantum state of qubit1 seen in Figure 718 using one buffer ion Two possible scenarios existqubit 1 is in |0〉 or |1〉 this figure explains the latter Pulse 1 targetsthe |0〉 rarr |e〉 transition of the qubit and does nothing since the ion isin |1〉 The buffer is then excited by pulse 2 which shifts the resonancefrequency of the readout ion Pulse 3 does nothing on the qubit since itwas never excited and pulse 4 is no longer in resonance with the readoution and thus no fluorescence is emitted In the other scenario the qubitis in |0〉 1 excites it the buffer becomes detuned pulse 2 is ineffectiveand the qubit is deexcited by pulse 3 In this case the readout ion isin resonance with pulse 4 and a strong fluorescence signal is detectedThe state of qubit 1 is thus detected based on a lowhigh fluorescencesignal from the readout ion This procedure can be repeated multipletimes in order to increase the collected photon statistics as long as thebuffer is reinitialized to |0〉

scheme also allows for the use of buffer ions also of the qubit ionelement to increase the readout fidelity at the price of shrinkingthe qubit chain by one for each buffer ion used

The readout scheme is presented in Figure 719 and is based onthe dipole blockade effect explained in Section 73 and Figure 713to connect the readout ion fluorescence via a buffer ion to thequbit state

Our group is currently investigating two possible candidates forthe readout ion First neodymium (Nd) placed in a high-finessemicro-cavity Its excited state lifetime can be shortened due to thePurcell effect in the cavity and is estimated in Paper VII to be-come around 200 ns which is necessary in order to produce enoughfluorescence during the limited lifetime of the qubit and buffer ionsDue to the high collection efficiency of the cavity a total detectionefficiency of about 10 is reasonable This approach is still in itsinfancy but additional experiments are planned

Second the 4f-5d transition in cerium (Ce) is inherently shortlived with a lifetime of 40 ns and its fluorescence can be detectedusing a confocal microscope setup Such a setup has been con-

76


Figure 721 Estimated fidelitiesof n-qubit GHZ states with orwithout readout In the readoutcase the results assume that allqubit ions can directly interactwith the buffer ion which isreasonable for up to at least fivequbits using a qubit dopingconcentration of 4

structed by our group [87] and is used in Section 75 and Pa-per VIII in an attempt to detect single Ce ions So far howeverno discrete single ion signal has been observed but the investiga-tion is ongoing

Simulated photon statistics of this readout scheme can be seenin Figure 720 where the results are valid for any type of readoution as long as the ratio of the lifetime and collection efficiencyremains constant The data assume a Eu3+Y2SiO5 crystal andphotons are collected during an optimum time period based on thenumber of collected photons per time interval and the Eu excitedstate lifetime

Paper VII also contains a thorough calculation of an estimatedCNOT fidelity without or with readout when using Eu as a qubition resulting in fidelities of 994 and 991 respectively Fur-thermore the scalability of the system is analyzed by calculat-ing the estimated fidelity of creating n-qubit Greenberger-Horne-Zeilinger (GHZ) states the results of which can be seen in Fig-ure 721 An important assumption during these calculations wasthat all qubits could directly interact with the buffer ion which isreasonable for a system of up to at least five qubits if a 4 qubitdoping concentration is used If the qubits cannot directly interactwith the buffer ion their quantum information can be switched toa qubit that can at a cost of reducing the readout fidelity Al-ternatively several readout ions with associated buffer ions canbe used to create islands of qubits that can be read out directlyThis however introduces the problem of differentiating betweenthe different readout ions during detection

Summary of Paper VII a single ion qubit scheme using a ded-icated readout ion and buffer ion(s) is presented together withfidelity calculations of the readout a CNOT and the system scal-ability

77


Figure 720 Simulation over the number of detected fluorescence pho-tons for the two qubit states |0〉 in blue and |1〉 in red Valid foreither Nd in a micro-cavity with a 200 ns lifetime and (a)-(b) 1 (c)10 collection efficiency or Ce in a confocal microscope setup with 40ns lifetime and collection efficiencies of (a)-(b) 02 (c) 2 (a) with-out the use of a buffer ion and low collection efficiency (b) one bufferion with 10 repetitions of the scheme shown in Figure 719 still usinglow collection efficiency (c) one buffer ion and only 3 repetitions but ahigh collection efficiency The probability of distinguishing between thetwo states is (a) 93 (b) 997 and (c) 9985

78


Figure 723 The width of aspectral hole created byredistributing ions in the Zeemanlevels is investigated as a functionof an external magnetic field Thedata is smoothed with a movingaverage corresponding to 4 MHzon the frequency axis The datahas been normalized such that themean value above 50 MHz is setto one

2F52

2F72

371 nm

5d(2D52

2D32)

B

B

Δ1205845119889

Δ1205844119891

Figure 724 Relevant energylevels of cerium The insets showsthe Zeeman splitting of theground and excited zero-phononline states as a function of theapplied magnetic field

Cryostat

SMF

BPPH

X

Z

Y

SPAD

DM

PDHECDLAOM

Lens

Crystal

Figure 722 Simplified laser and confocal microscope setup An ex-ternal cavity diode laser (ECDL) is locked to a cavity with the Pound-Drever-Hall (PDH) technique The laser light shown as purple linesgoes through a double pass acousto-optic modulator (AOM) beforepropagating through a single mode fiber (SMF) to the microscope setupA dichroic mirror (DM) reflects the laser light into the crystal holder (seefigure at upper left) situated in the cryostat The crystal can be trans-lated in the transverse directions x and y using atto-cubes and thelens can be moved in the z direction The fluorescence (turquoise lines)is collected and then transmitted through the dichroic mirror (DM) andthe band pass filters (BP) which suppress any reflected laser light Thesingle photon counting avalanche diode (SPAD) detects the fluorescenceafter it has passed through a pinhole (PH)

75 Cerium (Ce) as a dedicated readout ion(Paper VIII)

In an attempt to experimentally realize the single ion qubit schemediscussed in Section 74 and Paper VII much effort has been de-voted to detecting a single cerium ion in a Ce3+Y2SiO5 crystalSo far our group has not been able to detect any discrete fluo-rescence signals that could indicate single instances Neverthelessthis section provides an overview of the effort and results of Pa-per VIII where spectral hole burning in the Zeeman levels and apermanent trapping mechanism in cerium are analyzed The con-focal microscope setup used for these experiments can be seen andis explained in Figure 722

In Figure 723 the size of a spectral hole as a function of anexternal magnetic field is analyzed The hole is burned in theZeeman levels that are split due to the non-zero magnetic fieldsee Figure 724 As can be seen the hole becomes wider anddeeper for stronger magnetic fields but the reason why is unknownThe narrowest hole was achieved for an applied field of 005 mT

79

76 Investigating Eu3+Y2O3 nanocrystals

Figure 725 The Zeeman spectralhole area as a function of thewaiting time between the readoutand the hole burning Thelifetime is estimated to be 72plusmn 21ms at an external magnetic fieldstrength of 10 mT Data pointsare shown in blue where the errorbars mark one standard deviationThe red curve is an exponentialfit

Figure 726 The ground +excited state Zeeman splitting∆ν4f + ∆ν5d as a function of theexternal magnetic field is shownin blue where the error bars markone standard deviation A linearfit is displayed in red with a slopeof 434plusmn 17 MHzmT

(6 plusmn 4 MHz wide) yielding an upper limit of 3 plusmn 2 MHz on thehomogeneous linewidth of the 4f-5d zero-phonon line transition

The lifetime of the spectral hole was also measured by graphingthe hole area as a function of the wait time between the hole cre-ation and readout as seen in Figure 725 for an external magneticfield of 10 mT An exponential fit yields a lifetime of 72plusmn 21 ms

The combined Zeeman splitting of the ground and excitedstates as a function of applied magnetic field was measured asseen in Figure 726 The fitted red line represents a dependence∆ν4f+∆ν5d

∆B = 434plusmn 17 MHzmT Similar data yields an estimate

for only the ground state splitting ∆ν4f∆B = 179 plusmn 15 MHzmT

which is in agreement with the known ground state splitting rateof 19 MHzmT [98 99] Combining these results yielded an esti-mate for the excited state Zeeman splitting of ∆ν5d

∆B = 255 plusmn 23MHzmT

During these investigations it was observed that the fluores-cence signal decreased from its original high photon count rateas seen in Figure 727 Throughout this measurement the lasershone on the sample with constant intensity It was also locked andstable within the homogeneous linewidth of the transition Evenafter blocking the laser light for up to 40 minutes the fluorescencesignal never returned to its original high count rate This per-manent hole was measured to be around 70 MHz wide (FWHM)where a magnetic field of 02 mT was used to compensate for astray magnetic field present inside the cryostat at the time of theexperiment This relatively narrow linewidth suggests that it goesvia the 4f-5d zero phonon line as no other transition is that nar-row However the reason for the discrepancy of 70 MHz and thevalue of 3plusmn 2 MHz measured above is still unknown

A rate equation model with the 4f 5d and conduction bandlevels as well as a trapping state connected to the conductionband was used to simulate the decrease in the fluorescence signalThe result of this simulation can be seen in the red line of Fig-ure 727 and more details about the simulations can be found inPaper VIII

In conclusion Paper VIII investigates spectral hole burning inthe Zeeman levels determines the hole lifetime and sets an upperlimit on the homogeneous linewidth of the 4f-5d zero phonon linetransition Furthermore the Zeeman splitting coefficients for theground and excited state are also estimated Finally a permanenttrapping that reduces the cerium fluorescence signal was observed


In addition to trying to detect single cerium ions the confocal mi-croscope setup presented in Section 75 has been used to studyeuropium doped into yttrium oxide (Eu3+Y2O3) nanocrystals

80


Figure 727 A decrease in the fluorescence signal due to permanenttrapping as a function of time under continuous laser illumination ofthe Ce3+Y2SiO5 crystal can be seen in the blue line The strong fluo-rescence signal does not return even after blocking the laser for up to40 minutes indicating that the trap lifetime is at least hours The redline shows the results of a rate equation model as further explained inPaper VIII

However since the first draft of this work is not ready yet I willbe very brief about the experimental method and results

The confocal microscope setup seen in Figure 722 was mod-ified to the Eu3+Y2O3 wavelength of around 580 nm and used toprobe approximately 400 nm large nanocrystals with the dye lasersetup instead of the UV external cavity diode laser used for thecerium experiments For more information about the laser setupssee Appendix A The setup allows for high spatial and spectral res-olution data to be collected from a relatively small ensemble of ionsdistributed in only a few nanocrystals Different samplesubstratecombinations were analyzed in order to obtain a low backgroundfluorescence and reflection

A three-frequency measurement technique where the incom-ing light quickly alternates between three frequencies was used todetermine the ground hyperfine level orderings for both europiumisotopes Furthermore the homogeneous linewidth of the ions wasmeasured Since it was much broader than the natural linewidthhowever more investigations were performed which revealed in-teresting properties of the spin dynamics of the ions within thenanocrystals

81

77 Progress on DiVincenzorsquos criteria


This section summarizes the progress of rare-earth quantum com-puting with respect to the DiVincenzo criteria The discussion isbased on the single ion readout scheme presented in Section 74and Paper VII but several other equally likely approaches exist

A scalable physical system with well-characterized qubits

At the current stage well-characterized ensemble qubits have beenexperimentally realized In order to have a more scalable systemhowever single ion qubits are required Our grouprsquos inability toachieve single ion detection so far has resulted in poor experimentalprogress Nevertheless the basic qubit preparation and initializa-tion should be very similar for the two approaches

The ability to initialize the state of the qubits to asimple fiducial state such as |000〉Regardless of the ensemble or single ion qubit approach thescheme for initializing qubits is well-established [49] In Paper VItwo qubits were prepared at different optical frequencies with onlythe minor problem of off-resonant excitation causing the spectralhole of the first qubit to be partially refilled during the prepara-tion of the second qubit This was solved by preparing the qubitssimultaneously by alternating the hole burning between the twofrequency regions instead of creating one before moving on to thenext At this stage no major problems are expected when morequbits are initialized but potential difficulties include spectral dif-fusion and off-resonant excitation as mentioned above It will alsobe more difficult to analyze the preparation of the nearby absorp-tion structure in the single ion scheme where only the qubit stateis readily available compared to the ensemble approach wherethe vicinity of the qubit peak can be easily probed through anabsorption measurement

For error correcting quantum computation the initialization ofqubits used to measure the error should ideally be on the timescaleof the gate operations Currently the initialization step takes sev-eral lifetimes of the excited states since it uses optical pumpingShort-lived Stark levels could potentially be used to rapidly ini-tialize a qubit but has so far not been demonstrated

Long relevant decoherence times much longer than thegate operation time

As seen in Tables 41 and 42 the coherence times are up to mil-liseconds long for the optical transition and can be several hours forthe ground spin states in Eu3+Y2SiO5 The small hyperfine split-tings of Pr3+Y2SiO5 set the minimum pulse duration to the mi-

82


crosecond scale which results in single qubit gate fidelities of about96 due to the limited optical coherence time [60] Fortunately forEu3+Y2SiO5 estimated fidelities of 9996 seem reasonable seePaper VII but have not yet been proven experimentally Further-more the extremely good ground hyperfine coherence times meanthat quantum information can be stored for a very long time

A ldquouniversalrdquo set of quantum gates

With the use of two-color sechyp pulses arbitrary single qubitgates can be performed in a robust manner [89ndash91] The two-qubitCNOT operation will hopefully be demonstrated experimentallysoon with the initial steps being taken in Paper VI Together witharbitrary single qubit gates it forms a basis that is capable ofperforming any quantum gate However the fidelities need to beimproved for any practical quantum computation application

A qubit-specific measurement capability

Due to the lack of single ion detection this requirement is thecritical point at which more effort is needed in order to proceedwith the scalable single ion qubit scheme


In this last section the future of the quantum computationprojects is discussed

The electromagnetically induced transparency (EIT) projectPaper V measured the ground hyperfine coherence time for tem-peratures up to 11 K No experiments are currently planned forreaching higher temperatures But the effort devoted to creat-ing a real-time display of entire measurement curves has provenextremely useful and speeds up present-day experiments

With regard to the two-qubit interaction project presented inPaper VI additional experiments are planned in order to demon-strate a CNOT gate It seems likely that some parts of the CNOTgate will soon be demonstrated eg showing |00〉 rarr |00〉 and|10〉 rarr |11〉 The additional error introduced by placing the con-trol in a superposition and after the CNOT gate performing afull quantum state tomography might unfortunately be too largeto demonstrate a full entanglement between the two qubits Afterthese experiments have taken place the manuscript will be updatedand submitted for peer review and publication

The two single ion readout schemes discussed in Section 74 andPaper VII the Nd micro-cavity and the Ce confocal microscopeare still under investigation but no experimental realization of thescheme has yet been achieved

83


The Nd3+Y2O3 coherence time was recently measured in aceramic sample and the initial unpublished results are promisingThe next step is to set up the micro-cavity and look at Nd3+Y2O3

nanocrystals inside the cavityThe Ce approach has had its setbacks over the years Although

no single ion has yet been detected the results are progressingwith further investigation into both the permanent trapping Pa-per VIII and the problems we face in detecting single ions Ourcurrent effort is devoted to using micro-crystals instead of bulk inthe confocal microscope setup in order to reduce the fluorescencesignal obtained from ions far away from the focus along the laserbeam propagation direction Even though single ion detection ofrare-earth-ions have been achieved eg in references [100ndash106]no one has so far performed state selective readout of a qubit ionbased on the fluorescence from a single dedicated readout ion

An article about the Eu3+Y2O3 nanocrystals is currently un-der preparation and no further experiments are planned at thistime

84

Chapter 8

Conclusions

How light propagates and is absorbed in rare-earth-ion-doped crys-tals a few applications of slow light and some steps toward quan-tum computing that is the briefest summary I can make of mythesis work

With respect to the light-matter interaction results we havedrawn attention to how the polarization of light changes whilepropagating through birefringent crystals with off-axis absorptionor gain even if the incoming light is polarized along a principalaxis (Paper I) Furthermore a narrowband spectral filter has beencreated with a suppression of 534 plusmn 07 dB between light pulsesseparated in frequency by only 125 MHz (Paper II)

The slow light and DC Stark effects have been used to fre-quency shift light within plusmn4 MHz with efficiencies above 85 (Pa-per III) tune the group velocity of a pulse from 15 kms to 30kms ie a tunable range of a factor of 20 with efficiencies above80 and temporally compress an incoming pulse by a factor ofapproximately 2 (Paper IV)

From an initially mixed state we showed that a partial su-perposition can be achieved in microseconds between two groundnuclear states using an optical two-color pulse (Paper V) In thisstate half of the wave function is in the excited state whereasthe other is in a superposition of two ground states This can becompared to optical pumping methods which although they cangenerate a full superposition state take many times the excitedstate lifetime to complete ie up to several milliseconds Theoptical two-color method was used to measure the ground statehyperfine coherence time as a function of temperature reachingup to 11 K

With regard to quantum computing two ensemble qubits havebeen initialized simultaneously and qubit-qubit interactions havebeen shown in preparation of performing a CNOT gate (Paper VI)However due to poor scaling of the ensemble qubit approach an-

85


other scheme using single ion qubits has also been investigatedIn order to perform state selective readout of these single ions wesuggested to use a dedicated readout ion and buffer ions to reachhigh fidelity readouts (Paper VII) One potential candidate forthe dedicated readout ion is cerium and although we have notdetected a single cerium ion yet an upper bound on the homo-geneous linewidth 3 plusmn 2 MHz was obtained and a slow minutetimescale permanent spectral hole burning mechanism was ob-served (Paper VIII)


During my time here I have learned a great deal especially withrespect to light-matter interaction which actually was the nameof the course in which I came into contact with Stefan Kroll mysupervisor The grouprsquos ideas about ultrasound optical tomogra-phy (UOT) have recently evolved into todayrsquos full-fledged projectwith monthly meetings and collaboration between several scientif-ically diverse research teams My hope is that the effort devotedto the spectral filters and frequency shifters will benefit the futureof this project just as my knowledge of light-matter interactionshas helped during various discussions

As is often the case in science a lot of work is hidden beneaththe surface of the few results that eventually become an articleincluding entire projects For me the attempt on reaching highfidelity transfers in europium was such a project Similarly thedetection of a single cerium ion has continued to elude us and anew problem of permanent trapping was discovered instead Theelectromagnetically induced transparency article contains a quickmethod for generating coherence from a mixed state but we havenot identified a place for it in quantum computing so far In con-trast even though the qubit-qubit interaction project is not final-ized yet the initial results are promising In conclusion you mightsay that the quantum computing progress has been slowly stum-bling forward for the past few years Fortunately the group hasrecently expanded and new people and fresh insights will hopefullybring us beyond the single cerium detection step which has beenan obstacle for a long time into the uncharted territory of singleion quantum computing using rare-earth-ion-doped crystals

Learning theory running simulations and performing exper-iments have been the continuous cycle of working here and thefeeling when the last piece of knowledge falls into place when thecoding error is finally resolved or when the oscilloscope displaysthose expected but highly awaited results is according to me oneof the best things about working in science

Another great benefit is the diversity both in the people withwhom you work but also in what you do some days you are truly

86

Conclusions

a physicist trying to understand nature the next day you workas an (amateur) programmer debugging your code a (rookie)chemist mixing and preparing the dye solution for the laser a(make-believe) detective attempting to locate faulty equipmentby examining evidence ruling out unlikely candidates and inves-tigating the list of suspects until the culprit is found or finallyas a (novice) writer when results should be published or as inthis case when the words of your PhD should be composed into abook

87

Appendix A

Equipment

In order to concentrate on the physics of the papers included inthis thesis I have moved most descriptions of the experimentalsetups from the main chapters to this appendix In Section A1the setup used in all papers except Paper VIII is explained Eventhe papers that are mostly theoretical Papers I and VII havesome experiments associated with them and have used this setupSection A1 has three subsections the dye laser setup which in-cludes the pulse creation and part of the locking setup the restof the locking setup and the most common experimental setup onthe cryostat table as well as how it was modified for the variousexperiments

Brief information about the liquid helium bath cryostat is pro-vided in Section A2 and the cerium setup used for Paper VIII isexplained in Section A3

A1 Main setup

Thanks to present and previous members of our group a (mostly)well working system has been developed including a tunable dyelaser that is locked to an ultra low expansion glass cavity a Rabifrequency calibration system and high level coding such as ldquoCre-ate pitrdquo and ldquoCreate 170kHz qubitrdquo used to initialize a qubit asseen in Figures 72 and 73 in just two code lines The follow-ing subsections briefly review the equipment used For more in-formation see previous overviews of the system in for examplereferences [107ndash110] keeping in mind that some parts have beenupdated through the years eg we are no longer locking the laserto a crystal as was done when the system was initially developedbut to a cavity instead

89

A11 Dye laser setup

Dye laser table

λ2-plate

Fiber

AOM1

Wavemeter

EOM

AOM2

AOM3

λ2-plate

Polarizer

Fiber to locking platform

Fiber to experimental table

OD

Pumplaser

Dye laser

Filterwheel

MirrorBeam splitter

Lens

Figure A1 Experimental setup of the dye laser table see Section A11for more information

A11 Dye laser setup

The setup on the dye laser table which can be seen in Figure A1is inside a cleanroom in order to minimize disturbances from dustA 6W 532 nm Verdi-V6 solid state laser pumps the dye laser of typeCoherent 699-21 The dye is Rhodamine 6G mixed with ethyleneglycol pumped at 42 bar and cooled to 10degC with a tuning range ofapproximately 570-635 nm and possible output powers of sim1 W at580 nm (Eu3+Y2SiO5) and sim600 mW at 606 nm (Pr3+Y2SiO5)

The laser mode is cleaned up with a single mode polarizationmaintaining (PM) fiber where a half-wave plate as well as somemirrors that are not shown in the figure are used to align the incou-pling A small fraction of the transmitted light is split off using abeam splitter and sent towards the locking platform (dashed line)Another beam splitter reflects some light into a wavemeter (dot-ted line) HighFinesse WS-6IR whereas the transmission (dashedline) goes through a filter wheel and a double pass acousto-opticmodulator (AOM1) Brimrose GPF-1125-750-590 which is capa-ble of tuning the frequency by up to 15 GHz in the two passesAn optical Faraday rotator and two polarizers are then used asan optical diode (OD) to let light through in this direction onlyAn electro-optical modulator (EOM) New Focus 4002 generatessidebands at 50 MHz which are used for the laser locking Thelight is sent to the locking platform explained in Section A12via another fiber where the incoming beam polarization is alignedwith a half-wave plate and a polarizer

The majority of the light from the laser is transmitted through

90

Equipment

Locking platform

Fiber from dye laser

table

λ2-plate

Error detector

Transmissiondetector

CCD camera

λ4-plate

PBSLens

Cavity

Figure A2 Located on an actively stabilized platform is a cavity thatis used to lock the dye laser See Section A12 for more details

the first beam splitter (solid line) and is sent into a double passAOM2 AAST200B100A05-vis which can alter the frequencyphase andor amplitude of the transmitted light and is used tocreate pulses A single pass AOM3 AAST360B200A05-vis(recently changed to a 60 MHz AOM of type Isomet 1205C-2) isused to generate two-color pulses when needed by sending in anRF signal containing two frequencies usually separated by 102MHz ie the ground hyperfine splitting between |0〉 and |1〉 inPr3+Y2SiO5 Both AOMs are controlled by an arbitrary wave-form generator (Tektronix AWG520) through LabView and Mat-Lab scripts Finally the light is guided into a fiber connected tothe experimental table which is explained in Section A13

This setup has remained fairly unchanged during my thesiswork However updates have recently been made with regard tofiber noise stabilization in an effort to improve the laser linewidthat the crystal location None of these changes have been includedin Figure A1 since they were not used for any experiments in thisthesis

A12 Dye laser locking

A small fraction of the dye laser light is guided to the locking plat-form as seen in Figure A2 The platform is actively stabilizedagainst vibrations using a Table Stable TS-140 The light is lin-early polarized and contains sidebands at 50 MHz A half-waveplate makes sure that the polarization direction is vertical beforethe light is transmitted through a polarizing beam splitter cube(PBS) Using a few mirrors and a lens the light is coupled into anultra-stable cavity before which a quarter-wave plate is locatedThe cavity is kept at 10degC and a pressure of 10minus9 mbar to 10minus8

91

A13 Experimental table

mbar Any reflection from the cavity is rotated once more by thequarter-wave plate resulting in a total rotation of 90 degrees ieturning the light polarization from vertical to horizontal The PBSsplits off this horizontally polarized reflection (dotted line) whichis then focused onto a homebuilt error detector [110] to be usedin the Pound-Drever-Hall locking technique explanations of whichcan be found in references [107 111] The designed linewidth of thelocking system is 10 Hz but no experiments have been performedto verify it as it would require an equally stable reference

The cavity transmission can be examined with both a CCD(charge-coupled device) camera to look at the spatial mode andanother homebuilt detector [110]

Normally the frequency of a locked laser is limited to the dis-crete locations of the cavity modes separated in our case by amode spacing of roughly 33 GHz However by frequency shiftingthe light reaching the cavity using AOM1 shown in Figure A1the dye laser can be locked at almost any frequency Since thetuning range of the AOM is only approximately half that of thecavity mode spacing it needs to be realigned with the +1 or minus1order of the AOM to cover most of the frequency space Morecommonly the AOM is kept at one alignment at the price of beingable to reach only half of all available frequencies This option tolock at most frequencies is important for experiments that benefitfrom being in the center of the inhomogeneous absorption profile


In contrast to the setups described so far the equipment on theexperimental table is always changing However most of my ex-periments have had a fairly simple and common starting point asshown in Figure A3

Using AOM2 and AOM3 seen in Figure A1 arbitrary pulseswithin a wide frequency range can be created and are sent viaa fiber to the experimental table Figure A3 A beam splitterreflects a small fraction of the light (dashed line) which is thenfocused onto a reference detector Thorlabs PDB150A used tocalibrate for laser intensity fluctuations and the frequency depen-dent AOM efficiency which is not always perfectly compensatedfor by the calibration system The majority of the light (solidline) passes through a half-wave plate and a polarizer which areused to orient the linear polarization to the desired crystal axisAfterwards the light is focused into the cryostat and propagatesthrough the crystal before it is collimated and eventually focusedonto the transmission detector Thorlabs PDB150A In most ex-periments a focus size of approximately 100 microm was used insidethe crystal but for more precise details see the articles

The experiments performed in Paper I used a setup similarto the one described above but where the half-wave plate and

92

Equipment

Polarizer

λ2-plate

Transmission detector

Cryostat

Reference detector

Crystal


table

Lens

Experimentaltable

SE

N W

Figure A3 The most common components of the experimental setupon the cryostat table See Section A13 for more information

polarizer switched places and a second polarizer was added afterthe cryostat to measure polarization specific transmission

For the setup in Paper II a second polarizer was again addedafter the cryostat as well as another transmission detector Hama-matsu PMT (photomultiplier tube) R943-02 to measure the veryweak light when propagating through the high absorption regionsof the crystal A schematic view of this setup can be seen in Fig-ure 54 in Section 52

In Paper III a heterodyne detection scheme was used ieanother beam splitter was inserted before the cryostat to create areference beam that when overlapped with the transmitted beamafter the cryostat results in a beating pattern at the differencefrequency

The setup used in Papers IV-VI was precisely the one seen inFigure A3

Even though Paper VII is a theoretical article I performedexperiments on Eu3+Y2SiO5 using a similar setup as before butwith a pinhole after the cryostat to probe only the transmissionfrom ions radially centered in the focus in an attempt to reachtransfer fidelities above 99 Due to a lack of available laser powerie Rabi frequency the desired pulse scheme could not be fullytested and the results were not good enough for publication

Finally the signals were analyzed using an oscilloscope eitherTektronix TDS 540 or LeCroy WaveRunner HRO66Zi and all datatreatment was performed using LabView or MatLab

A2 Cryostat

The cryostat used for all experiments in my thesis has been a liquidhelium bath cryostat from Oxford Instruments of type spectromag-4000-8 The sample space is a 25 mm diameter cylinder inside

93

A3 Cerium (Ce) setup

λ4-plate

Ce laser table


CCDcamera

Cavity

PBS

Errordetector

EOM

λ2-plate

ECDL

OD

λ2-plateAOMPolarizer

Fiber to locking cavity

Fiber to confocal setup

Figure A4 The 371 nm external cavity diode laser setup with lockingand pulse creation can be seen and is explained further in Section A3

which the crystal is mounted in a holder at the end of an approxi-mately 1 m long sample rod The outer vacuum shield is normallypumped by either Leybold PT 70 F Compact or Pfeiffer VacuumHiCube 80Eco to around 10minus6 mbar to 10minus5 mbar

The cryostat is oriented as shown in Figure A3 in a northeastrarr southwest direction and the Earthrsquos magnetic field is north171 microT east 12 microT and down 474 microT (2017) [95] To com-pensate for this field three coils one in each direction have beenadded outside the cryostat but have been used only in Paper VThe cryostat also contains two superconducting magnets capableof reaching 8 T along the direction perpendicular to the light shownin Figure A3 but a much smaller field of about 10 mT to 100 mThas typically been used


The setup used for Paper VIII is different from the other articlesas it operates in the ultraviolet The 371 nm laser setup can beseen in Figure A4 and is mounted on a 75 cm by 90 cm mov-able optical table The external cavity diode laser (ECDL) usesa littrow configuration with a 371 nm laser diode Sacher SAL-372-10 The light passes through an optical diode (OD) ThorlabsIO-5-370-HP after which a half-wave plate is located The reflec-tion from a beam splitter (dashed line) passes through a telescopethat shrinks the beam size before it is sent into an AOM Isomet1250C-829A in double pass configuration Using a pickup mirrorthe transmitted light passes through a half-wave plate and a po-

94

Equipment

larizer before a fiber guides it to the confocal microscope setup asexplained in Section 75 Figure 722 and reference [87]

The main part of the light (solid line) passes through the beamsplitter after the OD and into a fiber that carries it to the lockingpart of the setup The polarization is rotated and cleaned up us-ing a half-wave plate and a polarizer before an EOM Linos PM25KDP generates sidebands at 20 MHz The light is transmittedthrough a polarizing beam splitter cube (PBS) and focused intoan ultra-stable cavity before which a quarter-wave plate is locatedThe transmitted light can be observed using both a normal detec-tor S5973-02 photo diode and Femto HCA-10M-100k amplifier[110] as well as a CCD camera PointGrey Firefly The reflectionfrom the cavity is once more rotated by the quarter-wave platesuch that the PBS reflects the light (dotted line) into the errordetector modified version of Femto HCA-S-200M-Si-FS [110]

This laser is also locked using the Pound-Drever-Hall lockingtechnique [107 111] More detailed information about the ceriumsetup can be found in references [87 110 112 113]

95

The authorrsquos contribution tothe papers

Note that the thesis author previously published under Adam NNilsson


Simulations experiments and the first draft of an article re-garding wave propagation through birefringent crystals withoff-axis absorption were initially performed by Mahmood Sa-booni The most important results are that the absorp-tion coefficient generally varies during propagation due tothe rotation of the polarization inside the crystal and thatsteady state polarizations exist ie polarizations that donot change during propagationI continued the research and performed new simulationswhich caused us to modify most of the article which Irewrote and draw new conclusions


A high suppression (534plusmn 07 dB) narrowband (125 MHz)spectral filter was created in Pr3+Y2SiO5 The filter wasanalyzed with respect to ultrasound optical tomography(UOT) and simulations were performed in order to studythe potential etendue of the filtersI performed all simulations experiments analysis and wrotethe manuscript

97

The authorrsquos contribution to the papers


Optical pulses were frequency shifted (plusmn4 MHz) using thelinear DC Stark and slow light effects in Pr3+Y2SiO5 Thedevice is controlled by an external voltage and we claim thatthe solid angle of acceptance could be close to 2πI contributed to discussions especially regarding the analy-sis simulations and manuscript


A group-velocity controller based on the DC Stark and slowlight effects in a Pr3+Y2SiO5 crystal was shown to alterthe velocity of light pulses from 15 kms to 30 kms withoutsignificant loss or distortion A proof of principle experimentwas also performed which temporally compressed a pulse bya factor of approximately 2Qian Li and I performed all experiments together and I wasthen included in discussions about the simulations analysisand manuscript


Using electromagnetically induced transparency (EIT) anall-optical Raman spin echo technique which is capable ofinducing coherence directly from a mixed state was analyzedexperimentally and in simulationsI performed the simulations joined Andreas Walther for theexperimental work and was involved in discussions aboutthe analysis and manuscript

98



Qubit-qubit interactions based on the dipole blockade ef-fect are studied using two ensemble qubits in Pr3+Y2SiO5To minimize the effect of off-resonant excitation an inter-leaved optical pumping scheme is used during the qubit ini-tialization Furthermore the number of interacting ions areshown to grow quadratically with the ion density per fre-quency channel and to decrease at a slower rate comparedto the background absorption when repeating the prepara-tion schemeI prepared and performed the experiments analyzed thedata and wrote the manuscript


A single ion readout scheme is presented where a buffer ion isused to improve qubit readout fidelity A CNOT-gate fidelitywith and without readout is calculated and the scalabilityto several qubits is investigatedI contributed to discussions The Eu3+Y2SiO5 high fidelitytransfer experiments which I performed should have beenincluded in this article but due to a lack of available Rabifrequency the fidelities were never publication worthy


A slow permanent trapping in Ce3+Y2SiO5 was analyzedas well as spectral hole burning in the Zeeman levels wherean upper bound on the homogeneous linewidth (3plusmn 2 MHz)of the 4f-5d transition was estimatedJenny Karlsson and I performed all experiments togetherand both of us analyzed the data and wrote the manuscript

99

Acknowledgements

As is often the case I have too many people to thank for theirhelp with respect to my PhD from my first teachers who laidthe foundation of my math and physics interest to the currentcolleagues with whom I perform experiments I will neverthelesstry to list a few by name and mention some of the assistance theyhave provided

First and foremost I thank my main supervisor Stefan Krollfor always being there to help always having time to discuss andallowing me to focus on what I find most interesting My co-supervisors Andreas Walther for all the discussions about quan-tum mechanics for ever-present help and for letting me share yourroom for a while and Peter Samuelsson for being a nice and warmperson who always says that things seem to be going well with myPhD

Lars Rippe for constantly talking be it about physics or othertopics it really contributes to the grouprsquos spirit Qian Li for lettingme be a part of the slow light experiments the general discussionsthe laughs and being an absolutely perfect office mate JennyKarlsson for the cerium experiments and the always happy smileYing Yan for laying the foundation of the cerium project beforeI started and recently returning to perform high fidelity transferexperiments with me Diana Serrano for all her knowledge aboutrare-earth crystals Mahmood Sabooni for letting me continuethe wave propagation simulations and manuscript and GerhardKristensson for many helpful discussions about the same project

To the students whom I have helped supervise Pernilla Helmerfor being the first one and an exemplary one at that Philip Dals-becker for efforts and discussions about spectral filters and VassilyKornienko for the constant enthusiasm about our work it lightsup the day

To the collaborators whom I have worked with Philippe Gold-ner Alban Ferrier John Bartholomew Karmel De-Oliveira LimaIvan Scheblykin and Axel Thuresson thank you so much for yourhelp and dedication

To all both previous and present group members whom I havehad the pleasure to meet I thank you for the work you are do-

101

ing and for continuing to make the quantum information group awonderful and happy workplace

A special thanks to all administrative and technical personalAnne Petersson Jungbeck for always having the answers to allmy questions Ake Johansson for on several occasions fixing mycomputer and helping with other equipment Jakob Testad fortaking care of the financial issues and Leif Magnusson and StefanSchmiedel for always having helium available even on very shortnotice

Thanks to Anders Persson for help with the teaching and fixinglasers and to Lars Engstrom for planning and keeping track of thedoctoral teaching

To the SPIEOSA chapter members thanks for all the eventsand fun activities

I would also like to thank the entire atomic division for theircompany during lunches at work Friday meetings and Mondaycakes

Finally I extend my gratitude and thanks to my friends andfamily especially my wife Linnea and daughter Vega for beingthere when I need it for believing in me and making me smile

102

References

1 R P Feynman R B Leighton and M Sands The FeynmanLectures on Physiscs - Volume 1 AddisonWesley (1963)



4 NASA Earth Observatory Planetary Motion (2017)URL httpsearthobservatorynasagovFeatures

OrbitsHistory

5 NASA 100 Years of General Relativity (2017) URLhttpsasdgsfcnasagovblueshiftindexphp

20151125100-years-of-general-relativity

6 J Gleick and H Roberts Genius The Life and Science ofRichard Feynman LSB (1995)

7 S M Blinder Radiation Pulse from an Accelerated PointCharge (2017) URL httpdemonstrationswolfram

comRadiationPulseFromAnAcceleratedPointCharge

8 R N Shakhmuratov The energy storage in the formationof slow light Journal of Modern Optics 57 PII 920439128(2010)

9 E Courtens Giant Faraday Rotations In Self-induced Trans-parency Physical Review Letters 21 3 (1968)

10 M Sabooni Q Li L Rippe R K Mohan and S KrollSpectral Engineering of Slow Light Cavity Line Narrowingand Pulse Compression Physical Review Letters 111 183602(2013)

11 R N Shakhmuratov A Rebane P Megret and J OdeursSlow light with persistent hole burning Physical Review A71 053811 (2005)

12 A Walther A Amari S Kroll and A Kalachev Experimen-tal superradiance and slow-light effects for quantum memo-

103

References

ries Physical Review A 80 012317 (2009)

13 R P Feynman Quote from The Character of Physical Law -Part 6 Probability and uncertainty - the quantum mechanicalview of nature (1965) URL httpsenwikipediaorg

wikiThe_Character_of_Physical_Law

14 J J Sakurai and J Napolitano Modern quantum mechanicsPearson Education (2011)

15 M Born Zur Quantenmechanik der Stoszligvorgange [On thequantum mechanics of collisions] Zeitschrift fur Physik 37863ndash867 (1926)

16 Wikipedia Copenhagen interpretation (2017) URL https

enwikipediaorgwikiCopenhagen_interpretation

17 F Bloch Nuclear Induction Physical Review 70 460ndash474(1946)

18 R P Feynman F L Vernon and R W Hellwarth Geomet-rical Representation of the Schrodinger Equation for Solv-ing Maser Problems Journal of Applied Physics 28 49ndash52(1957)

19 L Allen and J H Eberly Optical resonance and two-levelatoms Wiley New York (1975)

20 P W Milonni and J H Eberly Lasers John Wiley amp SonsNew York (1988)

21 E Fraval M J Sellars and J J Longdell Dynamic de-coherence control of a solid-state nuclear-quadrupole qubitPhysical Review Letters 95 030506 (2005)

22 E Fraval Minimising the decoherence of rare earth ion solidstate spin qubits PhD thesis The Australian National Uni-versity (2005)

23 M J Zhong M P Hedges R L Ahlefeldt J GBartholomew S E Beavan S M Wittig J J Longdelland M J Sellars Optically addressable nuclear spins in asolid with a six-hour coherence time Nature 517 177ndashU121(2015)

24 M F Pascual-Winter R C Tongning T Chaneliere andJ L Le Gouet Spin coherence lifetime extension inTm3+YAG through dynamical decoupling Physical ReviewB 86 184301 (2012)

25 A Arcangeli M Lovric B Tumino A Ferrier and P Gold-ner Spectroscopy and coherence lifetime extension of hyper-fine transitions in 151Eu3+Y2SiO5 Physical Review B 89184305 (2013)

26 G Heinze C Hubrich and T Halfmann Stopped Light and

104

References

Image Storage by Electromagnetically Induced Transparencyup to the Regime of One Minute Physical Review Letters111 033601 (2013)

27 P W Milonni Why Spontaneous Emission American Jour-nal of Physics 52 340ndash343 (1984)

28 H B Callen and T A Welton Irreversibility and GeneralizedNoise Physical Review 83 34ndash40 (1951)

29 R R Puri Spontaneous Emission - Vacuum Fluctuations orRadiation Reaction Journal of the Optical Society of Amer-ica B-optical Physics 2 447ndash450 (1985)

30 P W Milonni Spontaneous Emission - Vacuum Fluctuationsor Radiation Reaction by Puri - Comment Journal of theOptical Society of America B-optical Physics 2 1953ndash1954(1985)

31 M Pollnau What we thought we knew and what we ought toknow about stimulated and spontaneous emission Presenta-tion given at Lund University (15th November 2017)

32 G Steele The Physics of Spontaneous Emission and Quan-tum Mechanical Decay Department of Physics MIT MA02139 (2000)

33 L Fonda G C Ghirardi and A Rimini Decay Theory ofUnstable Quantum Systems Reports on Progress in Physics41 587ndash631 (1978)

34 G C Ghirardi C Omero T Weber and A Rimini Small-time Behavior of Quantum Non-decay Probability and ZenoParadox in Quantum-mechanics Nuovo Cimento Della So-cieta Italiana Di Fisica A-nuclei Particles and Fields 52421ndash442 (1979)

35 J H Eberly N B Narozhny and J J SanchezmondragonPeriodic Spontaneous Collapse and Revival in a SimpleQuantum Model Physical Review Letters 44 1323ndash1326(1980)

36 Wikipedia Spontaneous emission (2017) URL httpsen

wikipediaorgwikiSpontaneous_emission

37 H Walther B T H Varcoe B G Englert and T BeckerCavity quantum electrodynamics Reports on Progress inPhysics 69 1325ndash1382 (2006)

38 Y C Sun Spectroscopic Properties of Rare Earths in OpticalMaterials - Chapter 7 Springer Series in Material Science(2005)

39 R M Macfarlane and R M Shelby Coherent transientand holeburning spectroscopy of rare earth ions in solids In

105

References

A Kaplyanskii and R Macfarlane editors Spectroscopy ofsolids containing rare earth ions North-Holland Amsterdam(1987)

40 A J Freeman and R E Watson Theoretical Investigationof Some Magnetic and Spectroscopic Properties of Rare-earthIons Physical Review 127 2058 (1962)

41 F Konz Y Sun C W Thiel R L Cone R W EquallR L Hutcheson and R M Macfarlane Temperature andconcentration dependence of optical dephasing spectral-holelifetime and anisotropic absorption in Eu3+ Y2SiO5 Phys-ical Review B 68 085109 (2003)

42 W M Yen A L Schawlow and W C Scott Phonon-induced Relaxation in Excited Optical States of TrivalentPraseodymium in Laf3 Physical Review A-general Physics136 A271 (1964)

43 S B Altner G Zumofen U P Wild and M MitsunagaPhoton-echo attenuation in rare-earth-ion-doped crystalsPhysical Review B 54 17493ndash17507 (1996)

44 Z Q Sun M S Li and Y C Zhou Thermal propertiesof single-phase Y2SiO5 Journal of the European CeramicSociety 29 551ndash557 (2009)

45 L Petersen High-resolution spectroscopy of praseodymiumions in a solid matrix towards single-ion detection sensitiv-ity PhD thesis ETH Zurich (2011)

46 G H Dieke and H M Crosswhite The Spectra of the Doublyand Triply Ionized Rare Earths Applied Optics 2 675ndash686(1963)

47 W T Carnall G L Goodman K Rajnak and R S Rana ASystematic Analysis of the Spectra of the Lanthanides Dopedinto Single-crystal Laf3 Journal of Chemical Physics 903443ndash3457 (1989)

48 R W Equall R L Cone and R M Macfarlane Ho-mogeneous Broadening and Hyperfine-structure of Optical-transitions In Pr3+Y2SiO5 Physical Review B 52 3963ndash3969 (1995)

49 M Nilsson L Rippe S Kroll R Klieber and D SuterHole-burning techniques for isolation and study of individ-ual hyperfine transitions in inhomogeneously broadened solidsdemonstrated in Pr3+Y2SiO5 Physical Review B 70 214116(2004)

50 T Aitasalo J Holsa M Lastusaari J Legendziewicz J Ni-ittykoski and F Pelle Delayed luminescence of Ce3+ dopedY2SiO5 Optical Materials 26 107ndash112 (2004)

106

References

51 R Beach M D Shinn L Davis R W Solarz andW F Krupke Optical-absorption and Stimulated-emission ofNeodymium In Yttrium Orthosilicate Ieee Journal of Quan-tum Electronics 26 1405ndash1412 (1990)

52 R C Hilborn Einstein Coefficients Cross-sections F Val-ues Dipole-moments and All That American Journal ofPhysics 50 982ndash986 (1982)

53 F R Graf A Renn G Zumofen and U P Wild Photon-echo attenuation by dynamical processes in rare-earth-ion-doped crystals Physical Review B 58 5462ndash5478 (1998)

54 R W Equall Y Sun R L Cone and R M Macfarlane Ul-traslow Optical Dephasing In Eu3+Y2SiO5 Physical ReviewLetters 72 2179ndash2181 (1994)

55 F R Graf A Renn U P Wild and M Mitsunaga Site in-terference in Stark-modulated photon echoes Physical ReviewB 55 11225ndash11229 (1997)

56 S E Beavan E A Goldschmidt and M J Sellars Demon-stration of a dynamic bandpass frequency filter in a rare-earthion-doped crystal Journal of the Optical Society of AmericaB-optical Physics 30 1173ndash1177 (2013)

57 M P Hedges High performance solid state quantum memoryPhD thesis Australian National University (2011)

58 K Holliday M Croci E Vauthey and U P Wild Spec-tral Hole-burning and Holography in an Y2SiO5Pr3+ Crys-tal Physical Review B 47 14741ndash14752 (1993)

59 H de Riedmatten and M Afzelius Quantum Light Stor-age in Solid State Atomic Ensembles In A Predojevic andM MW editors Engineering the Atom-Photon InteractionSpringer New York (2015)

60 L Rippe B Julsgaard A Walther Y Ying and S Kroll Ex-perimental quantum-state tomography of a solid-state qubitPhysical Review A 77 022307 (2008)

61 H Zhang M Sabooni L Rippe C Kim S Kroll L VWang and P R Hemmer Slow light for deep tissue imagingwith ultrasound modulation Applied Physics Letters 100131102 (2012)

62 M J Digonnet Rare-earth-doped fiber lasers and amplifiersrevised and expanded CRC press (2001)

63 C W Thiel R L Cone and T Boettger Laser linewidthnarrowing using transient spectral hole burning Journal ofLuminescence 152 84ndash87 (2014)

64 A Walther L Rippe L V Wang S Andersson-Engels and

107

References

S Kroll Analysis of the potential for non-invasive imaging ofoxygenation at heart depth using ultrasound optical tomogra-phy (UOT) or photo-acoustic tomography (PAT) BiomedicalOptics Express 8 4523ndash4536 (2017)

65 C Clausen F Bussieres M Afzelius and N Gisin QuantumStorage of Heralded Polarization Qubits in Birefringent andAnisotropically Absorbing Materials Physical Review Letters108 190503 (2012)

66 J H Wesenberg K Moslashlmer L Rippe and S Kroll Scal-able designs for quantum computing with rare-earth-ion-dopedcrystals Physical Review A 75 012304 (2007)

67 A Orieux and E Diamanti Recent advances on integratedquantum communications Journal of Optics 18 083002(2016)

68 R P Feynman Simulating Physics with Computers Inter-national Journal of Theoretical Physics 21 467ndash488 (1982)

69 S Lloyd Universal quantum simulators Science 273 1073ndash1078 (1996)

70 R P Poplavskii Thermodynamic models of information pro-cesses Soviet Physics Uspekhi 18 222 (1975)

71 Y Manin Computable and Uncomputable (in Russian)Sovetskoye Radio Moscow (1980)

72 Wikipedia Timeline of quantum computing (2017) URLhttpsenwikipediaorgwikiTimeline_of_quantum_

computing

73 P W Shor Algorithms for Quantum Computation - Dis-crete Logarithms and Factoring 35th Annual Symposium onFoundations of Computer Science Proceedings pages 124ndash134 (1994)

74 L K Grover A fast quantum mechanical algorithm fordatabase search In Proceedings of the twenty-eighth annualACM symposium on Theory of computing pages 212ndash219ACM (1996)

75 T Kadowaki and H Nishimori Quantum annealing in thetransverse Ising model Physical Review E 58 5355 (1998)

76 IBM Quantum Computing (2017) URL httpswww

researchibmcomibm-q

77 Google Quantum AI (2017) URL httpsresearch

googlecompubsQuantumAIhtml

78 Microsoft Quantum Computing (2017) URL httpswww

microsoftcomen-usquantum

108

References

79 Intel Quantum Computing (2017) URL httpsnewsroom

intelcompress-kitsquantum-computing

80 D-Wave The Quantum Computing Company (2017) URLhttpswwwdwavesyscomhome

81 D P Divincenzo Topics in quantum computers MesoscopicElectron Transport 345 657ndash677 (1997)

82 D P DiVincenzo The physical implementation of quantumcomputation Fortschritte Der Physik-progress of Physics 48771ndash783 (2000)

83 Quantum Technologies Roadmap 2015 edition of theEuropean roadmap for Quantum Information Processing andCommunication (2015) URL httpwwwosaorgen-us

corporate_gatewaypublicationspublication_report_

libraryquantum_technologies_roadmap

84 European Commission Quantum Technol-ogy Flagship (2017) URL httpsec

europaeudigital-single-marketenpolicies

quantum-technologies

85 T Zhong J M Kindem J Rochman and A Faraon Inter-facing broadband photonic qubits to on-chip cavity-protectedrare-earth ensembles Nature Communications 8 (2017)

86 T Zhong J M Kindem J G Bartholomew J RochmanI Craiciu E Miyazono M Bettinelli E Cavalli V VermaS W Nam F Marsili M D Shaw A D Beyer andA Faraon Nanophotonic rare-earth quantum memory withoptically controlled retrieval Science 357 1392ndash1395 (2017)

87 J Karlsson L Rippe and S Kroll A confocal optical mi-croscope for detection of single impurities in a bulk crystal atcryogenic temperatures Review of Scientific Instruments 87033701 (2016)

88 R M Macfarlane High-resolution laser spectroscopy of rare-earth doped insulators a personal perspective Journal ofLuminescence 100 PII S0022ndash2313(02)00450ndash7 (2002)

89 M S Silver R I Joseph C N Chen V J Sank and D IHoult Selective-population Inversion in Nmr Nature 310681ndash683 (1984)

90 M S Silver R I Joseph and D I Hoult Selective SpinInversion in Nuclear Magnetic-resonance and Coherent Op-tics through an Exact Solution of the Bloch-riccati EquationPhysical Review A 31 2753ndash2755 (1985)

91 I Roos and K Moslashlmer Quantum computing with an inhomo-geneously broadened ensemble of ions Suppression of errorsfrom detuning variations by specially adapted pulses and co-

109

References

herent population trapping Physical Review A 69 022321(2004)

92 B Lauritzen N Timoney N Gisin M Afzelius H de Ried-matten Y Sun R M Macfarlane and R L Cone Spectro-scopic investigations of Eu3+Y2SiO5 for quantum memoryapplications Physical Review B 85 115111 (2012)

93 P Helmer High fidelity operations in a europium doped in-organic crystal Bachelorrsquos thesis Lund University (2015)

94 M A Nielsen and I L Chuang Quantum Computation andQuantum Information Cambridge University Press 10th an-niversary edition (2010)

95 National centers for environmental information Geomag-netism (2017) URL httpswwwngdcnoaagovgeomag

magfieldshtml

96 B S Ham M S Shahriar M K Kim and P R HemmerFrequency-selective time-domain optical data storage by elec-tromagnetically induced transparency in a rare-earth-dopedsolid Optics Letters 22 1849ndash1851 (1997)

97 D Serrano Y Yan J Karlsson L Rippe A WaltherS Kroll A Ferrier and P Goldner Impact of the ion-ionenergy transfer on quantum computing schemes in rare-earthdoped solids Journal of Luminescence 151 93ndash99 (2014)

98 I Kurkin and K Chernov EPR and spin-lattice relaxation ofrare-earth activated centres in Y2SiO5 single crystals PhysicaB+C 101 233ndash238 (1980)

99 M S Buryi V V Laguta D V Savchenko and M NiklElectron Paramagnetic Resonance study of Lu2SiO5 andY2SiO5 scintillators doped by cerium Advanced Science En-gineering and Medicine 5 573ndash576 (2013)

100 R Kolesov K Xia R Reuter R Stohr A Zappe J MeijerP R Hemmer and J Wrachtrup Optical detection of a singlerare-earth ion in a crystal Nature Communications 3 1029(2012)

101 R Kolesov K Xia R Reuter M Jamali R Stohr T InalP Siyushev and J Wrachtrup Mapping Spin Coherence of aSingle Rare-Earth Ion in a Crystal onto a Single Photon Po-larization State Physical Review Letters 111 120502 (2013)

102 P Siyushev K Xia R Reuter M Jamali N Zhao N YangC Duan N Kukharchyk A D Wieck R Kolesov andJ Wrachtrup Coherent properties of single rare-earth spinqubits Nature Communications 5 3895 (2014)

103 K Xia R Kolesov Y Wang P Siyushev R Reuter T Ko-rnher N Kukharchyk A D Wieck B Villa S Yang and

110

References

J Wrachtrup All-Optical Preparation of Coherent DarkStates of a Single Rare Earth Ion Spin in a Crystal PhysicalReview Letters 115 093602 (2015)

104 T Utikal E Eichhammer L Petersen A Renn S Goet-zinger and V Sandoghdar Spectroscopic detection and statepreparation of a single praseodymium ion in a crystal NatureCommunications 5 3627 (2014)

105 E Eichhammer T Utikal S Gotzinger and V SandoghdarSpectroscopic detection of single Pr3+ ions on the 3H4-1D2

transition New Journal of Physics 17 083018 (2015)

106 I Nakamura T Yoshihiro H Inagawa S Fujiyoshi andM Matsushita Spectroscopy of single Pr3+ ion in LaF3 crys-tal at 15 K Scientific Reports 4 7364 (2014)

107 L Rippe Quantum Computing With Naturally Trapped Sub-Nanometre-Spaced Ions PhD thesis Lund University (2006)

108 A Walther Coherent Processes In Rare-Earth-Ion-DopedSolids PhD thesis Lund University (2009)

109 M Sabooni Efficient Quantum Memories Based on Spec-tral Engineering of Rare-Earth-Ion-Doped Solids PhD thesisLund University (2013)

110 J Karlsson Cerium as a quantum state probe for rare-earthqubits in a crystal PhD thesis Lund University (2015)

111 E D Black An introduction to Pound-Drever-Hall laser fre-quency stabilization American Journal of Physics 69 79ndash87(2001)

112 X Zhao Diode laser frequency stabilization onto an opticalcavity Masterrsquos thesis Lund University Lund Reports onAtomic Physics LRAP-473 (2013)

113 Y Yan Towards single Ce ion detection in a bulk crystal forthe development of a single-ion qubit readout scheme PhDthesis Lund University (2013)

111

AD

AM

KIN

OS

Light-M


Com

puting in Rare-E

arth-Ion-Doped C

rystals 2018

978

9177

5354

30







Prin

ted

by M

edia

-Try

ck L

und

2018

N

ORD

IC S

WA

N E

CO

LABE

L 3

041

0903


Abstract

Populaumlrvetenskaplig sammanfattning


Abbreviations

Contents

1 Introduction




2 Understanding light-matter interaction








221 Slow light




3 Understanding the quantum mechanical nature of atoms



32 Light-matter interaction in quantum mechanics

321 Bloch sphere



33 Decays

331 Coherence time

332 Lifetime

34 DC Stark effect

4 Rare-earth-ion-doped crystals






44 An overview of praseodymium (Pr) and europium (Eu) properties in Y2SiO5


5 Light-matter interaction in rare-earth-ion-doped crystals


52 Development and characterization of high suppression and high eacutetendue narrowband spectral filters (Paper II)




6 A brief overview of quantum computation


62 DiVincenzos criteria


7 Quantum computation in rare-earth-ion-doped crystals








75 Cerium (Ce) as a dedicated readout ion (Paper VIII)


77 Progress on DiVincenzos criteria


8 Conclusions


A Equipment

A1 Main setup

A11 Dye laser setup



A2 Cryostat


The authors contribution to the papers

Acknowledgements

AD

AM

KIN

OS

Light-M


Com

puting in Rare-E

arth-Ion-Doped C

rystals 2018

978

9177

5354

30







Prin

ted

by M

edia

-Try

ck L

und

2018

N

ORD

IC S

WA

N E

CO

LABE

L 3

041

0903




Adam Kinos

Doctoral Thesis

2018





Organization

LUND UNIVERSITY


Document name




Author

Adam Kinos

Title and subtitle


Abstract




Language

English

ISSN and key title


ISBN

978-91-7753-543-0


219

Price



















pulses










Adam Kinos

Doctoral Thesis

2018







Abstract





iii


iv






v




vi








vii











PLOS One 10 e0121905 (2015)




viii







Sci Rep 6 30410 (2016)



ACS Nano 10 (11) 9823-9830 (2016)

ix

Abbreviations


BP Band Pass filter


CNOT Controlled NOT

DM Dichroic Mirror





OD Optical Diode



PH Pinhole





Ce Cerium

Eu Europium

Nd Neodymium

Pr Praseodymium

Y2O3 Yttrium Oxide


xi

Contents

Abstract iii


















Contents















Contents

Papers









Chapter 1

Introduction






1










2

Introduction









3

Chapter 2





5











E(r) =1

4πε0

q

r2er (21)




6




7


(a)

(b)

120579











9


















10


119812in t 119851

119911

119909

120596 1205960







qeE0


eminusiωt (25)



exx(tminus rc) (26)




4πε0c2rx0exe

minusiω(tminusrc)

asymp q2eω

2

4πε0c2rme

1


(27)




11










12


119876

119875

120588119903

119911119889120588

120578


Re

Im

1205790 =120596119911

119888

radius =119888

120596








ρ=0

qeω2

4πε0c2x0exe

minusiω(tminusrc)

rη2πρdρ (28)



Eradtot(t r) =qeω

2ηx0exeminusiωt

2ε0c2

int r=infin

r=z

eiωrcdr (29)







13


|119812in|

|119812rad tot|



119812in

119812rad tot

119812after



Eradtot(t r) =q2eη

2ε0cme

iω









14





asymp(


c

)E0exe

minusiω(tminuszc)


(211)


nsparse asymp 1 +1

2

q2eN

ε0me

1





Eafter(t r) = exp

[minus q2

eN

2ε0meγc∆z

]Ein(t r)

= exp[minusα

2∆z]Ein(t r)

(213)


15







n1 sin θ1 = n2 sin θ2 (214)








16



221 Slow light









partω(215)

17


a

b

c No dispersion

d Dispersion

Pulse location

Pulse location

Pulse location






18






vg =c

1 + Uhost+Ures

Uvac

(216)


n = 1 +Uhost

Uvac(217)



vg asymp2πΓ

α(218)





19



P(t r) =Nq2

e








P(t r) =Nαpol(ω)



αpol(ω) =q2e

ε0me

sum

k

fkω2



nabla middotE =ρ



partt

c2nablatimesB =

(j

ε0+partE

partt

)(222)


nabla =(partpartx

partparty

partpartz



partt

20




nabla middotB = 0


partt

c2nablatimesB =part

partt

(P

ε0+ E

)(223)



k2Eavg =ω2

c2Eavg +

ω2

ε0c2P

k2 =ω2

c2

(1 +

Nαpol(ω)


) (224)


n2 = 1 +Nαpol(ω)







3

(n2 minus 1

n2 + 2

)=sum

j

Njαpol j(ω) (226)

21







22

Chapter 3








23






F = ma (31)






i~partΨ(t r)




24


0 L 119909

119881 = infin119881 = 0

119881 = infin











H0 =minus~2

2m

part2

partx2+ V (x) (35)


2mv2x = 1





0 lt x lt L (36)


where A =radic


25



En =~2

2m

π2n2

L2(37)




Φn(px) =1radic2π~

intΨn(x)eminus

i~pxxdx (38)


∆x∆px ge~2

(39)

∆t∆E ge ~2

(310)



26








HI(t) = minus1

2qer middotE0e



|Ψ〉 = α|e〉+ β|g〉|α|2 + |β|2 = 1

(312)


|Ψ〉 = cosθ

2|e〉+ eiφ sin

θ

2|g〉 (313)


ρ = |Ψ〉〈Ψ| (314)


(10

)and |g〉 =

(01

) this becomes

ρ =

(αβ

)(αlowast βlowast

)=

(|α|2 αβlowast

αlowastβ |β|2)

=

(ρee ρegρge ρgg

)(315)

27

321 Bloch sphere

w

z

v

R

|e

uy

x



u = ρeg + ρge


(316)


u = minus u

T2minus δv + ΩImw

v = δuminus v

T2+ ΩRew


T1

(317)




~(318)



321 Bloch sphere


28


R2

|g

R1

W1

W2

z |e

yx





R = W timesR (319)







29





|Ψ(t)〉 = α|e〉+ βeiω0t|g〉 (320)



radic18 β =


radic2 β = 1

radic2 and case 4


radic2


30



radic18 β =


radic2

β = 1radic

2 and case 4 α = 1 β = 0

31








0 + iEIm0





(part

partz+n

c

part

partt

)ΩRe =

α0

2π

int +infin

minusinfing(δ)vdδ

(part

partz+n

c

part

partt

)ΩIm =

α0

2π

int +infin

minusinfing(δ)udδ

(322)





32


|g

R

(a)

W

z |e

yx

|g

(b)

R

W

z |e

yx

|g

R

(c)

W

z |e

yx



∆ΩRe =α0


∆ΩIm =α0


(323)





33

33 Decays

R2

|g

z |e

y

R1

x


33 Decays


331 Coherence time



Γhom =1

πT2(324)


34


a b





332 Lifetime


35

332 Lifetime






36





34 DC Stark effect



37

Chapter 4









39


21Sc

39Y








P (ωp) =1

exp(

~ωp

kBT

)minus 1

(41)


40






41


50

48

46

44

42

40

38

36

34

32

30

28

26

24

22

20

18

16

14

12

10

8

6

4

2

0

Ener

gy [103

cm-1

]

1500

1440

1380

1320

1260

1200

1140

1080

1020

960

900

840

780

720

660

600

540

480

420

360

300

240

180

120

60

0

Frequ

ency [TH

z]200

300

400

500

600

700

8009001000

2000

300040005000

Wavelen

gth [n

m]


42


plusmn12g

plusmn32g

plusmn52g

plusmn12e

plusmn32e

plusmn52e

606 nm

495 THz

102 MHz

173 MHz

46 MHz

48 MHz


10265degab

c

1041 nm

79deg

C2 (b)

D1

D2











43


Δ1206412static

124deg

Δ1206414static

Δ1206411static

Δ1206413static

C2 (b)


i in








∆ω =(microstatic


~=


(42)




44









45


Frequency [MHz]

046

94102

148196

275321

369

Ab

sorp

tio

n

~kHz


Frequency

Ab

sorp

tio

n

~GHz

Crystal










46

Chapter 5









47













48


1198631

1198632

120641119890119892119916in


(d) Propagation

1198631

1198632

(b) Absorption

119916in

119916P

119916res

1198631

1198632

(c) Projection on 1198631 1198632

119916res

1198631

1198632

(e) Propagation

746 plusmn 19deg

1198632

1198631




R =n2D2minus n2

D1

χabs(51)


49


Ele

ctric

fiel

d al

ong

(a)

D2

Ele

ctric

fiel

d al

ong

(c)

D2

Ele

ctric

fiel

d al

ong

(b)

D2





D2minus n2

D1= (nD2

+ nD1)(nD2

minus nD1) asymp











50







ID2= 10minus5







51


(Paper II)


Double pass AOM

Polarizer Polarizer


PD2


Cryostat

Shutter

119887

1198632

1198631


PD1

Pr3+ Y2SiO5









52










53











54



0

1

2

3

Ab

sorp

tio

n

L (a) - 15 V- 5 V0 V5 V15 V

-4 -3 -2 -1 0 1 2 3 4Frequency [MHz]

FF

T In

ten

sity

(b)





55








56











57












58










59

Chapter 6








61















62















63



64

|0rang

|1rang

|auxrang

|119890rang

606 nm

495 THz

102 MHz

173 MHz

46 MHz

48 MHz

1205960 Ω0 1206010

1205961 Ω1 1206011



Chapter 7








65














66



|0

R0

Re

z |e

yx






tFWHM asymp176

β(72)

νwidth =microβ

π(73)






67


|1

|B

|D

z |0

y

x








|B〉 =1radic2


)

|D〉 =1radic2


) (74)



68


|1

R0

R

|D

z |0

y

x












ρmixed =1

2(ρ0 + ρ1) =

1

2(ρB + ρD) (76)


69


Time

1205872 120587 read

Spin echo sequence

|119890⟩

1

|0⟩

|119890⟩

1

|0⟩

|119890⟩

1

|0⟩

ൗ120591 2 ൗ120591 2



2


) In other





70


y

|1

x1

z |0(a)

y

|1

x2

z |0(b)

y

|1

1x2

z |0(c)








71












72



|0rang

|1rang

|119890rang

1 3

(a)

|0rang

|1rang

|119890rang

2 2

Control Target

|0rang

|1rang

|119890rangΔ120584

2 2

|0rang

|1rang

|119890rang

1 3

(b)

Control Target




73


|0rang

|1rang

|119890rang

1 3

Control

|auxrang

|0rang

|1rang

|32119890rang

2

Target

|auxrang

4

|12119890rang

|52119890rang

4648

102

173

[MHz]










74










75


q7

q6

q5 q3

q4

q1

q2

Buffer

Readout


|0rang

|1rang

|119890rang

1 3

Qubit 1

|0rang

|1rang

|119890rang

2

Buffer

Δ120584

4

Readout






76







77



78



2F52

2F72

371 nm

5d(2D52

2D32)

B

B

Δ1205845119889

Δ1205844119891


Cryostat

SMF

BPPH

X

Z

Y

SPAD

DM

PDHECDLAOM

Lens

Crystal





79
















80






81










82












83






84

Chapter 8

Conclusions






85








86

Conclusions


87

Appendix A

Equipment



A1 Main setup


89

A11 Dye laser setup

Dye laser table

λ2-plate

Fiber

AOM1

Wavemeter

EOM

AOM2

AOM3

λ2-plate

Polarizer



OD

Pumplaser

Dye laser

Filterwheel

MirrorBeam splitter

Lens


A11 Dye laser setup




90

Equipment

Locking platform


table

λ2-plate

Error detector


CCD camera

λ4-plate

PBSLens

Cavity






91









92

Equipment

Polarizer

λ2-plate


Cryostat

Reference detector

Crystal


table

Lens

Experimentaltable

SE

N W








A2 Cryostat


93


λ4-plate

Ce laser table


CCDcamera

Cavity

PBS

Errordetector

EOM

λ2-plate

ECDL

OD









94

Equipment




95







97








98








99

Acknowledgements







101







102

References





OrbitsHistory











103

References


















104

References
















105

References













106

References















107

References










computing





researchibmcomibm-q





108

References



















109

References






magfieldshtml









110

References













111

AD

AM

KIN

OS

Light-M


Com

puting in Rare-E

arth-Ion-Doped C

rystals 2018

978

9177

5354

30







Prin

ted

by M

edia

-Try

ck L

und

2018

N

ORD

IC S

WA

N E

CO

LABE

L 3

041

0903


Abstract



Abbreviations

Contents

1 Introduction












221 Slow light








321 Bloch sphere



33 Decays

331 Coherence time

332 Lifetime

34 DC Stark effect































8 Conclusions


A Equipment

A1 Main setup

A11 Dye laser setup



A2 Cryostat



Acknowledgements



Adam Kinos

Doctoral Thesis

2018





Organization

LUND UNIVERSITY


Document name




Author

Adam Kinos

Title and subtitle


Abstract




Language

English

ISSN and key title


ISBN

978-91-7753-543-0


219

Price



















pulses










Adam Kinos

Doctoral Thesis

2018







Abstract





iii


iv






v




vi








vii











PLOS One 10 e0121905 (2015)




viii







Sci Rep 6 30410 (2016)



ACS Nano 10 (11) 9823-9830 (2016)

ix

Abbreviations


BP Band Pass filter


CNOT Controlled NOT

DM Dichroic Mirror





OD Optical Diode



PH Pinhole





Ce Cerium

Eu Europium

Nd Neodymium

Pr Praseodymium

Y2O3 Yttrium Oxide


xi

Contents

Abstract iii


















Contents















Contents

Papers









Chapter 1

Introduction






1










2

Introduction









3

Chapter 2





5











E(r) =1

4πε0

q

r2er (21)




6




7


(a)

(b)

120579











9


















10


119812in t 119851

119911

119909

120596 1205960







qeE0


eminusiωt (25)



exx(tminus rc) (26)




4πε0c2rx0exe

minusiω(tminusrc)

asymp q2eω

2

4πε0c2rme

1


(27)




11










12


119876

119875

120588119903

119911119889120588

120578


Re

Im

1205790 =120596119911

119888

radius =119888

120596








ρ=0

qeω2

4πε0c2x0exe

minusiω(tminusrc)

rη2πρdρ (28)



Eradtot(t r) =qeω

2ηx0exeminusiωt

2ε0c2

int r=infin

r=z

eiωrcdr (29)







13


|119812in|

|119812rad tot|



119812in

119812rad tot

119812after



Eradtot(t r) =q2eη

2ε0cme

iω









14





asymp(


c

)E0exe

minusiω(tminuszc)


(211)


nsparse asymp 1 +1

2

q2eN

ε0me

1





Eafter(t r) = exp

[minus q2

eN

2ε0meγc∆z

]Ein(t r)

= exp[minusα

2∆z]Ein(t r)

(213)


15







n1 sin θ1 = n2 sin θ2 (214)








16



221 Slow light









partω(215)

17


a

b

c No dispersion

d Dispersion

Pulse location

Pulse location

Pulse location






18






vg =c

1 + Uhost+Ures

Uvac

(216)


n = 1 +Uhost

Uvac(217)



vg asymp2πΓ

α(218)





19



P(t r) =Nq2

e








P(t r) =Nαpol(ω)



αpol(ω) =q2e

ε0me

sum

k

fkω2



nabla middotE =ρ



partt

c2nablatimesB =

(j

ε0+partE

partt

)(222)


nabla =(partpartx

partparty

partpartz



partt

20




nabla middotB = 0


partt

c2nablatimesB =part

partt

(P

ε0+ E

)(223)



k2Eavg =ω2

c2Eavg +

ω2

ε0c2P

k2 =ω2

c2

(1 +

Nαpol(ω)


) (224)


n2 = 1 +Nαpol(ω)







3

(n2 minus 1

n2 + 2

)=sum

j

Njαpol j(ω) (226)

21







22

Chapter 3








23






F = ma (31)






i~partΨ(t r)




24


0 L 119909

119881 = infin119881 = 0

119881 = infin











H0 =minus~2

2m

part2

partx2+ V (x) (35)


2mv2x = 1





0 lt x lt L (36)


where A =radic


25



En =~2

2m

π2n2

L2(37)




Φn(px) =1radic2π~

intΨn(x)eminus

i~pxxdx (38)


∆x∆px ge~2

(39)

∆t∆E ge ~2

(310)



26








HI(t) = minus1

2qer middotE0e



|Ψ〉 = α|e〉+ β|g〉|α|2 + |β|2 = 1

(312)


|Ψ〉 = cosθ

2|e〉+ eiφ sin

θ

2|g〉 (313)


ρ = |Ψ〉〈Ψ| (314)


(10

)and |g〉 =

(01

) this becomes

ρ =

(αβ

)(αlowast βlowast

)=

(|α|2 αβlowast

αlowastβ |β|2)

=

(ρee ρegρge ρgg

)(315)

27

321 Bloch sphere

w

z

v

R

|e

uy

x



u = ρeg + ρge


(316)


u = minus u

T2minus δv + ΩImw

v = δuminus v

T2+ ΩRew


T1

(317)




~(318)



321 Bloch sphere


28


R2

|g

R1

W1

W2

z |e

yx





R = W timesR (319)







29





|Ψ(t)〉 = α|e〉+ βeiω0t|g〉 (320)



radic18 β =


radic2 β = 1

radic2 and case 4


radic2


30



radic18 β =


radic2

β = 1radic

2 and case 4 α = 1 β = 0

31








0 + iEIm0





(part

partz+n

c

part

partt

)ΩRe =

α0

2π

int +infin

minusinfing(δ)vdδ

(part

partz+n

c

part

partt

)ΩIm =

α0

2π

int +infin

minusinfing(δ)udδ

(322)





32


|g

R

(a)

W

z |e

yx

|g

(b)

R

W

z |e

yx

|g

R

(c)

W

z |e

yx



∆ΩRe =α0


∆ΩIm =α0


(323)





33

33 Decays

R2

|g

z |e

y

R1

x


33 Decays


331 Coherence time



Γhom =1

πT2(324)


34


a b





332 Lifetime


35

332 Lifetime






36





34 DC Stark effect



37

Chapter 4









39


21Sc

39Y








P (ωp) =1

exp(

~ωp

kBT

)minus 1

(41)


40






41


50

48

46

44

42

40

38

36

34

32

30

28

26

24

22

20

18

16

14

12

10

8

6

4

2

0

Ener

gy [103

cm-1

]

1500

1440

1380

1320

1260

1200

1140

1080

1020

960

900

840

780

720

660

600

540

480

420

360

300

240

180

120

60

0

Frequ

ency [TH

z]200

300

400

500

600

700

8009001000

2000

300040005000

Wavelen

gth [n

m]


42


plusmn12g

plusmn32g

plusmn52g

plusmn12e

plusmn32e

plusmn52e

606 nm

495 THz

102 MHz

173 MHz

46 MHz

48 MHz


10265degab

c

1041 nm

79deg

C2 (b)

D1

D2











43


Δ1206412static

124deg

Δ1206414static

Δ1206411static

Δ1206413static

C2 (b)


i in








∆ω =(microstatic


~=


(42)




44









45


Frequency [MHz]

046

94102

148196

275321

369

Ab

sorp

tio

n

~kHz


Frequency

Ab

sorp

tio

n

~GHz

Crystal










46

Chapter 5









47













48


1198631

1198632

120641119890119892119916in


(d) Propagation

1198631

1198632

(b) Absorption

119916in

119916P

119916res

1198631

1198632

(c) Projection on 1198631 1198632

119916res

1198631

1198632

(e) Propagation

746 plusmn 19deg

1198632

1198631




R =n2D2minus n2

D1

χabs(51)


49


Ele

ctric

fiel

d al

ong

(a)

D2

Ele

ctric

fiel

d al

ong

(c)

D2

Ele

ctric

fiel

d al

ong

(b)

D2





D2minus n2

D1= (nD2

+ nD1)(nD2

minus nD1) asymp











50







ID2= 10minus5







51


(Paper II)


Double pass AOM

Polarizer Polarizer


PD2


Cryostat

Shutter

119887

1198632

1198631


PD1

Pr3+ Y2SiO5









52










53











54



0

1

2

3

Ab

sorp

tio

n

L (a) - 15 V- 5 V0 V5 V15 V

-4 -3 -2 -1 0 1 2 3 4Frequency [MHz]

FF

T In

ten

sity

(b)





55








56











57












58










59

Chapter 6








61















62















63



64

|0rang

|1rang

|auxrang

|119890rang

606 nm

495 THz

102 MHz

173 MHz

46 MHz

48 MHz

1205960 Ω0 1206010

1205961 Ω1 1206011



Chapter 7








65














66



|0

R0

Re

z |e

yx






tFWHM asymp176

β(72)

νwidth =microβ

π(73)






67


|1

|B

|D

z |0

y

x








|B〉 =1radic2


)

|D〉 =1radic2


) (74)



68


|1

R0

R

|D

z |0

y

x












ρmixed =1

2(ρ0 + ρ1) =

1

2(ρB + ρD) (76)


69


Time

1205872 120587 read

Spin echo sequence

|119890⟩

1

|0⟩

|119890⟩

1

|0⟩

|119890⟩

1

|0⟩

ൗ120591 2 ൗ120591 2



2


) In other





70


y

|1

x1

z |0(a)

y

|1

x2

z |0(b)

y

|1

1x2

z |0(c)








71












72



|0rang

|1rang

|119890rang

1 3

(a)

|0rang

|1rang

|119890rang

2 2

Control Target

|0rang

|1rang

|119890rangΔ120584

2 2

|0rang

|1rang

|119890rang

1 3

(b)

Control Target




73


|0rang

|1rang

|119890rang

1 3

Control

|auxrang

|0rang

|1rang

|32119890rang

2

Target

|auxrang

4

|12119890rang

|52119890rang

4648

102

173

[MHz]










74










75


q7

q6

q5 q3

q4

q1

q2

Buffer

Readout


|0rang

|1rang

|119890rang

1 3

Qubit 1

|0rang

|1rang

|119890rang

2

Buffer

Δ120584

4

Readout






76







77



78



2F52

2F72

371 nm

5d(2D52

2D32)

B

B

Δ1205845119889

Δ1205844119891


Cryostat

SMF

BPPH

X

Z

Y

SPAD

DM

PDHECDLAOM

Lens

Crystal





79
















80






81










82












83






84

Chapter 8

Conclusions






85








86

Conclusions


87

Appendix A

Equipment



A1 Main setup


89

A11 Dye laser setup

Dye laser table

λ2-plate

Fiber

AOM1

Wavemeter

EOM

AOM2

AOM3

λ2-plate

Polarizer



OD

Pumplaser

Dye laser

Filterwheel

MirrorBeam splitter

Lens


A11 Dye laser setup




90

Equipment

Locking platform


table

λ2-plate

Error detector


CCD camera

λ4-plate

PBSLens

Cavity






91









92

Equipment

Polarizer

λ2-plate


Cryostat

Reference detector

Crystal


table

Lens

Experimentaltable

SE

N W








A2 Cryostat


93


λ4-plate

Ce laser table


CCDcamera

Cavity

PBS

Errordetector

EOM

λ2-plate

ECDL

OD









94

Equipment




95







97








98








99

Acknowledgements







101







102

References





OrbitsHistory











103

References


















104

References
















105

References













106

References















107

References










computing





researchibmcomibm-q





108

References



















109

References






magfieldshtml









110

References













111

AD

AM

KIN

OS

Light-M


Com

puting in Rare-E

arth-Ion-Doped C

rystals 2018

978

9177

5354

30







Prin

ted

by M

edia

-Try

ck L

und

2018

N

ORD

IC S

WA

N E

CO

LABE

L 3

041

0903


Abstract



Abbreviations

Contents

1 Introduction












221 Slow light








321 Bloch sphere



33 Decays

331 Coherence time

332 Lifetime

34 DC Stark effect































8 Conclusions


A Equipment

A1 Main setup

A11 Dye laser setup



A2 Cryostat



Acknowledgements

Organization

LUND UNIVERSITY


Document name




Author

Adam Kinos

Title and subtitle


Abstract




Language

English

ISSN and key title


ISBN

978-91-7753-543-0


219

Price



















pulses










Adam Kinos

Doctoral Thesis

2018







Abstract





iii


iv






v




vi








vii











PLOS One 10 e0121905 (2015)




viii







Sci Rep 6 30410 (2016)



ACS Nano 10 (11) 9823-9830 (2016)

ix

Abbreviations


BP Band Pass filter


CNOT Controlled NOT

DM Dichroic Mirror





OD Optical Diode



PH Pinhole





Ce Cerium

Eu Europium

Nd Neodymium

Pr Praseodymium

Y2O3 Yttrium Oxide


xi

Contents

Abstract iii


















Contents















Contents

Papers









Chapter 1

Introduction






1










2

Introduction









3

Chapter 2





5











E(r) =1

4πε0

q

r2er (21)




6




7


(a)

(b)

120579











9


















10


119812in t 119851

119911

119909

120596 1205960







qeE0


eminusiωt (25)



exx(tminus rc) (26)




4πε0c2rx0exe

minusiω(tminusrc)

asymp q2eω

2

4πε0c2rme

1


(27)




11










12


119876

119875

120588119903

119911119889120588

120578


Re

Im

1205790 =120596119911

119888

radius =119888

120596








ρ=0

qeω2

4πε0c2x0exe

minusiω(tminusrc)

rη2πρdρ (28)



Eradtot(t r) =qeω

2ηx0exeminusiωt

2ε0c2

int r=infin

r=z

eiωrcdr (29)







13


|119812in|

|119812rad tot|



119812in

119812rad tot

119812after



Eradtot(t r) =q2eη

2ε0cme

iω









14





asymp(


c

)E0exe

minusiω(tminuszc)


(211)


nsparse asymp 1 +1

2

q2eN

ε0me

1





Eafter(t r) = exp

[minus q2

eN

2ε0meγc∆z

]Ein(t r)

= exp[minusα

2∆z]Ein(t r)

(213)


15







n1 sin θ1 = n2 sin θ2 (214)








16



221 Slow light









partω(215)

17


a

b

c No dispersion

d Dispersion

Pulse location

Pulse location

Pulse location






18






vg =c

1 + Uhost+Ures

Uvac

(216)


n = 1 +Uhost

Uvac(217)



vg asymp2πΓ

α(218)





19



P(t r) =Nq2

e








P(t r) =Nαpol(ω)



αpol(ω) =q2e

ε0me

sum

k

fkω2



nabla middotE =ρ



partt

c2nablatimesB =

(j

ε0+partE

partt

)(222)


nabla =(partpartx

partparty

partpartz



partt

20




nabla middotB = 0


partt

c2nablatimesB =part

partt

(P

ε0+ E

)(223)



k2Eavg =ω2

c2Eavg +

ω2

ε0c2P

k2 =ω2

c2

(1 +

Nαpol(ω)


) (224)


n2 = 1 +Nαpol(ω)







3

(n2 minus 1

n2 + 2

)=sum

j

Njαpol j(ω) (226)

21







22

Chapter 3








23






F = ma (31)






i~partΨ(t r)




24


0 L 119909

119881 = infin119881 = 0

119881 = infin











H0 =minus~2

2m

part2

partx2+ V (x) (35)


2mv2x = 1





0 lt x lt L (36)


where A =radic


25



En =~2

2m

π2n2

L2(37)




Φn(px) =1radic2π~

intΨn(x)eminus

i~pxxdx (38)


∆x∆px ge~2

(39)

∆t∆E ge ~2

(310)



26








HI(t) = minus1

2qer middotE0e



|Ψ〉 = α|e〉+ β|g〉|α|2 + |β|2 = 1

(312)


|Ψ〉 = cosθ

2|e〉+ eiφ sin

θ

2|g〉 (313)


ρ = |Ψ〉〈Ψ| (314)


(10

)and |g〉 =

(01

) this becomes

ρ =

(αβ

)(αlowast βlowast

)=

(|α|2 αβlowast

αlowastβ |β|2)

=

(ρee ρegρge ρgg

)(315)

27

321 Bloch sphere

w

z

v

R

|e

uy

x



u = ρeg + ρge


(316)


u = minus u

T2minus δv + ΩImw

v = δuminus v

T2+ ΩRew


T1

(317)




~(318)



321 Bloch sphere


28


R2

|g

R1

W1

W2

z |e

yx





R = W timesR (319)







29





|Ψ(t)〉 = α|e〉+ βeiω0t|g〉 (320)



radic18 β =


radic2 β = 1

radic2 and case 4


radic2


30



radic18 β =


radic2

β = 1radic

2 and case 4 α = 1 β = 0

31








0 + iEIm0





(part

partz+n

c

part

partt

)ΩRe =

α0

2π

int +infin

minusinfing(δ)vdδ

(part

partz+n

c

part

partt

)ΩIm =

α0

2π

int +infin

minusinfing(δ)udδ

(322)





32


|g

R

(a)

W

z |e

yx

|g

(b)

R

W

z |e

yx

|g

R

(c)

W

z |e

yx



∆ΩRe =α0


∆ΩIm =α0


(323)





33

33 Decays

R2

|g

z |e

y

R1

x


33 Decays


331 Coherence time



Γhom =1

πT2(324)


34


a b





332 Lifetime


35

332 Lifetime






36





34 DC Stark effect



37

Chapter 4









39


21Sc

39Y








P (ωp) =1

exp(

~ωp

kBT

)minus 1

(41)


40






41


50

48

46

44

42

40

38

36

34

32

30

28

26

24

22

20

18

16

14

12

10

8

6

4

2

0

Ener

gy [103

cm-1

]

1500

1440

1380

1320

1260

1200

1140

1080

1020

960

900

840

780

720

660

600

540

480

420

360

300

240

180

120

60

0

Frequ

ency [TH

z]200

300

400

500

600

700

8009001000

2000

300040005000

Wavelen

gth [n

m]


42


plusmn12g

plusmn32g

plusmn52g

plusmn12e

plusmn32e

plusmn52e

606 nm

495 THz

102 MHz

173 MHz

46 MHz

48 MHz


10265degab

c

1041 nm

79deg

C2 (b)

D1

D2











43


Δ1206412static

124deg

Δ1206414static

Δ1206411static

Δ1206413static

C2 (b)


i in








∆ω =(microstatic


~=


(42)




44









45


Frequency [MHz]

046

94102

148196

275321

369

Ab

sorp

tio

n

~kHz


Frequency

Ab

sorp

tio

n

~GHz

Crystal










46

Chapter 5









47













48


1198631

1198632

120641119890119892119916in


(d) Propagation

1198631

1198632

(b) Absorption

119916in

119916P

119916res

1198631

1198632

(c) Projection on 1198631 1198632

119916res

1198631

1198632

(e) Propagation

746 plusmn 19deg

1198632

1198631




R =n2D2minus n2

D1

χabs(51)


49


Ele

ctric

fiel

d al

ong

(a)

D2

Ele

ctric

fiel

d al

ong

(c)

D2

Ele

ctric

fiel

d al

ong

(b)

D2





D2minus n2

D1= (nD2

+ nD1)(nD2

minus nD1) asymp











50







ID2= 10minus5







51


(Paper II)


Double pass AOM

Polarizer Polarizer


PD2


Cryostat

Shutter

119887

1198632

1198631


PD1

Pr3+ Y2SiO5









52










53











54



0

1

2

3

Ab

sorp

tio

n

L (a) - 15 V- 5 V0 V5 V15 V

-4 -3 -2 -1 0 1 2 3 4Frequency [MHz]

FF

T In

ten

sity

(b)





55








56











57












58










59

Chapter 6








61















62















63



64

|0rang

|1rang

|auxrang

|119890rang

606 nm

495 THz

102 MHz

173 MHz

46 MHz

48 MHz

1205960 Ω0 1206010

1205961 Ω1 1206011



Chapter 7








65














66



|0

R0

Re

z |e

yx






tFWHM asymp176

β(72)

νwidth =microβ

π(73)






67


|1

|B

|D

z |0

y

x








|B〉 =1radic2


)

|D〉 =1radic2


) (74)



68


|1

R0

R

|D

z |0

y

x












ρmixed =1

2(ρ0 + ρ1) =

1

2(ρB + ρD) (76)


69


Time

1205872 120587 read

Spin echo sequence

|119890⟩

1

|0⟩

|119890⟩

1

|0⟩

|119890⟩

1

|0⟩

ൗ120591 2 ൗ120591 2



2


) In other





70


y

|1

x1

z |0(a)

y

|1

x2

z |0(b)

y

|1

1x2

z |0(c)








71












72



|0rang

|1rang

|119890rang

1 3

(a)

|0rang

|1rang

|119890rang

2 2

Control Target

|0rang

|1rang

|119890rangΔ120584

2 2

|0rang

|1rang

|119890rang

1 3

(b)

Control Target




73


|0rang

|1rang

|119890rang

1 3

Control

|auxrang

|0rang

|1rang

|32119890rang

2

Target

|auxrang

4

|12119890rang

|52119890rang

4648

102

173

[MHz]










74










75


q7

q6

q5 q3

q4

q1

q2

Buffer

Readout


|0rang

|1rang

|119890rang

1 3

Qubit 1

|0rang

|1rang

|119890rang

2

Buffer

Δ120584

4

Readout






76







77



78



2F52

2F72

371 nm

5d(2D52

2D32)

B

B

Δ1205845119889

Δ1205844119891


Cryostat

SMF

BPPH

X

Z

Y

SPAD

DM

PDHECDLAOM

Lens

Crystal





79
















80






81










82












83






84

Chapter 8

Conclusions






85








86

Conclusions


87

Appendix A

Equipment



A1 Main setup


89

A11 Dye laser setup

Dye laser table

λ2-plate

Fiber

AOM1

Wavemeter

EOM

AOM2

AOM3

λ2-plate

Polarizer



OD

Pumplaser

Dye laser

Filterwheel

MirrorBeam splitter

Lens


A11 Dye laser setup




90

Equipment

Locking platform


table

λ2-plate

Error detector


CCD camera

λ4-plate

PBSLens

Cavity






91









92

Equipment

Polarizer

λ2-plate


Cryostat

Reference detector

Crystal


table

Lens

Experimentaltable

SE

N W








A2 Cryostat


93


λ4-plate

Ce laser table


CCDcamera

Cavity

PBS

Errordetector

EOM

λ2-plate

ECDL

OD









94

Equipment




95







97








98








99

Acknowledgements







101







102

References





OrbitsHistory











103

References


















104

References
















105

References













106

References















107

References










computing





researchibmcomibm-q





108

References



















109

References






magfieldshtml









110

References













111

AD

AM

KIN

OS

Light-M


Com

puting in Rare-E

arth-Ion-Doped C

rystals 2018

978

9177

5354

30







Prin

ted

by M

edia

-Try

ck L

und

2018

N

ORD

IC S

WA

N E

CO

LABE

L 3

041

0903


Abstract



Abbreviations

Contents

1 Introduction












221 Slow light








321 Bloch sphere



33 Decays

331 Coherence time

332 Lifetime

34 DC Stark effect































8 Conclusions


A Equipment

A1 Main setup

A11 Dye laser setup



A2 Cryostat



Acknowledgements



Adam Kinos

Doctoral Thesis

2018







Abstract





iii


iv






v




vi








vii











PLOS One 10 e0121905 (2015)




viii







Sci Rep 6 30410 (2016)



ACS Nano 10 (11) 9823-9830 (2016)

ix

Abbreviations


BP Band Pass filter


CNOT Controlled NOT

DM Dichroic Mirror





OD Optical Diode



PH Pinhole





Ce Cerium

Eu Europium

Nd Neodymium

Pr Praseodymium

Y2O3 Yttrium Oxide


xi

Contents

Abstract iii


















Contents















Contents

Papers









Chapter 1

Introduction






1










2

Introduction









3

Chapter 2





5











E(r) =1

4πε0

q

r2er (21)




6




7


(a)

(b)

120579











9


















10


119812in t 119851

119911

119909

120596 1205960







qeE0


eminusiωt (25)



exx(tminus rc) (26)




4πε0c2rx0exe

minusiω(tminusrc)

asymp q2eω

2

4πε0c2rme

1


(27)




11










12


119876

119875

120588119903

119911119889120588

120578


Re

Im

1205790 =120596119911

119888

radius =119888

120596








ρ=0

qeω2

4πε0c2x0exe

minusiω(tminusrc)

rη2πρdρ (28)



Eradtot(t r) =qeω

2ηx0exeminusiωt

2ε0c2

int r=infin

r=z

eiωrcdr (29)







13


|119812in|

|119812rad tot|



119812in

119812rad tot

119812after



Eradtot(t r) =q2eη

2ε0cme

iω









14





asymp(


c

)E0exe

minusiω(tminuszc)


(211)


nsparse asymp 1 +1

2

q2eN

ε0me

1





Eafter(t r) = exp

[minus q2

eN

2ε0meγc∆z

]Ein(t r)

= exp[minusα

2∆z]Ein(t r)

(213)


15







n1 sin θ1 = n2 sin θ2 (214)








16



221 Slow light









partω(215)

17


a

b

c No dispersion

d Dispersion

Pulse location

Pulse location

Pulse location






18






vg =c

1 + Uhost+Ures

Uvac

(216)


n = 1 +Uhost

Uvac(217)



vg asymp2πΓ

α(218)





19



P(t r) =Nq2

e








P(t r) =Nαpol(ω)



αpol(ω) =q2e

ε0me

sum

k

fkω2



nabla middotE =ρ



partt

c2nablatimesB =

(j

ε0+partE

partt

)(222)


nabla =(partpartx

partparty

partpartz



partt

20




nabla middotB = 0


partt

c2nablatimesB =part

partt

(P

ε0+ E

)(223)



k2Eavg =ω2

c2Eavg +

ω2

ε0c2P

k2 =ω2

c2

(1 +

Nαpol(ω)


) (224)


n2 = 1 +Nαpol(ω)







3

(n2 minus 1

n2 + 2

)=sum

j

Njαpol j(ω) (226)

21







22

Chapter 3








23






F = ma (31)






i~partΨ(t r)




24


0 L 119909

119881 = infin119881 = 0

119881 = infin











H0 =minus~2

2m

part2

partx2+ V (x) (35)


2mv2x = 1





0 lt x lt L (36)


where A =radic


25



En =~2

2m

π2n2

L2(37)




Φn(px) =1radic2π~

intΨn(x)eminus

i~pxxdx (38)


∆x∆px ge~2

(39)

∆t∆E ge ~2

(310)



26








HI(t) = minus1

2qer middotE0e



|Ψ〉 = α|e〉+ β|g〉|α|2 + |β|2 = 1

(312)


|Ψ〉 = cosθ

2|e〉+ eiφ sin

θ

2|g〉 (313)


ρ = |Ψ〉〈Ψ| (314)


(10

)and |g〉 =

(01

) this becomes

ρ =

(αβ

)(αlowast βlowast

)=

(|α|2 αβlowast

αlowastβ |β|2)

=

(ρee ρegρge ρgg

)(315)

27

321 Bloch sphere

w

z

v

R

|e

uy

x



u = ρeg + ρge


(316)


u = minus u

T2minus δv + ΩImw

v = δuminus v

T2+ ΩRew


T1

(317)




~(318)



321 Bloch sphere


28


R2

|g

R1

W1

W2

z |e

yx





R = W timesR (319)







29





|Ψ(t)〉 = α|e〉+ βeiω0t|g〉 (320)



radic18 β =


radic2 β = 1

radic2 and case 4


radic2


30



radic18 β =


radic2

β = 1radic

2 and case 4 α = 1 β = 0

31








0 + iEIm0





(part

partz+n

c

part

partt

)ΩRe =

α0

2π

int +infin

minusinfing(δ)vdδ

(part

partz+n

c

part

partt

)ΩIm =

α0

2π

int +infin

minusinfing(δ)udδ

(322)





32


|g

R

(a)

W

z |e

yx

|g

(b)

R

W

z |e

yx

|g

R

(c)

W

z |e

yx



∆ΩRe =α0


∆ΩIm =α0


(323)





33

33 Decays

R2

|g

z |e

y

R1

x


33 Decays


331 Coherence time



Γhom =1

πT2(324)


34


a b





332 Lifetime


35

332 Lifetime






36





34 DC Stark effect



37

Chapter 4









39


21Sc

39Y








P (ωp) =1

exp(

~ωp

kBT

)minus 1

(41)


40






41


50

48

46

44

42

40

38

36

34

32

30

28

26

24

22

20

18

16

14

12

10

8

6

4

2

0

Ener

gy [103

cm-1

]

1500

1440

1380

1320

1260

1200

1140

1080

1020

960

900

840

780

720

660

600

540

480

420

360

300

240

180

120

60

0

Frequ

ency [TH

z]200

300

400

500

600

700

8009001000

2000

300040005000

Wavelen

gth [n

m]


42


plusmn12g

plusmn32g

plusmn52g

plusmn12e

plusmn32e

plusmn52e

606 nm

495 THz

102 MHz

173 MHz

46 MHz

48 MHz


10265degab

c

1041 nm

79deg

C2 (b)

D1

D2











43


Δ1206412static

124deg

Δ1206414static

Δ1206411static

Δ1206413static

C2 (b)


i in








∆ω =(microstatic


~=


(42)




44









45


Frequency [MHz]

046

94102

148196

275321

369

Ab

sorp

tio

n

~kHz


Frequency

Ab

sorp

tio

n

~GHz

Crystal










46

Chapter 5









47













48


1198631

1198632

120641119890119892119916in


(d) Propagation

1198631

1198632

(b) Absorption

119916in

119916P

119916res

1198631

1198632

(c) Projection on 1198631 1198632

119916res

1198631

1198632

(e) Propagation

746 plusmn 19deg

1198632

1198631




R =n2D2minus n2

D1

χabs(51)


49


Ele

ctric

fiel

d al

ong

(a)

D2

Ele

ctric

fiel

d al

ong

(c)

D2

Ele

ctric

fiel

d al

ong

(b)

D2





D2minus n2

D1= (nD2

+ nD1)(nD2

minus nD1) asymp











50







ID2= 10minus5







51


(Paper II)


Double pass AOM

Polarizer Polarizer


PD2


Cryostat

Shutter

119887

1198632

1198631


PD1

Pr3+ Y2SiO5









52










53











54



0

1

2

3

Ab

sorp

tio

n

L (a) - 15 V- 5 V0 V5 V15 V

-4 -3 -2 -1 0 1 2 3 4Frequency [MHz]

FF

T In

ten

sity

(b)





55








56











57












58










59

Chapter 6








61















62















63



64

|0rang

|1rang

|auxrang

|119890rang

606 nm

495 THz

102 MHz

173 MHz

46 MHz

48 MHz

1205960 Ω0 1206010

1205961 Ω1 1206011



Chapter 7








65














66



|0

R0

Re

z |e

yx






tFWHM asymp176

β(72)

νwidth =microβ

π(73)






67


|1

|B

|D

z |0

y

x








|B〉 =1radic2


)

|D〉 =1radic2


) (74)



68


|1

R0

R

|D

z |0

y

x












ρmixed =1

2(ρ0 + ρ1) =

1

2(ρB + ρD) (76)


69


Time

1205872 120587 read

Spin echo sequence

|119890⟩

1

|0⟩

|119890⟩

1

|0⟩

|119890⟩

1

|0⟩

ൗ120591 2 ൗ120591 2



2


) In other





70


y

|1

x1

z |0(a)

y

|1

x2

z |0(b)

y

|1

1x2

z |0(c)








71












72



|0rang

|1rang

|119890rang

1 3

(a)

|0rang

|1rang

|119890rang

2 2

Control Target

|0rang

|1rang

|119890rangΔ120584

2 2

|0rang

|1rang

|119890rang

1 3

(b)

Control Target




73


|0rang

|1rang

|119890rang

1 3

Control

|auxrang

|0rang

|1rang

|32119890rang

2

Target

|auxrang

4

|12119890rang

|52119890rang

4648

102

173

[MHz]










74










75


q7

q6

q5 q3

q4

q1

q2

Buffer

Readout


|0rang

|1rang

|119890rang

1 3

Qubit 1

|0rang

|1rang

|119890rang

2

Buffer

Δ120584

4

Readout






76







77



78



2F52

2F72

371 nm

5d(2D52

2D32)

B

B

Δ1205845119889

Δ1205844119891


Cryostat

SMF

BPPH

X

Z

Y

SPAD

DM

PDHECDLAOM

Lens

Crystal





79
















80






81










82












83






84

Chapter 8

Conclusions






85








86

Conclusions


87

Appendix A

Equipment



A1 Main setup


89

A11 Dye laser setup

Dye laser table

λ2-plate

Fiber

AOM1

Wavemeter

EOM

AOM2

AOM3

λ2-plate

Polarizer



OD

Pumplaser

Dye laser

Filterwheel

MirrorBeam splitter

Lens


A11 Dye laser setup




90

Equipment

Locking platform


table

λ2-plate

Error detector


CCD camera

λ4-plate

PBSLens

Cavity






91









92

Equipment

Polarizer

λ2-plate


Cryostat

Reference detector

Crystal


table

Lens

Experimentaltable

SE

N W








A2 Cryostat


93


λ4-plate

Ce laser table


CCDcamera

Cavity

PBS

Errordetector

EOM

λ2-plate

ECDL

OD









94

Equipment




95







97








98








99

Acknowledgements







101







102

References





OrbitsHistory











103

References


















104

References
















105

References













106

References















107

References










computing





researchibmcomibm-q





108

References



















109

References






magfieldshtml









110

References













111

AD

AM

KIN

OS

Light-M


Com

puting in Rare-E

arth-Ion-Doped C

rystals 2018

978

9177

5354

30







Prin

ted

by M

edia

-Try

ck L

und

2018

N

ORD

IC S

WA

N E

CO

LABE

L 3

041

0903


Abstract



Abbreviations

Contents

1 Introduction












221 Slow light








321 Bloch sphere



33 Decays

331 Coherence time

332 Lifetime

34 DC Stark effect































8 Conclusions


A Equipment

A1 Main setup

A11 Dye laser setup



A2 Cryostat



Acknowledgements







Abstract





iii


iv






v




vi








vii











PLOS One 10 e0121905 (2015)




viii







Sci Rep 6 30410 (2016)



ACS Nano 10 (11) 9823-9830 (2016)

ix

Abbreviations


BP Band Pass filter


CNOT Controlled NOT

DM Dichroic Mirror





OD Optical Diode



PH Pinhole





Ce Cerium

Eu Europium

Nd Neodymium

Pr Praseodymium

Y2O3 Yttrium Oxide


xi

Contents

Abstract iii


















Contents















Contents

Papers









Chapter 1

Introduction






1










2

Introduction









3

Chapter 2





5











E(r) =1

4πε0

q

r2er (21)




6




7


(a)

(b)

120579











9


















10


119812in t 119851

119911

119909

120596 1205960







qeE0


eminusiωt (25)



exx(tminus rc) (26)




4πε0c2rx0exe

minusiω(tminusrc)

asymp q2eω

2

4πε0c2rme

1


(27)




11










12


119876

119875

120588119903

119911119889120588

120578


Re

Im

1205790 =120596119911

119888

radius =119888

120596








ρ=0

qeω2

4πε0c2x0exe

minusiω(tminusrc)

rη2πρdρ (28)



Eradtot(t r) =qeω

2ηx0exeminusiωt

2ε0c2

int r=infin

r=z

eiωrcdr (29)







13


|119812in|

|119812rad tot|



119812in

119812rad tot

119812after



Eradtot(t r) =q2eη

2ε0cme

iω









14





asymp(


c

)E0exe

minusiω(tminuszc)


(211)


nsparse asymp 1 +1

2

q2eN

ε0me

1





Eafter(t r) = exp

[minus q2

eN

2ε0meγc∆z

]Ein(t r)

= exp[minusα

2∆z]Ein(t r)

(213)


15







n1 sin θ1 = n2 sin θ2 (214)








16



221 Slow light









partω(215)

17


a

b

c No dispersion

d Dispersion

Pulse location

Pulse location

Pulse location






18






vg =c

1 + Uhost+Ures

Uvac

(216)


n = 1 +Uhost

Uvac(217)



vg asymp2πΓ

α(218)





19



P(t r) =Nq2

e








P(t r) =Nαpol(ω)



αpol(ω) =q2e

ε0me

sum

k

fkω2



nabla middotE =ρ



partt

c2nablatimesB =

(j

ε0+partE

partt

)(222)


nabla =(partpartx

partparty

partpartz



partt

20




nabla middotB = 0


partt

c2nablatimesB =part

partt

(P

ε0+ E

)(223)



k2Eavg =ω2

c2Eavg +

ω2

ε0c2P

k2 =ω2

c2

(1 +

Nαpol(ω)


) (224)


n2 = 1 +Nαpol(ω)







3

(n2 minus 1

n2 + 2

)=sum

j

Njαpol j(ω) (226)

21







22

Chapter 3








23






F = ma (31)






i~partΨ(t r)




24


0 L 119909

119881 = infin119881 = 0

119881 = infin











H0 =minus~2

2m

part2

partx2+ V (x) (35)


2mv2x = 1





0 lt x lt L (36)


where A =radic


25



En =~2

2m

π2n2

L2(37)




Φn(px) =1radic2π~

intΨn(x)eminus

i~pxxdx (38)


∆x∆px ge~2

(39)

∆t∆E ge ~2

(310)



26








HI(t) = minus1

2qer middotE0e



|Ψ〉 = α|e〉+ β|g〉|α|2 + |β|2 = 1

(312)


|Ψ〉 = cosθ

2|e〉+ eiφ sin

θ

2|g〉 (313)


ρ = |Ψ〉〈Ψ| (314)


(10

)and |g〉 =

(01

) this becomes

ρ =

(αβ

)(αlowast βlowast

)=

(|α|2 αβlowast

αlowastβ |β|2)

=

(ρee ρegρge ρgg

)(315)

27

321 Bloch sphere

w

z

v

R

|e

uy

x



u = ρeg + ρge


(316)


u = minus u

T2minus δv + ΩImw

v = δuminus v

T2+ ΩRew


T1

(317)




~(318)



321 Bloch sphere


28


R2

|g

R1

W1

W2

z |e

yx





R = W timesR (319)







29





|Ψ(t)〉 = α|e〉+ βeiω0t|g〉 (320)



radic18 β =


radic2 β = 1

radic2 and case 4


radic2


30



radic18 β =


radic2

β = 1radic

2 and case 4 α = 1 β = 0

31








0 + iEIm0





(part

partz+n

c

part

partt

)ΩRe =

α0

2π

int +infin

minusinfing(δ)vdδ

(part

partz+n

c

part

partt

)ΩIm =

α0

2π

int +infin

minusinfing(δ)udδ

(322)





32


|g

R

(a)

W

z |e

yx

|g

(b)

R

W

z |e

yx

|g

R

(c)

W

z |e

yx



∆ΩRe =α0


∆ΩIm =α0


(323)





33

33 Decays

R2

|g

z |e

y

R1

x


33 Decays


331 Coherence time



Γhom =1

πT2(324)


34


a b





332 Lifetime


35

332 Lifetime






36





34 DC Stark effect



37

Chapter 4









39


21Sc

39Y








P (ωp) =1

exp(

~ωp

kBT

)minus 1

(41)


40






41


50

48

46

44

42

40

38

36

34

32

30

28

26

24

22

20

18

16

14

12

10

8

6

4

2

0

Ener

gy [103

cm-1

]

1500

1440

1380

1320

1260

1200

1140

1080

1020

960

900

840

780

720

660

600

540

480

420

360

300

240

180

120

60

0

Frequ

ency [TH

z]200

300

400

500

600

700

8009001000

2000

300040005000

Wavelen

gth [n

m]


42


plusmn12g

plusmn32g

plusmn52g

plusmn12e

plusmn32e

plusmn52e

606 nm

495 THz

102 MHz

173 MHz

46 MHz

48 MHz


10265degab

c

1041 nm

79deg

C2 (b)

D1

D2











43


Δ1206412static

124deg

Δ1206414static

Δ1206411static

Δ1206413static

C2 (b)


i in








∆ω =(microstatic


~=


(42)




44









45


Frequency [MHz]

046

94102

148196

275321

369

Ab

sorp

tio

n

~kHz


Frequency

Ab

sorp

tio

n

~GHz

Crystal










46

Chapter 5









47













48


1198631

1198632

120641119890119892119916in


(d) Propagation

1198631

1198632

(b) Absorption

119916in

119916P

119916res

1198631

1198632

(c) Projection on 1198631 1198632

119916res

1198631

1198632

(e) Propagation

746 plusmn 19deg

1198632

1198631




R =n2D2minus n2

D1

χabs(51)


49


Ele

ctric

fiel

d al

ong

(a)

D2

Ele

ctric

fiel

d al

ong

(c)

D2

Ele

ctric

fiel

d al

ong

(b)

D2





D2minus n2

D1= (nD2

+ nD1)(nD2

minus nD1) asymp











50







ID2= 10minus5







51


(Paper II)


Double pass AOM

Polarizer Polarizer


PD2


Cryostat

Shutter

119887

1198632

1198631


PD1

Pr3+ Y2SiO5









52










53











54



0

1

2

3

Ab

sorp

tio

n

L (a) - 15 V- 5 V0 V5 V15 V

-4 -3 -2 -1 0 1 2 3 4Frequency [MHz]

FF

T In

ten

sity

(b)





55








56











57












58










59

Chapter 6








61















62















63



64

|0rang

|1rang

|auxrang

|119890rang

606 nm

495 THz

102 MHz

173 MHz

46 MHz

48 MHz

1205960 Ω0 1206010

1205961 Ω1 1206011



Chapter 7








65














66



|0

R0

Re

z |e

yx






tFWHM asymp176

β(72)

νwidth =microβ

π(73)






67


|1

|B

|D

z |0

y

x








|B〉 =1radic2


)

|D〉 =1radic2


) (74)



68


|1

R0

R

|D

z |0

y

x












ρmixed =1

2(ρ0 + ρ1) =

1

2(ρB + ρD) (76)


69


Time

1205872 120587 read

Spin echo sequence

|119890⟩

1

|0⟩

|119890⟩

1

|0⟩

|119890⟩

1

|0⟩

ൗ120591 2 ൗ120591 2



2


) In other





70


y

|1

x1

z |0(a)

y

|1

x2

z |0(b)

y

|1

1x2

z |0(c)








71












72



|0rang

|1rang

|119890rang

1 3

(a)

|0rang

|1rang

|119890rang

2 2

Control Target

|0rang

|1rang

|119890rangΔ120584

2 2

|0rang

|1rang

|119890rang

1 3

(b)

Control Target




73


|0rang

|1rang

|119890rang

1 3

Control

|auxrang

|0rang

|1rang

|32119890rang

2

Target

|auxrang

4

|12119890rang

|52119890rang

4648

102

173

[MHz]










74










75


q7

q6

q5 q3

q4

q1

q2

Buffer

Readout


|0rang

|1rang

|119890rang

1 3

Qubit 1

|0rang

|1rang

|119890rang

2

Buffer

Δ120584

4

Readout






76







77



78



2F52

2F72

371 nm

5d(2D52

2D32)

B

B

Δ1205845119889

Δ1205844119891


Cryostat

SMF

BPPH

X

Z

Y

SPAD

DM

PDHECDLAOM

Lens

Crystal





79
















80






81










82












83






84

Chapter 8

Conclusions






85








86

Conclusions


87

Appendix A

Equipment



A1 Main setup


89

A11 Dye laser setup

Dye laser table

λ2-plate

Fiber

AOM1

Wavemeter

EOM

AOM2

AOM3

λ2-plate

Polarizer



OD

Pumplaser

Dye laser

Filterwheel

MirrorBeam splitter

Lens


A11 Dye laser setup




90

Equipment

Locking platform


table

λ2-plate

Error detector


CCD camera

λ4-plate

PBSLens

Cavity






91









92

Equipment

Polarizer

λ2-plate


Cryostat

Reference detector

Crystal


table

Lens

Experimentaltable

SE

N W








A2 Cryostat


93


λ4-plate

Ce laser table


CCDcamera

Cavity

PBS

Errordetector

EOM

λ2-plate

ECDL

OD









94

Equipment




95







97








98








99

Acknowledgements







101







102

References





OrbitsHistory











103

References


















104

References
















105

References













106

References















107

References










computing





researchibmcomibm-q





108

References



















109

References






magfieldshtml









110

References













111

AD

AM

KIN

OS

Light-M


Com

puting in Rare-E

arth-Ion-Doped C

rystals 2018

978

9177

5354

30







Prin

ted

by M

edia

-Try

ck L

und

2018

N

ORD

IC S

WA

N E

CO

LABE

L 3

041

0903


Abstract



Abbreviations

Contents

1 Introduction












221 Slow light








321 Bloch sphere



33 Decays

331 Coherence time

332 Lifetime

34 DC Stark effect































8 Conclusions


A Equipment

A1 Main setup

A11 Dye laser setup



A2 Cryostat



Acknowledgements


iv






v




vi








vii











PLOS One 10 e0121905 (2015)




viii







Sci Rep 6 30410 (2016)



ACS Nano 10 (11) 9823-9830 (2016)

ix

Abbreviations


BP Band Pass filter


CNOT Controlled NOT

DM Dichroic Mirror





OD Optical Diode



PH Pinhole





Ce Cerium

Eu Europium

Nd Neodymium

Pr Praseodymium

Y2O3 Yttrium Oxide


xi

Contents

Abstract iii


















Contents















Contents

Papers









Chapter 1

Introduction






1










2

Introduction









3

Chapter 2





5











E(r) =1

4πε0

q

r2er (21)




6




7


(a)

(b)

120579











9


















10


119812in t 119851

119911

119909

120596 1205960







qeE0


eminusiωt (25)



exx(tminus rc) (26)




4πε0c2rx0exe

minusiω(tminusrc)

asymp q2eω

2

4πε0c2rme

1


(27)




11










12


119876

119875

120588119903

119911119889120588

120578


Re

Im

1205790 =120596119911

119888

radius =119888

120596








ρ=0

qeω2

4πε0c2x0exe

minusiω(tminusrc)

rη2πρdρ (28)



Eradtot(t r) =qeω

2ηx0exeminusiωt

2ε0c2

int r=infin

r=z

eiωrcdr (29)







13


|119812in|

|119812rad tot|



119812in

119812rad tot

119812after



Eradtot(t r) =q2eη

2ε0cme

iω









14





asymp(


c

)E0exe

minusiω(tminuszc)


(211)


nsparse asymp 1 +1

2

q2eN

ε0me

1





Eafter(t r) = exp

[minus q2

eN

2ε0meγc∆z

]Ein(t r)

= exp[minusα

2∆z]Ein(t r)

(213)


15







n1 sin θ1 = n2 sin θ2 (214)








16



221 Slow light









partω(215)

17


a

b

c No dispersion

d Dispersion

Pulse location

Pulse location

Pulse location






18






vg =c

1 + Uhost+Ures

Uvac

(216)


n = 1 +Uhost

Uvac(217)



vg asymp2πΓ

α(218)





19



P(t r) =Nq2

e








P(t r) =Nαpol(ω)



αpol(ω) =q2e

ε0me

sum

k

fkω2



nabla middotE =ρ



partt

c2nablatimesB =

(j

ε0+partE

partt

)(222)


nabla =(partpartx

partparty

partpartz



partt

20




nabla middotB = 0


partt

c2nablatimesB =part

partt

(P

ε0+ E

)(223)



k2Eavg =ω2

c2Eavg +

ω2

ε0c2P

k2 =ω2

c2

(1 +

Nαpol(ω)


) (224)


n2 = 1 +Nαpol(ω)







3

(n2 minus 1

n2 + 2

)=sum

j

Njαpol j(ω) (226)

21







22

Chapter 3








23






F = ma (31)






i~partΨ(t r)




24


0 L 119909

119881 = infin119881 = 0

119881 = infin











H0 =minus~2

2m

part2

partx2+ V (x) (35)


2mv2x = 1





0 lt x lt L (36)


where A =radic


25



En =~2

2m

π2n2

L2(37)




Φn(px) =1radic2π~

intΨn(x)eminus

i~pxxdx (38)


∆x∆px ge~2

(39)

∆t∆E ge ~2

(310)



26








HI(t) = minus1

2qer middotE0e



|Ψ〉 = α|e〉+ β|g〉|α|2 + |β|2 = 1

(312)


|Ψ〉 = cosθ

2|e〉+ eiφ sin

θ

2|g〉 (313)


ρ = |Ψ〉〈Ψ| (314)


(10

)and |g〉 =

(01

) this becomes

ρ =

(αβ

)(αlowast βlowast

)=

(|α|2 αβlowast

αlowastβ |β|2)

=

(ρee ρegρge ρgg

)(315)

27

321 Bloch sphere

w

z

v

R

|e

uy

x



u = ρeg + ρge


(316)


u = minus u

T2minus δv + ΩImw

v = δuminus v

T2+ ΩRew


T1

(317)




~(318)



321 Bloch sphere


28


R2

|g

R1

W1

W2

z |e

yx





R = W timesR (319)







29





|Ψ(t)〉 = α|e〉+ βeiω0t|g〉 (320)



radic18 β =


radic2 β = 1

radic2 and case 4


radic2


30



radic18 β =


radic2

β = 1radic

2 and case 4 α = 1 β = 0

31








0 + iEIm0





(part

partz+n

c

part

partt

)ΩRe =

α0

2π

int +infin

minusinfing(δ)vdδ

(part

partz+n

c

part

partt

)ΩIm =

α0

2π

int +infin

minusinfing(δ)udδ

(322)





32


|g

R

(a)

W

z |e

yx

|g

(b)

R

W

z |e

yx

|g

R

(c)

W

z |e

yx



∆ΩRe =α0


∆ΩIm =α0


(323)





33

33 Decays

R2

|g

z |e

y

R1

x


33 Decays


331 Coherence time



Γhom =1

πT2(324)


34


a b





332 Lifetime


35

332 Lifetime






36





34 DC Stark effect



37

Chapter 4









39


21Sc

39Y








P (ωp) =1

exp(

~ωp

kBT

)minus 1

(41)


40






41


50

48

46

44

42

40

38

36

34

32

30

28

26

24

22

20

18

16

14

12

10

8

6

4

2

0

Ener

gy [103

cm-1

]

1500

1440

1380

1320

1260

1200

1140

1080

1020

960

900

840

780

720

660

600

540

480

420

360

300

240

180

120

60

0

Frequ

ency [TH

z]200

300

400

500

600

700

8009001000

2000

300040005000

Wavelen

gth [n

m]


42


plusmn12g

plusmn32g

plusmn52g

plusmn12e

plusmn32e

plusmn52e

606 nm

495 THz

102 MHz

173 MHz

46 MHz

48 MHz


10265degab

c

1041 nm

79deg

C2 (b)

D1

D2











43


Δ1206412static

124deg

Δ1206414static

Δ1206411static

Δ1206413static

C2 (b)


i in








∆ω =(microstatic


~=


(42)




44









45


Frequency [MHz]

046

94102

148196

275321

369

Ab

sorp

tio

n

~kHz


Frequency

Ab

sorp

tio

n

~GHz

Crystal










46

Chapter 5









47













48


1198631

1198632

120641119890119892119916in


(d) Propagation

1198631

1198632

(b) Absorption

119916in

119916P

119916res

1198631

1198632

(c) Projection on 1198631 1198632

119916res

1198631

1198632

(e) Propagation

746 plusmn 19deg

1198632

1198631




R =n2D2minus n2

D1

χabs(51)


49


Ele

ctric

fiel

d al

ong

(a)

D2

Ele

ctric

fiel

d al

ong

(c)

D2

Ele

ctric

fiel

d al

ong

(b)

D2





D2minus n2

D1= (nD2

+ nD1)(nD2

minus nD1) asymp











50







ID2= 10minus5







51


(Paper II)


Double pass AOM

Polarizer Polarizer


PD2


Cryostat

Shutter

119887

1198632

1198631


PD1

Pr3+ Y2SiO5









52










53











54



0

1

2

3

Ab

sorp

tio

n

L (a) - 15 V- 5 V0 V5 V15 V

-4 -3 -2 -1 0 1 2 3 4Frequency [MHz]

FF

T In

ten

sity

(b)





55








56











57












58










59

Chapter 6








61















62















63



64

|0rang

|1rang

|auxrang

|119890rang

606 nm

495 THz

102 MHz

173 MHz

46 MHz

48 MHz

1205960 Ω0 1206010

1205961 Ω1 1206011



Chapter 7








65














66



|0

R0

Re

z |e

yx






tFWHM asymp176

β(72)

νwidth =microβ

π(73)






67


|1

|B

|D

z |0

y

x








|B〉 =1radic2


)

|D〉 =1radic2


) (74)



68


|1

R0

R

|D

z |0

y

x












ρmixed =1

2(ρ0 + ρ1) =

1

2(ρB + ρD) (76)


69


Time

1205872 120587 read

Spin echo sequence

|119890⟩

1

|0⟩

|119890⟩

1

|0⟩

|119890⟩

1

|0⟩

ൗ120591 2 ൗ120591 2



2


) In other





70


y

|1

x1

z |0(a)

y

|1

x2

z |0(b)

y

|1

1x2

z |0(c)








71












72



|0rang

|1rang

|119890rang

1 3

(a)

|0rang

|1rang

|119890rang

2 2

Control Target

|0rang

|1rang

|119890rangΔ120584

2 2

|0rang

|1rang

|119890rang

1 3

(b)

Control Target




73


|0rang

|1rang

|119890rang

1 3

Control

|auxrang

|0rang

|1rang

|32119890rang

2

Target

|auxrang

4

|12119890rang

|52119890rang

4648

102

173

[MHz]










74










75


q7

q6

q5 q3

q4

q1

q2

Buffer

Readout


|0rang

|1rang

|119890rang

1 3

Qubit 1

|0rang

|1rang

|119890rang

2

Buffer

Δ120584

4

Readout






76







77



78



2F52

2F72

371 nm

5d(2D52

2D32)

B

B

Δ1205845119889

Δ1205844119891


Cryostat

SMF

BPPH

X

Z

Y

SPAD

DM

PDHECDLAOM

Lens

Crystal





79
















80






81










82












83






84

Chapter 8

Conclusions






85








86

Conclusions


87

Appendix A

Equipment



A1 Main setup


89

A11 Dye laser setup

Dye laser table

λ2-plate

Fiber

AOM1

Wavemeter

EOM

AOM2

AOM3

λ2-plate

Polarizer



OD

Pumplaser

Dye laser

Filterwheel

MirrorBeam splitter

Lens


A11 Dye laser setup




90

Equipment

Locking platform


table

λ2-plate

Error detector


CCD camera

λ4-plate

PBSLens

Cavity






91









92

Equipment

Polarizer

λ2-plate


Cryostat

Reference detector

Crystal


table

Lens

Experimentaltable

SE

N W








A2 Cryostat


93


λ4-plate

Ce laser table


CCDcamera

Cavity

PBS

Errordetector

EOM

λ2-plate

ECDL

OD









94

Equipment




95







97








98








99

Acknowledgements







101







102

References





OrbitsHistory











103

References


















104

References
















105

References













106

References















107

References










computing





researchibmcomibm-q





108

References



















109

References






magfieldshtml









110

References













111

AD

AM

KIN

OS

Light-M


Com

puting in Rare-E

arth-Ion-Doped C

rystals 2018

978

9177

5354

30







Prin

ted

by M

edia

-Try

ck L

und

2018

N

ORD

IC S

WA

N E

CO

LABE

L 3

041

0903


Abstract



Abbreviations

Contents

1 Introduction












221 Slow light








321 Bloch sphere



33 Decays

331 Coherence time

332 Lifetime

34 DC Stark effect































8 Conclusions


A Equipment

A1 Main setup

A11 Dye laser setup



A2 Cryostat



Acknowledgements

light-matter interaction and quantum computing in rare-earth-ion-doped crystals kinos, adam

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