Matter wave scattering on complex potentials

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To cite this version:

Pierrick Cheiney. Matter wave scattering on complex potentials. Quantum Gases

[cond-mat.quant-gas]. Université Paul Sabatier - Toulouse III, 2013. English. �tel-00875153�

Letravailde thèse présentéi i aété possible grâ e àde nombreuses personnes qu'il

est temps de remer ier.

Je voudrais ommen er par remer ier les dire teurs su essifs du laboratoire :

Ja ques Vigué et Béatri e Chatel ainsi que Jean-Mar L'Hermite pour m'avoir

a - ueilli dans lelaboratoirependant es trois ans.

Jetiensensuiteàremer iervivementlesmembresdemonjurydethèse: Christophe

Salomon et Ia opo Carusotto en tant que rapporteurs ainsi que Christoph W

est-brook, Bruno Laburthe-Tolra et Sylvain Capponi pour leurs remarques très

on-stru tives.

Je voudrais remer ier très haleureusement toute l'équipe des atomes froids et

en premier lieu mon dire teur de thèse David Guéry-Odelin qui m'a a ueilli dans

l'équipe à partir mon stage de M2. Mer i pour ta gentillesse tes onseils et ton

optimismeàtouteépreuvedurant estroisans,ainsiquedutempsquetuas onsa ré

à larele ture de e manus rit. Mer i également du soutient sans faille que tum'as

apporté durant ette thèse etpour la suite.

Mer i à Renaud Mathevet et Olivier Carraz, ave qui j'ai beau oup appris en

travaillant sur le ralentisseur Zeeman. Dorénavant, haque nouvelle ma hine devra

omporter au moins un aimant permanent! La vidéo du nain de jardin restera

également dans les annales! Thierry Lahaye ensuite, 'était un grand plaisir de

travaillerave toi,tagrande onnaissan ede laphysiqueettonexigen eontététrès

pré ieuses.

Unsouvenirtoutparti ulierpourCharlotteFabreetFrançoisVermers h,d'unan

mesaînésdans etteaventure. Charlottes,les(longues) prisesde donnéesn'auraient

pas été aussi fun sans toi. François, il faut qu'on se fasse la ronde du foie gras un

jour! Je vous souhaite bonne han e à tous les deux dans la suite de vos aventures

enseignantes. Mer i enn à tous les autres membres de l'équipe ave qui j'ai eu le

plaisirdetravailler: Gianlu aGattobigio,StéphaneFaure,GabrielCondon,François

Damon,AélaFortun,GabrielHetetetJulietteBilly,ainsiqu'auxservi este hniques,

aux ateliers d'éle tronique et de mé aniques. Un grand mer i aux se rétaires du

département: ChristinneSou asse, SylvieBoukhari etMarie-Fran eRolland.

Je tiens également à remer ier les her heurs permanents du laboratoire ainsi

quelesdo torantsave quij'aipartagé es troisans: Simon,Jessi a,Renaud,Hugo,

Mi hael, Steven, Wesley, Jonathan, Muammar, Adrien et tous eux que j'ai pu

oublier. Je n'ai qu'une hose à dire: DUB!

Pour nir sur une note peu originale, mais pourtant bien sin ère, mer i à tous

les autres, qui m'ont soutenu. Il y a ma famille, mes amis et bien sûr Chiara qui

;-) bien sur.

Introdu tion 3

1 Design and realization of a permanent magnet Zeeman slower 9

1.1 Introdu tion . . . 10

1.2 Theoreti al framework . . . 10

1.2.1 Prin iple . . . 10

1.2.2 Simplemodel, magneti eld shape . . . 11

1.2.3 The adiabati following ondition . . . 12

1.2.4 Atomi stru ture . . . 13

1.2.5 Choi eof the magneti eld parameters. . . 15

1.2.6 Longitudinalor transverse magneti eld . . . 16

1.3 The dipole onguration . . . 17

1.3.1 Choi eand modelizationof the permanentmagnets . . . 17

1.3.2 Position of the magnets . . . 18

1.3.3 Measured magneti eld . . . 20

1.4 Slowed atomi beam . . . 22

1.4.1 Experimental setup . . . 22

1.4.2 Slowed beamvelo ity distribution . . . 24

1.4.3 Flux asa fun tion of the detuning . . . 25

1.5 The Halba h onguration . . . 26

1.5.1 The prin iple . . . 26

1.5.2 Field al ulations . . . 26

1.5.3 Magnets layout . . . 27

1.6 Me hani s and eld measurements. . . 28

1.6.1 Me hani aldesign. . . 28

1.6.2 Shielding. . . 28

1.6.3 Fieldparameters and measurement . . . 30

1.7 Performan es . . . 31

1.7.1 Needed laser powers . . . 32

1.7.2 Repumper . . . 33

1.7.3 MOT loading . . . 34

2.1 Introdu tion . . . 37

2.2 Condensation in anharmoni trap . . . 40

2.3 Pre- ooled atomi sour e . . . 41

2.3.1 Oven and Zeeman Slower. . . 41

2.3.2 The 2D-MOT . . . 42 2.4 Imaging system . . . 43 2.4.1 Absorptionimaging . . . 43 2.4.2 Setup . . . 43 2.4.3 MOT mode . . . 44 2.4.4 BEC mode. . . 44 2.4.5 Quantitativemeasurements . . . 45

2.5 The horizontaldipole guide . . . 45

2.5.1 Dipolepotential . . . 46

2.5.2 The spatial mode . . . 47

2.5.3 Setting up of the horizontal beam . . . 48

2.5.4 Power issues . . . 49

2.5.5 Experimentalsequen e . . . 51

2.5.6 Lifetime . . . 52

2.6 Crossed dipoletrap and evaporation . . . 53

2.6.1 Evaporation onstraints . . . 53

2.6.2 Chara teristi s of the rossedpotential . . . 56

2.6.3 Evaporation traje tory . . . 57

2.6.4 Spin distillation . . . 59

2.7 Con lusion . . . 61

3 Latti e setup and hara terization 63 3.1 Introdu tion . . . 63

3.2 Setup of the opti allatti e . . . 64

3.2.1 Potential and notations. . . 64

3.2.2 Experimentalsetup . . . 65

3.2.3 Methodto ross the latti e beams . . . 68

3.3 Calibration . . . 68

3.3.1 Kapitza-Dira dira tion . . . 69

3.3.2 Experimentalrealization . . . 70

3.3.3 Beyond the Raman-Nathregime . . . 71

3.4 Robustness against misalignments . . . 72

3.5 Con lusion . . . 73

4 A quantum parti le in a periodi potential 75 4.1 Introdu tion . . . 75

4.2.2 Computationof the band stru ture . . . 76

4.3 Mathieu equation . . . 78

4.3.1 Stability regions. . . 79

4.3.2 Computationof the hara teristi exponent . . . 80

4.4 Interpretation of as attering experiment . . . 81

4.5 Con lusion . . . 83

5 Bragg ree tion on a nite size latti e 85 5.1 Introdu tion . . . 85

5.2 Ee t of the envelope . . . 85

5.2.1 Naive pi ture . . . 85

5.2.2 Contributions of the Landau-Zenertransitions . . . 86

5.3 Preparation of the wavepa ket . . . 88

5.3.1 Out ouplingof the ondensate . . . 88

5.3.2 A eleration of the wavepa ket. . . 90

5.4 S attering on the latti e . . . 91

5.5 Experimental sequen e . . . 91

5.5.1 Time-resolved s attering . . . 92

5.5.2 Probingof the transmission diagram . . . 92

5.6 Cavity ee ts . . . 94

5.7 Con lusion . . . 95

6 Realization of tunnel barriers for matter waves using spatial gaps 97 6.1 Introdu tion . . . 97

6.2 Lo alband stru ture model . . . 98

6.2.1 Position-dependent band stru ture . . . 98

6.2.2 Transmission probabilities . . . 99

6.2.3 Comparisonwith a repulsive barrier . . . 100

6.3 Experiment . . . 101

6.3.1 Preparationof the initialwavepa ket . . . 101

6.3.2 Propagationinside the latti e . . . 102

6.3.3 Properties of the tunneled pa kets . . . 104

6.4 Dephasing between energy omponents . . . 105

6.4.1 Semi- lassi altraje tories . . . 105

6.4.2 Optimizationof the envelopeshape . . . 107

6.5 Con lusion . . . 109

7 S attering on an amplitude-modulated latti e 111 7.1 Introdu tion . . . 111

7.2 The experiment . . . 112

7.2.1 Proto ol . . . 112

7.3 Interband transitions inthe vanishingdepth limit . . . 115

7.4 The Floquet-Blo hframework . . . 118

7.4.1 Prin iple . . . 118

7.4.2 Computationof the Floquet-Blo hband diagram . . . 119

7.4.3 Ree tion on asquare envelope latti e . . . 120

7.5 Traje tories inthe semi- lassi al model . . . 121

7.5.1 Validationof the semi lassi al model . . . 121

7.5.2 Analysis ofthe ree tion me hanism . . . 123

7.6 Appli ation:velo ity lter . . . 125 7.7 Con lusion . . . 126 Con lusion 127 Résumé de la thèse 131 Appendi es 157 A Published papers 159 A.1 Zeeman slowers made simple with permanent magnets in a Halba h onguration . . . 159

A.2 Realization of a distributed Bragg ree tor for propagating guided matter waves . . . 159

A.3 Guidedatomlaser: transversemodequalityandlongitudinal momen-tum distribution. . . 160

A.4 Matter wave s attering onan amplitude-modulatedopti al latti e . . 160

A.5 Realizationof tunnelbarriers for matterwaves using spatial gaps . . 161

B More on the out oupling dynami s 163

C Magneti material hara terization 167

D Atomi ux from absorption measurements 169

E Re ir ulating oven 171

F Ideal Halba h ylinder 173

Atom opti s

The wave-parti le dualityis one of the keystones of quantum me hani s. Whilethe

wavenatureoflightwasre ognizedalreadyinthe 17th enturyby Huygens,theidea

of the wave nature of massive parti les was rst formulated mu h later in 1923 by

Louisde Broglie[1℄. Inhisthesis, deBroglieproposedthat allmassiveparti lesalso

possess wave properties hara terizedby awavelength

λ

DB

=

h

mv

(1)

where

h

is the Plan k onstant and

m

and

v

are the parti lemass and velo ity. Thewavenatureofparti leswasrst onrmedin1927byDavissonandGermer

[2℄with the observation of dira tionof ele trons on aNi kel rystal.

ThreeYearslater,thedira tionofHeliumatomsfromfreshly leaved NaCland

LiF rystals wasdemonstrated byImmanuelEstermann and Otto Stern[3℄. In this

experiment,the Heliumatomsdid notpenetrate intothe rystalandwere dira ted

bythe periodi stru tureofthe surfa e. Dira tion wasalsoobserved withneutrons

in1947 by Enri o Fermiand Leona Marshall[4℄.

These experiments set the beginning of atom opti s i.e the possibility to treat

atomsaswavesandtoperformonatomsexperimentsthatwerepreviouslyrestri ted

tolight,su hasdira tionorandinterferen es.. Numerousexperimentsdire tly

in-spiredbyopti shavethenbeenperformedwiththermalatomi beams;forexample,

the dira tion of a thermal beam by a transmission grating in the group of D.R

Prit hards in1988 [5℄ and the Young double-slit experiment realize with Helium in

1991by O.CarnalandJ. Mlynek[6℄. Theseexperiments werete hni ally very

di- ult due tothe la k of appropriate atom-opti alelements tomanipulate the atomi

beamsandtothesmalldeBrogliewavelengthasso iatedwiththe atoms. Indeed,at

atemperature

T

, the mean parti lemomentum is su hthat the thermalde Broglie wavelength de reases as mass in reases:

λ

T

= h/

√

2πmk

B

T

so

λ

T

is approximately

0.5

Å for Helium atoms atroomtemperature.

The eld of atom opti s experien ed a renewal with the development of the

Figure 1: First eviden e of

dira tion of He atoms on

a NaCl rystal. Figure

re-produ ed from [3℄.

1997 awardedtoStevenChu,ClaudeCohen-TannoudjiandWilliamD.Phillips"for

developmentofmethodsto oolandtrapatomswithlaserlight"[7,8,9℄. Bybringing

the temperature tothe sub mi rokelvinregime, the de Brogliewavelength raises in

the range of

100 nm − 1 µm

whi hmakesit relatively easyto observe interferen es withatoms. Furthermore,manyatom-opti alelementshavebeenrealizedleveraging

the use of the intera tion between light and matter. Sin e light is often the most

onvenient tooltomanipulateatoms, theroles of matterandlight anbestrikingly

reversed, and therefore, many atom opti s experiments are inspired dire tly by the

analogy with opti s. This, in parti ular will be the ase for the work presented in

this thesis.

New atomi sour es: BEC and atom lasers

Combining laser ooling te hniques and evaporative ooling, a new page of atom

opti s opens up in 1995 with the a hievement of the Bose ondensation in dilute

atomi gases in the groups of E. A. Cornell and C. E. Wiemann with Rubidium

and W. Ketterle with Sodium [10, 11℄. Based on works of Satyendra Nath Bose,

Albert Einstein predi ted in 1925 that below a riti al temperature, a ma ros opi

proportionofatomsinatrap ondenseinthegroundstate. WithintheBose-Einstein

ondensate,theindividualdeBrogliewavelengths oftheatomsoverlap,thusforming

a single oherent matter wave that extends a ross the system. Soon after the rst

experimental realization of a Bose-Einstein ondensate, these oheren e properties

were demonstrated by allowing two separated ondensates to overlap and interfere

[12℄.

Figure 2: Interferen e

pat-tern between two initially

separated Bose-Einstein

ondensate. Figure

repro-du ed from [12℄

phenomenon, however, is quite dierent: in a laser, the key ingredients are the

stimulated emission phenomena and the feedba k of light in a avity, resulting in

the ampli ation of one opti al mode of the avity. The photons, however, do not

intera t dire tly with ea h other and do not rea h thermal equilibrium. In a

Bose-Einstein ondensate the a umulation of parti les in the ground state results from

the Bose statisti sfor parti le of integer spin at thermalequilibrium. Re ently, the

analogy was pushed further with the realization of a Bose-Einstein ondensate of

light [13℄. In this experiment, photons thermalized through the intera tion with a

dye and the photon number was onserved usinga mi ro opti al avity.

Following this analogy, atom lasers were developed. The aimwas to produ e a

quasi- ontinuous oherent matter wave that possesses similar oheren e properties

as that of a ontinous opti al laser. This is done by extra ting atoms oherently

from a ondensate. Several out oupling me hanisms have been onsidered: using

a Radio frequen y spin ip [14℄, Raman beams [15℄, or by lowering progressively

the depth of the trapping potential in one dire tion [16, 17℄. Up to now, all atom

lasers operate in a pulsed mode limited by the depletion of the ondensate. If the

extra tion from the ondensate is oherent, it is possible to produ e an atom laser

whosetemporal oheren e isFourierlimited[18℄. Stri tlyspeaking, aBose-Einstein

Figure 3: Sket h of an atomi interferometer with separable arms using Bragg

dira tion on an opti allatti e. Figure reprodu ed from [30℄.

Atom opti s toolbox

Tomanipulate oherentmatterwaves, awide varietyof atom-opti alelementshave

been developed. For example, atomi mirrors an be realized using a repulsive

potentials produ ed by an evanes ent opti al wave [19, 20℄, a fo used laser beam

[20℄oramagneti eld [19℄using theStern-Gerla hfor e. Anappropriatemagneti

eld onguration reates an atomi lens [21℄. One an alsouse the dira tionon

an opti allatti e torealize anatomi lens [22, 23℄.

In this thesis, we fo used on the study of the intera tion of matter wave with a

periodi potential, whi h isin lose analogy with diele tri opti s.

In1933,KapitzaandDira [24℄predi tedthatanele tronbeam ouldbedira ted

byastandingwavelighteld. The rst experimentswere arriedoutwith ele trons

in 1965 onwards [25, 26℄ and later in 1983 with neutral atoms in the group of D.

E. Prit hard [27℄. The use of opti al latti es potentials yields many developments

in parti ular in atomi interferometry. In this ase, the dira tion is asso iated

with the absorption ofa photoninone mode of the eld followedbythe stimulated

emission inan other mode yielding a momentum hange:

∆p = 2n~k

L

, where

λ

L

is the lightwavelength and

n

the dira tion order. By dira tingatoms inthe Bragg regime orresponding to a small latti e depth and a long intera tion time [28℄, it

is possible torealize a superposition of only two spatially separated atomi beams.

In this regime,the dira tionprobabilityis importantif the in omingatomi beam

fullls the Bragg ondition [29℄

2d sin Θ = nλ

DB

,

(2)

where

Θ

is the angle between the in oming atomi beam and the latti e (

d

being the latti e spa ing). The weights of the two dira tionorders an be ontrolled by

arms [31, 32, 30℄. In these setups, the in oming atomi beam impinges on a rst

opti al latti e at the Bragg angle. Two additional latti es rst ree t ea h beam

andre ombinethemtoformanMa h-Zehnderinterferometer (seeFig.3). Be ause

the two arms ofthe interferometersare spatially separated, itis possibleto apply a

perturbation ononly one path of the interferometer.

In su hinterferometers,itis advantageoustoin rease the en losed area and the

propagation time in the arms. In this prospe t, slow atoms are favorable sin e the

Bragg angle in reases as the in ident velo ity de reases, and be ause for a given

propagation length, atoms an a umulate a phase for a longer time. In order to

maximizethe intera tion time and toprevent the atoms from fallingundergravity,

isisthenne essary touse anatomi waveguiderealizedby apotentialthat onnes

atomstoone dire tion. This potential an berealizedeithermagneti ally[33℄orby

a red detuned laser, or alternatively using a hollow- ore opti al ber guide [34℄. A

waveguidepossessestwoadvantages: rstby onningtheatomstoonedimension,it

redu esthespreadingofthewavepa ketwithtime,whi h anbeusefultopropagate

awavepa ketforalong time. Se ond,if thewaveguide ishorizontal,it ompensates

for the gravity and prevents the atoms from falling. In this way, as the atoms are

not a elerated,

λ

DB

an remain large inthe ourseof the propagation.

Atomi beamsplittersintheguided ongurationhavebeenbuilt inaYand aX

ongurationwiththermalatomi beams[35,36,37,38℄. Duringthepastyears,our

group has alsorealized abeam splitterfor guided atom lasers ona X onguration

[39℄. During this thesis, we have designed new atom opti s tools adapted to the

onned geometry. First,we havedemonstrated the realizationof a Braggree tor

thatistheexa tanalogousofaberBragggrating[40℄inopti s. Inthisexperiment,

the s atteringof amatter waveonanite size latti erealizesinparti ular dierent

typesofvelo itylter. Usingthesamenitesizelatti e,westudiedthetransposition

of Landau-Zener tunneling to position spa e. We have shown that this system

produ estunnelbarrier that are tosomeextent equivalent tosubmi roni repulsive

barrier. Finally,wehave studiedthe problemof s attering of a matterwave onto a

latti e whose depth is modulated with time. Be ause of this additionnal degree of

freedom,this systemisveryri hand allows toextendthe notionof Braggree tion

and to make the Bragg ondition tunable. This s attering an be used to realize

performanttunablevelo ityltersinguidedenvironment. Interestingly,thisvelo ity

lterdoesnot relyonany spe i internal stru ture and an thusbe used withany

atom.

Outline

•

Inthe rst hapter,I reportindetailanexperimentaldevelopmentona sepa-ratesetup. It onsistsinthe designand realization ofaZeemanSlower whose

su es-in Review of S ienti Instrument in 2011 [41℄. This Zeeman slower is the

rst step totherealization ofase ond oldatom experimentinToulouse.

Af-ter the realization of this independent proje t, I moved onto the older setup

experiment.

•

Inthese ond hapter,Idetailtheexperimentalsetupthatallowsustoprodu e RubidiumBose-Einstein ondensates.

•

Inthethird,Ipresentthesetupoftheattra tivenitesizeopti allatti ethatis atthe heart of the performedexperiments. I alsopresent the hara terization

of the latti e using Kapitza-Dira dira tion.

•

In the fourth hapter,I presenttheoreti al toolsuseful todes ribethe motion of a parti le in a periodi potential. I present the well known Blo h band

stru ture, the Blo h states and their extensionthrough the Mathieu equation

to the ase where the energy liesin a gap of the band stru ture.

•

In the fth hapter, I present the experimental realization of a Bragg mirror forguidedmatterwave. Apropagatingmatter-waveimpingesonthenitesize

latti e that a ts in ertainregions of parameteras aBragg mirrorin onned

environment. Theseresultshavebeen publishedinPhysi alReviewLettersin

2011 [42℄

•

In the sixth hapter, I present the realization of a Bragg avity for atoms trapped in the enter of the latti e between two Bragg-mirrors. Interestingly,

this system allows to realize a tunnel barrier in real spa e whi h is to some

extentequivalenttoasubmi rometri repulsivetunnelbarrier. This workhas

been re ently submitted to Physi alReview Letters and is available onarXiv

[43℄.

•

In the last hapter, I des ribe a s attering experiment on a time-modulated latti e. In this experiment, we use the modulation with time asa new degree

of freedom whi h allows to perform omplex momentum engineering on the

impinging wavepa ket. One important feature of the method is that it does

not imply internal degrees of freedom. As a onsequen e, it is of very general

useandisappli abletoanyatom. ThisworkwaspublishedinPhysi alReview

A in 2013 [44℄.

Design and realization of a

permanent magnet Zeeman slower

Contents

1.1 Introdu tion . . . 10

1.2 Theoreti al framework. . . 10

1.2.1 Prin iple . . . 10

1.2.2 Simplemodel, magneti eld shape . . . 11

1.2.3 Theadiabati following ondition . . . 12

1.2.4 Atomi stru ture . . . 13

1.2.5 Choi eofthe magneti eldparameters. . . 15

1.2.6 Longitudinal or transverse magneti eld . . . 16

1.3 The dipole onguration . . . 17

1.3.1 Choi eand modelization ofthepermanent magnets . . . 17

1.3.2 Position ofthemagnets. . . 18

1.3.3 Measuredmagneti eld . . . 20

1.4 Slowed atomi beam . . . 22

1.4.1 Experimental setup . . . 22

1.4.2 Slowed beam velo itydistribution . . . 24

1.4.3 Fluxasafun tion of thedetuning. . . 25

1.5 The Halba h onguration . . . 26

1.5.1 Theprin iple . . . 26

1.5.2 Field al ulations . . . 26

1.5.3 Magnetslayout . . . 27

1.6 Me hani s and eldmeasurements . . . 28

1.6.1 Me hani al design . . . 28

1.6.2 Shielding . . . 28

1.6.3 Fieldparameters and measurement . . . 30

1.7 Performan es . . . 31

1.7.1 Neededlaserpowers . . . 32

1.8 Con lusion . . . 35

1.1 Introdu tion

Manyatomi physi sexperimentsonquantumdegenerategasesrequirealargeinitial

sample of old atoms, usually aptured in a Magneto-Opti al Trap (MOT). The

MOT anbeloadeddire tlyfromahotatomi vapor,however, thiste hniquelimits

the quality of the va uum that an be a hieved sin e only the slowest atoms are

aptured by the MOT. In order to load e iently a MOT, it is useful to send a

slow atomi beam (typi ally less than a few tenth of m/s) onto the MOT. A wide

variety of experimental te hniques have been developed for slowing atomi beam

su h as hirped slowers [45℄ or 2D-MOTs [46, 47℄. In parti ular, sin e their rst

realisation [48℄, the Zeeman slowers (ZSs) have be ome very popular for loading

MOTs. ZSs require a spe i inhomogeneous magneti eld, whi h is generated in

most asesusing urrentsrunninginwireswoundaroundtheatomi beam. Re ently,

several Zeeman slowers using permanent magnets have been built for Sr atoms by

Y. Ov hinnikov [49℄ and more re ently by G. Reinaudi and olleagues [50℄. The

advantage of this method is to simplify the onstru tion and the fun tioning of

the ZS. In this hapter, we will present two su essive realisations of permanent

magnetsZSsfor

87

Rb,thedipoleandthe Halba h onguration. Bothdemonstrated

fullysatisfa toryoperationandhaveimportantadvantages omparedtowire-wound

setups. We will rst present the basi s of the theoreti al framework and explain

the hoi e of the dierent parameters. Then, we will present the two designs, the

experimentalrealizations and their performan es.

1.2 Theoreti al framework

1.2.1 Prin iple

AZeemanslowerusestheradiationpressurefromaresonantlaserpropagating

ba k-wardfromtheatomi beam. Ea htimeaphotonisabsorbed,theatomvelo ityalong

the beam axis de reases by

v

R

= ~k/m = 5.9 mm/s

for

87

Rb (

k

denotes the laser waveve tor and

m

the atomi mass). The photon is then re-emitted spontaneously in a random dire tion. The spontaneous emission generates a random walk in the

velo ity spa e with avanishing averageand with a nite dispersion. To slowatoms

by

∆v = 300 m/s

, about

N = m∆v/~k ≃ 50000

absorption-emission y les are

ne essary. However, as the atoms are slowed down, the Doppler ee t shifts the

laser frequen y away fromthe atomi transition, the Doppler shift reads:

where

v

is the atom velo ity. An atom an then be slowed e iently by only

∆v > Γ/k ∼ 5 m/s

, (

Γ

being the transition linewidth),whi h islargely insu ient.

To keep atoms onresonan e, the hange inthe Dopplershift is ompensated for by

anopposite hangeoftheZeemanshiftindu edbyamagneti eld

B

inhomogeneous along the dire tionof propagation

z

.

1.2.2 Simple model, magneti eld shape

Inatwo-levelatom,themeanradiationpressurefor eresultingfromthe

absorption-emission y les reads:

F =

~

kΓ

2

I/I

0

1 + 4δ

2

_/Γ

2

_{+ I/I}

0

,

(1.2)

where the detuning

δ

takes into a ounts the Doppler and Zeemanshifts

δ = ω

L

− ω

A

= δ

0

− k · v −

µB

~

,

(1.3)

where

δ

0

is the shift-free laser detuning from the transition.

I

0

and

µ

are respe -tivelythesaturationintensityandthe magneti momentasso iatedwiththe ooling

transition. In pra ti e, one shapes the magneti eld su h that the Zeeman shift

ompensates forthe Doppler shiftatalltimes (then

δ

is onstant). To ompute the orresponding eld, one usually hoose rst the apture velo ity

v

c

. This velo ity orresponds to the maximum speed above whi h an an atom annot be brought to

rest. We now suppose that an atom initially at

v

c

experien es a onstant de eler-ation. We al ulate the orresponding traje tory and dedu e in a self onsistent

mannerthe magneti eldthat keeps the transitiononresonan e.

An atom an experien e a maximum a eleration

a

0

= Γ~k/(2m) = 1.1 ×

10

5

_m/s

2

rea hed for a detuning

δ = 0

and a saturated transition

s

0

= I/I

0

≫ 1

. To keep a safety margin, the magneti eld is al ulated for only a fra tion

η

of

a

0

in the ZS design. The atom traje tory thus reads:

v(z) =

pv

2

c

+ 2az

. Ideally, the

atom velo ity must be lose to zero when the atom leaves the ZS. The apparatus

length

ℓ

isthendi tatedbythe apturevelo ityandthede eleration:

ℓ = v

2

c

/(2ηa

0

)

. The ondition

δ = cte

impliesthat the magneti eld takes the followingform:

1

B(z) = B

b

+ ∆B



1 −

p1 − z/ℓ



.

(1.4)

The inhomogeneous part ompensates for the Doppler shift. In pra ti e, for

multi-levelatoms,auniformbias eld

B

b

maybesuperimposedtoavoidlevel rossingsas we shall see in Se . 1.2.4 withthe ase of

87

Rb.

The atomi motion an be elegantly des ribed representing the a eleration in

1

z (m)

v

(m

/s

)

0

0.5

1

0

100

200

300

400

500

0.2

0.4

0.6

0.8

0

0.5

1

0

100

200

300

400

500

z (m)

v

(m

/s

)

v

c

Figure 1.1: (a) A eleration divided by the maximum a eleration in phase spa e.

(b)Simulationofafewtraje tories. Atomswithaninitial velo ity

v

i

< v

c

are slowed down followingthe resonan e ondition.

the phase spa e

(z, v)

(see Fig. 1.1(a)). We see that inphase spa e there isa thin linewhereanatomisslowedthat orrespondstotheresonan e ondition. Changing

the slowing laser detuning

δ

0

shiftsthis lineup and down. Figure1.1 (b) depi ts a few traje tories for various initial velo ities. An atom initially out of resonan e is

not slowed down, therefore, itstraje toryis anhorizontalstraightline. If the atom

initialvelo ity

v

i

< v

c

,itstraje toryrea hestheresonan e lineataposition

z

inside the slower. Theatomthenstartstobede elerated. Ifthemagneti eldvariationis

small enough (see Se . 1.2.3 for ana urate denition), the traje torythen follows

the resonan e ondition

δ

0

+ kv −

µB(z)

~

= cte.

(1.5)

Atoms with initialvelo ities

v

i

> v

c

are never slowed. At the ZSoutput, the atoms go abruptlyout of resonan e and propagate with a onstant velo ity.

Role of the parameters

Figure1.2showshowtheslowingpro essdependsonthemagneti eldandthelaser

parameters. The apture velo ity is related to the total magneti eld variation

through

∆B = ~kv

c

/µ

. The bias eld together with the laser detuning to the transitionat rest

δ

0

are related tothe nal output velo ity through

v

f

= −

1

k



δ

0

−

µB

b

~



.

(1.6)

1.2.3 The adiabati following ondition

When an atom rea hes the resonan e line, the atom traje tory lings to it and the

atom isslowed underaspe i ondition. Fora given laser intensity,the maximum

0

δ

₀

ℓ

z

ω

_L

ω

_A

0

µB

b

/h

kv(z)

kv

_f

µB(z)/h

µ∆B/h

Figure 1.2: The magneti eld variation

∆B

di tates the apture velo ity. The detuning at rest

δ

0

must almost ompensate for the Zeeman shift at the ZS output

µB

f

= µ(B

b

+ ∆B)

. The dieren e between the two provides a non-zero output

velo ity.

I

0

/I)

. It is ne essary that the a eleration hosen to determine the magneti eld parametersissmaller than

a

max

:

a < a

max

,whi himpliesthe following ondition on the laser intensity:

2

I/I

0

> η/(1 − η).

(1.7)

Besides, by taking the derivative of the resonan e ondition (1.5), we see that the

velo ity of an atom that follows the resonan e ondition must respe ts d

v/

d

z =

(µ/~k)

d

B/

d

z

. For a given laser intensity, the a eleration annot ex eed the

on-resonan e a eleration

a = v

d

v/

d

z < a

max

. This leads to the adiabati following ondition onthe magneti eld gradient:









µ

d

B

d

z









< a

0

~

_k

v

s

0

1 + s

0

,

(1.8)

thatsetsalimitonthelevelofadmissibleimperfe tions(that ausesuddenvariations

of

B

).

1.2.4 Atomi stru ture

Letus nowdetermine quantitativelythe eld parameters. First,we need toremind

the main lines of interest in the

87

Rb atomi stru ture. The ground state

5S

1/2

is omposed of two hyperne states

F = 1

and

F = 2

separated by

6.8 GHz

. There

2

Figure 1.3: Hyperne stru ture of

87

Rb. We use the transition

F = 2 → F

′

_{= 3}

as

ooling transition. Figure reprodu ed from referen e [51℄.

is a ne stru ture in the rst ex ited levels

5P

1/2

and

5P

3/2

at 795 and 780 nm respe tively. In the following, we are only interested by the

5P

3/2

manifold that omprises four hyperne states (see Fig. 1.3).

In the presen e of an external magneti eld, the dierent magneti levels are

split andthetransitionsareshiftedbytheZeemanee t(seeFig.1.4). The ooling

transition must be a losed transition so atoms perform a great number of y les

without being pumped into an un oupled state. In the ase of

87

Rb, two hoi es

ara available:

m

F

= 2 → m

F

′

= 3

and

m

F

= −2 → m

F

′

= −3

that are asso iated respe tivelywithade reasingorin reasingmagneti eld.

3

EventhoughtherstZS

used ade reasingeld [48℄,this hoi eisingeneralless favorableand makesithard

Figure1.4: Hyperne stru tures in an external magneti eld: ground levels

5

2

_S

1/2

(left pannel) and ex ited levels

5

2

_P

3/2

(right pannel). We use the transition

m

F

=

−2 → m

F

′

= −3

as ooling transition. Figure reprodu ed from referen e [51℄.

toobtain lowoutput velo ity [52℄. 4

We thus set the slowing laser on the transition

m

F

= −2 → m

F

′

= −3

that has a magneti moment

µ = −µ

B

. This transition

orresponds to an angular momentum variation

∆m = −1

. As a onsequen e, we need touse

σ

−

polarizedlight.

In addition, we observe in Fig. 1.4 that rossings between dierent hyperne

stateso uratabout

100 G

. These rossingsmustbeavoidedtopreventdepumping

tothe

F = 1

groundstate; this is a hieved by addinga bias magneti eld

B

b

.

The following table remindsome relevantphysi al onstant for

87

Rb.

m

_{1.44 × 10}

−25

_kg

µ

_{−h · 1.4 MHz/G}

Γ

_{2π · 6.1 MHz}

I

0

1.6 mW/cm

2

1.2.5 Choi e of the magneti eld parameters

The magneti eld shape has two freeparameters,

∆B

and

B

b

. The eld variation is related to the apture velo ity through

∆B = ~kv

c

/µ

. In order to apture an important fra tion of the thermal beam, the apture velo ity must be on the same

order of magnitude as the typi al thermal velo ity (

v

th

∼ 300 m/s

for an eusive beamof Rbat

130

◦

_C

)ofthe unslowedatomi beam. Wethus hoose

v

c

= 330 m/s

, this impliesthat

∆B = 300 G

. Furthermore,the bias eld

B

b

must behigh enough to avoid the level rossings around

100 G

. We hose

B

b

= 200 G

(the eld at the output of the ZS is thus

B

f

= 500 G

). The length of the ZS is set at 1 m,

4

Inade reasing eld ZS,theoutput velo ityvaries sharply asafun tion of the detuning for

Figure 1.5: (a) Conventional onguration of a wired wound ZS. The eld is

longi-tudinal, and the light polarization is ir ular. (b) ZS using permanent magnets and

a transverse eld. In this onguration, the eld is transverse. We hoose a

polar-ization orthogonalto thequantization axis that an be de omposed into two ir ular

polarization. A repumper beam is then ne essary.

whi h orresponds toana eleration about half ofthe maximuma eleration. The

followingtable presents the dierent eld parameters.

∆B

300 G

B

b

200 G

ℓ

1 m

v

c

330 m/s

η

0.48

1.2.6 Longitudinal or transverse magneti eld

In the onventional realisation of ZSs, the magneti eld is generated by urrents

running in a solenoid wrapped around the atomi beam. The magneti eld is

then ollinear with the light wave ve tor

k

. In this longitudinal onguration, it is possible to produ e lightin any polarizationstate (we dene the polarizationin

regard to the quantization axis xed by the magneti eld dire tion). However, as

we shall see, in our ase, the magneti eld isorthogonal to

k

. Thus, any in oming polarization possesses a priori

π

and

σ

±

omponents: it is not possible to reate

a pure

σ

−

polarizationstate (see Fig. 1.5 and Ref. [53℄). As a onsequen e, even

if the light with the wrong polarization is out of resonan e, the

σ

+

and espe ially

π

omponents ex ite atoms to the

m

F

′

= −2

and

−1

states respe tively. From these states, they an fall into the

F = 1

ground state. Repumping light is thus ne essary between the

F = 1

and

F = 2

manifolds.

5

The detrimentalee t of the

unwanted polarization omponents is minimizedwhen the in omingpolarizationis

perpendi ular tothe magneti eld,be ause thereis no

π

ontributioninthat ase. However, we should emphasize that if the magneti eld is transverse, half of the

light ne essarilyhas the wrongpolarizationand islost.

5

1.3 The dipole onguration

We will now present ongurations of magnets that produ e the desired magneti

eld. The rst onguration, that we all the dipole onguration, is inspired by

referen e [49℄.

1.3.1 Choi e and modelization of the permanent magnets

Letusrstdes ribethemagneti materials. Sin etheyears1980,rareearthmagnets

NdFeB have be ome progressively ompa t and heap and are now of ommon use

in industry for their ex ellent magneti properties: a residualmagneti eld

B

r

on the order of 10kG and a oer ive eld

H

ci

≃ 1, 1 kA/mm

. Spe i ally, weused the grade NE35 from HKCM that has a higher maximum operation temperature than

other grades. Su h rare earth material is very hard from a magneti point of view

so that, despite demagnetizingelds that an be as high as

H

dem

= −0.75 kA/mm

(see Apendix), the magnetization is hardly ae ted. Therefore, eld al ulations

areparti ularlysimpleand reliablebe ausethe ontributionsofea hmagnet an be

added independently.

Modelization of the magnets

It is possible to model the magnets by simple urrent distributions. Indeed, the

ve torpotential

A(x)

produ ed by the distributionof magnetization

M

(x

′

₎

reads

A(r) =

µ

0

4π

Z

V

M

(r

′

_{) × (r − r}

′

₎

|r − r

′

_|

3

d

3

_r

′

_.

(1.9)

This an be rewritten as

A(r) =

µ

0

4π

Z

V

∇ ×

 M (r

′

₎

|r − r

′

_|



+

∇ ×

M

(r

′

₎

|r − r

′

_|

d

3

_r

′

.

(1.10)

The se ond term an els out be ause the magnetization is homogeneous and the

rst term an be repla ed by the surfa e integral

A

(r) =

µ

0

4π

Z

S

n

_×

M

(r

′

₎

|r − r

′

_|

d

S,

(1.11)

where we re ognize the ve tor potential produ ed by a surfa e urrent

J

s

running aroundthe magnet(see Fig. 1.6).

Togenerate with asingle oilthe same magneti eld asthe one produ edby

a5 mmthi k magnet,one would need a urrent as high as5000 A.

From now on, we will onsider the eld produ ed by the equivalent solenoid in

Figure 1.6: A re tangular magnet is equivalent to a sheet of urrent orthogonal to

the magnetization.

and note their size

2a × 2b × 2c

. Integrating the Biot and Savart's law on one oil and then along the axis

y

, we nd forthe eld produ edby are tangular solenoid:

B

sol

(x, y, z) = µ

0

J

s

1

X

n,p,q=0

(−1)

n+p+q

B

_{(x − (−1)}

n

_{a, y − (−1)}

p

_{b, z − (−1)}

q

_{c) ,}

(1.12)

where

B

isa ve torialeld whose omponents reads:

B

x

(X, Y, Z) =

1

2

ln

√

X

2

_{+ Y}

2

_{+ Z}

2

_{− Z}

√

X

2

_{+ Y}

2

_{+ Z}

2

_{+ Z}

!

B

y

(X, Y, Z) = − arctan

Y

√

X

2

_{+ Y}

2

_{+ Z}

2

X Z

!

B

z

(X, Y, Z) = B

x

(Z, Y, X).

We used this formulain allthe al ulationson the dipole onguration.

1.3.2 Position of the magnets

IntheproposalfromY.B.Ov hinnikov[49℄,themagnetsaredisposedintworowson

ea hside ofthe entralpipe. Themagnetizationisdire tedtoward the axisand the

size and position of ea h magnetis adjustable(see Fig.1.5(b)). Weused asimilar

geometry in our rst prototype with a major improvement. In the Ov hinnikov

proposal, the magnets have a ylindri al shape (they an be roughly onsidered as

point-likedipoles)andare losetotheatomi beam. Asa onsequen e,themagneti

eld varies ona ross se tionby up to 100 Gon ase tion of CF-16pipe.

Inour onguration,weusedelongatedmagnets(

100×20×5 mm

)andwepla ed them further away from the pipe to guarantee a better transverse homogeneity.

However, we then need touse a largerof amountof magnetsto ompensate for the

eld de ay with the distan e from the axis. We xed a toleran e threshold of 12G

on the transverse homogeneity. 6

In pra ti e, the magnetsthen havetobepla edat

0

500

1000

1500

0

10

20

30

40

0

200

400

600

800

1000 1200

−200

−100

0

100

200

z(mm)

z(mm)

Thi kness (mm) Distan e to axis (mm)

Figure 1.7: Thi kness and position of the magnets in the nal onguration. The

pipe isdepi ted as the entral bla k line.

0

500

1000

1500

−100

0

100

200

300

400

500

0

500

1000

0

5

10

15

z(mm)

z(mm)

B

(G)

δB

(G) (a) (b)

Figure 1.8: (a) Target and al ulated magneti eld (dashed and plain line

respe -tively). The dieren e between the two is less than 5 G in the region of interest

200 mm < z < 1200 mm

. (b)Deviation on atransverse se tion as a fun tionof the

longitudinal oordinate.

about 10 m fromthe axis.

In order to keep the ma hiningas simple as possible,we hose a support inone

pie e,and themagnetswere allpositionedatabout10 mfromtheaxes. Toobtain

the desired longitudinal variation, we varied essentially the magnets thi kness (by

sta kingthem)andonly hangedthedistan etoaxistoobtainaneadjustment. We

used a 3D numeri al al ulation relying on the Eq. (1.12)to optimize the magnets

thi knesses and positions. In the end, we used 240 of our elongated magnets whose

positions are given inFig. 1.7.

Figure1.8(a) depi tsthe ideal and al ulated eldsonthe axis. The transverse

inhomogeneityin reasesneartheSloweroutput. AttheendoftheZS,thetransverse

inhomogeneity isbelow12 G(see Fig. 1.8(b)). To hold the magnets, webuilt two

stru tures in aluminum in whi h we drilled gaps every three m atdierent depths

Figure 1.9: Photo of the dipole onguration ZS: [A℄ Aluminum stru ture, [s℄ gaps

withvariabledepth,[p℄plexiglassprote tions,[R℄reinfor ingplates. Dimensionsare

in mm.

by addingplexiglass sli es under the magnets(see Fig. 1.9).

1.3.3 Measured magneti eld

We measured the magneti eld on the axes with a Hall probe and ompared it

with the al ulated eld. We found that the a tual eld is

12 %

larger than the simulated one, probablybe ausewe underestimated the value of the magnetization

in the al ulations. The apture velo ity is then slightly in reased, and it does not

ausefurther problems. Apart fromthis globalfa tor,the standarddeviationtothe

simulationis

8.4 G

,whi hrepresents

2.4%

ofthemeanmagneti eld(seeFig.1.10). Similarly,wemeasured theeldonatransversese tionlo atedat

5 cm

fromthe output (see Fig. 1.11). The deviation from the simulation is below

4 G

. We thus expe tthat theslowingpro ess o urse ientlyonthe wholese tionoftheSlower.

Magneti shielding

Sin e we use an important volume of magnets at a relatively large distan e from

the axis, the magneti eld de ays on

∼ 10 cm

after the output of the Slower (see Fig. 1.10). This stray magneti eld an perturb the MOT downstream the ZS;

to ir umvent this limitation,we use a magneti shielding. As a matter of fa t, a

magneti shieldingis parti ularlye ientwhenitformsasmallanglewiththe eld

lines. Wethusbuiltasoftironshieldingwhoseshapehas beenoptimizedtoforman

angleofapproximately

15

◦

50

100

150

0

100

200

300

400

500

−50

0

50

100

150

1.08

1.1

1.12

1.14

1.16

B

(G

)

B

ex

p

/B

th

z (cm)

z (cm)

Figure1.10: Left: target (dashedline), al ulated(plain line)andmeasured( ir les)

magneti eld. Right: ratio of the measured to the al ulated elds. There is an

overalldieren e of about 12 G.

−6

0

6

10

5

0

−5

−10

484

486

488

490

492

−6

0

6

10

5

0

−5

−10

1.112

1.114

1.116

1.118

1.12

1.122

1.124

B

exp

(G)

B

exp

/B

th

position(mm) position (mm) p osition (mm) p osition (mm)

Figure1.11: Left: measuredeldinatransversese tion. Right: ratioofthemeasured

to al ulated eld.

thestrayeldontheMOT hamberside,weinstalledtwoadditionalplateswith1 m

between thematthe outputofthe ZS.The residualstray eldisthe onsequen eof

the unavoidablehole needed for the pipe inthe shielding. Thanks to the shielding,

we measured an important de rease of the magneti eld at the hamber position

that falls at a few G level (see Fig. 1.12 (b)). The shielding also in reases the

eld inside the Slower. This an be simply understood using the method of image

harges. Foramagneti shield withaninnitemagneti permeability,theeldlines

are parallel to the shielding. Su h eld lines an also be produ ed by sour es of

magneti eld symmetri with respe t to the shielding. The magneti eld is then

-200

0

200

400

600

800

1000

1200

1400

-100

0

100

200

300

400

500

600

700

100 G

1er et dernier aimant

B (G)

z (mm)

rstandlastmagnet

z(mm)

B

(G

)

Figure 1.12: (a) Magneti shielding on eived to make a small angle with the eld

lines. (b) Without shielding, the magneti eld is on the 10 G range in the MOT

hamber. It falls at the G level with the shielding. Inside of the Slower, the eld is

reinfor ed by

∼ 100 G

by the shielding.

1.4 Slowed atomi beam

In this se tion,we present the experimental performan es obtained with the dipole

onguration.

1.4.1 Experimental setup

Va uum system

Figure 1.13 presents a sket h of the setup. At one end, the MOT hamber is a

spheri alo tagonfromKimballphysi s(MCF600-SO200800). It has two horizontal

CF-100 windows and eight CF-40 ports. It is pumped by a 20 L/s ion pump.

One CF-40 port is onne ted to the 1200 mm-long CF-16 pipe around whi h the

Zeeman slower is set. At the other end, one nds a rst six-way ross, used to

onne t a40L/sionpump,athermoele tri ally ooled oldngerandsix-way ross

that holds another old nger, an angle valve for initialeva uation of the hamber

and a stepper-motor-a tuated beam shutter. Finally, the in-line port holds the

re ir ulating oven (see Appendix E).

Slowing lasers

Asoutlinedinthese tion1.2.4,theZeemansloweroperatesonthe losedtransition

σ

−

_{, F = 2, m}

Figure1.13: Sket h of the overall experimental setup. [RO℄ re ir ulating oven, [BS℄

beam shutter, [CF℄ old nger, [ZS℄ Zeeman slower, [MOT℄ MOT hamber, [ZB℄

Zeeman y ling and repumping beams, [PB℄

θ = 56

◦

probe beam.

45

◦

and

90

◦

probe

beams are sent through the horizontal windows depi ted on the MOT hamber.

Di-mensions in mm, not rigorouslyto s ale.

also a

π

omponent that populates the

m

F

′

= −2

state from whi h spontaneous emissionpopulates

F = 1

groundstate. Repumpinglight isthusne essary between

the

F = 1

and

F

′

_{= 2}

manifolds. The detuning of the y ling light below the

transition orresponding to our eld parameters is

δ

0

≃ −800 MHz

. Su h rather largedetunings arerealizedby sendingamasterlaserthrough two

200 MHz

AOM's in a double pass onguration before lo king on a resonan e line using saturation

spe tros opy. The repumper is simply lo ked onthe red-detuned side of the broad

Doppler absorption prole. The two master lasers are free-running diodes with a

low

∼ 5 MHz

linewidth(SanyoDL7140-201S).Beamsarere ombinedona ubeand

pass through a polarizer. Thenthey are sent with the same polarizationintoa 1W

Tapered Amplier. A total power of more than

250 mW

is available on the atoms afterber oupling. Thebeamisexpanded toabout

23 mm

(fullwidthat

1/e

2

)and

fo usedinthe vi inity of theoven outputaperturefor better transverse ollimation

of the atomi beam.

Probe beam

Probe beams on the

F = 2 → F

′

_{= 3}

transition an be sent in the hamber

through the dierentwindows and absorptionis measuredin this way at

45

◦

or

90

◦

fromthe atomi beam (see Fig. 1.13). Probing the beam with an angle makes the

absorption or uores en e velo ity-dependent. Thus, by s anning the detuning to

the transition

∆

of the probe, we measure the velo ity distribution of the atomi beam. The orresponden e being

v =

∆

√

2

k

.

(1.13)

The uores en e is olle ted through a CF-40 port by a large aperture ondenser

lens and fo used on a

1 cm

2

PIN photodiode. Photo urrent is measured with a

-600

-400

-200

0

200

400

600

-20

0

20

40

60

80

100

120

140

160

180

fluorescence (mV)

v (m/s)

10

20

30

40

50

60

70

80

0

1

2

3

4

5

6

flux (10

10

at/s)

v

_f

(m/s)

Figure 1.14: (a) Typi al uores en e signal. The peak orresponding to the slowed

atoms issplit byZeeman ee t be ause of thestray eldin the hamber. (b)Flux as

a fun tion of the output velo ity. At

30 m/s

, the output ux is

Φ ∼ 4 × 10

10

_at/s

.

(StanfordResear hSystemsSR560)used withamoderategain(

G = 5

)and a

3 kHz

low-passlter. Absorptionis olle tedon astandard photodiode. A frequen y s an

without atoms is re orded and subtra ted tothe absorption signal. The frequen y

s ans are averagedfor 8-16 runs. During the measurements,the repumperbeamon

the

F = 1 → F

′

_{= 2}

transitionmay be turnedon.

1.4.2 Slowed beam velo ity distribution

Figure 1.14 shows a typi al uores en e signal. A narrow peak appears at

v

m

=

30 m/s

that has a velo ity spread

δv = ±10 m/s

hardly distinguishable from the

natural width of the transition. The smaller peak orresponds to the transition

F = 2 → F

′

_{= 2}

. Note that the main peak is split into two peaks separated by a

few MHz. This splitting isdue tothe stray residualmagneti eld fromthe Slower

in the MOT hamber. Adding a ompensation oil parallel to the large window of

the va uum hamber, we an merge the two peaksand estimatethat the stray eld

is about 3 G.

In order to extra t a quantitative estimate of the atomi ux, we use the

ab-sorption signal ratherthan the uores en e. In this way weeliminate allgeometri

errors. Fromthe absorption signal

A(v)

,we extra tthe maximumabsorption

A

max

, the meanvelo ity

v

¯

andvelo ityspread

δv

. Theatomuxthenreads(see Appendix D)

Φ = C sin θ cos θD

A

max

vδv

¯

λΓσ

0

,

(1.14)

0

100

200

300

760

800

840

880

920

final velocity (m/s)

2

π

δ

0

(MHz)

Figure1.15: Con atenateduores en esignalsaswevarythedetuningoftheslowing

light

δ

0

. The output velo ity varies linearly with

δ

0

. Beyond

100 m/s

, a wider peak appears due to a premature out oupling.

1.4.3 Flux as a fun tion of the detuning

Wemeasuredtheoutputvelo ityfordierentdetuningofthe y linglight. The

out-putvelo ityvariesroughlylinearlywiththedetuningwithaslope

α = 0, 95 m.s

−1

_/MHz

(see Fig. 1.15). This slope does not t with the naive model presented in the rst

se tion that predi ts a slope

α = 0, 75 m.s

−1

_/MHz

. In Appendix B, we present

further theoreti al workto explain this dieren e.

For large output velo ities, a wider peak appears, this is probably the

onse-quen e of the magneti eld imperfe tions. Indeed equation (1.8) shows that

out- oupling an o ur be ause of a bump of the magneti eld and that this ee t is

ampliedathigh velo ity.

Finally,we al ulate the total atomi ux using Eq. (1.14). The ux is roughly

onstantforoutputvelo ity

v

out

> 40 m/s

andde reases atlowvelo itybe ausethe beam be omes divergent and atoms make ollisions with the walls (see Fig. 1.14

(b)).

Φ ∼ 4 × 10

10

_at/s

at

30 m/s

whi hissu ient toload aMOT very e iently.

Thedipole ongurationZSprodu esverysatisfa toryatomux. Inthismagnet

onguration, a good transverse homogeneity is obtained at the expense of a large

distan e of the magnetsto the axis. Therefore, a large amount of magnets (

V

mag

=

2.4 L

) is ne essary whi h in reases the size of the ZS and makes it hard to an el

1.5 The Halba h onguration

1.5.1 The prin iple

The way to get a homogeneous magneti eld in a transverse ross se tion is to

distribute the magneti material all around the atomi beam to make a so- alled

Halba h ylinder. In the ontext of atom physi s, elds with a linear or quadrati

dependen e have been used to realize refra tive atom-opti al omponents [54℄. For

a ZS, a highly uniform eld is required. Following [55℄, we onsider a magnetized

rim su h that the magnetization

M

at an angle

θ

from the

y

-axis makes an angle

2θ

withrespe t tothe sameaxis(Fig.1.16 (a)). Usingsu hideal onguration, the magneti eld ishomogeneous inside the rim and an els out outside. It reads (see

Appendix F):

B

_Hal

(r) =







0

for

r > R

ext

,

B

R

ln

 R

ext

R

int



ˆ

y

for

r < R

int

,

Numeri al investigations (see next se tion) indi ate that a

8

-pole Halba h-like onguration as depi ted in Fig. 1.16 (b) is able to produ e elds on the order of

600 G

with ahomogeneitybetter than

1 G

overa

16 mm

ross se tion. Higher eld

strength and/orbeam diametersare easyto a hieve if ne essary.

More detailed studies demonstratethat deviationson a typi al

600 G

magneti eld stay below the

±1 G

limit for

±0.2 mm

mispositioningof the magnets whi h is a ommonrequirementonma hining. Likewise, the samevariationsare observed

for

±2.5%

dispersioninthestrength ofthe magnets. Arough statisti alanalysison

a sampleof

20 − 30

magnets onvin ed usthat it is the rightorder of magnitude.

1.5.2 Field al ulations

Magnets modeling

The whole setup is based on elongated

6 × 6 × 148 mm

3

NdFeB magnets(HKCM,

part number: Q148x06x06Zn-30SH)with the magnetizationorthogonal to the long

axes. Be ause these magnets have a square ross se tion (

2a × 2a

), it is possible to use a dipole approximation. The long magnets an be de omposed in a set of

ubi magnets. Then, one easily he ks numeri ally that when the distan e to the

magnetis largerthat twi e the side,the eld ofthe asso iated dipoleis ana urate

approximationofthatof thea tualmagnettobetterthan

2%

. 7

Thisapproximation

makesthe al ulations veryfast on a onventional personal omputer.

7

Figure1.16: (a) Notations for Halba h ylinder. (b) Transverse ross se tion

show-ing a 8-pole approximation of the Halba h onguration.

1.5.3 Magnets layout

In prin iple, the eld magnitude an be adjusted varying the amount, the density

and/or the position of the magneti material. The availability of very elongated

magnets (

c/a ≈ 25

) dire ted us toward a simple layout. We vary only the distan e totheaxis

d(z)

. Atrst approximationthe elongatedmagnets anbe onsideredas innite. This is equivalent to an innite array of magneti dipoles. The magneti

eld strength then de reases as the inverse of the distan e to the magnet squared.

Thus, to produ ethe eld

B(z)

a goodansatz for

d(z)

is:

d(z) = d(ℓ)

s

B(ℓ)

B(z)

.

(1.15)

As a matterof fa t, this guess turns out tobe both very e ient and lose to a

linear fun tion. Numeri al al ulations show (see Fig. 1.17) that a linear

approxi-mation of Eq. ( 1.15) an be optimized to give a eld within

±3 G

from the ideal one over themost part ofthe slower. Su hdeviationsare ompletelyirrelevant

on- erningthelongitudinalmotion. Magnetsare thenpositionedonthegeneratri esof

a one and the me hani sis then straightforward (Se . 1.6.1).

Naturally,the agreement isnot sogoodatbothends where the ideal prolehas

sharpedgeswhile thea tualeld spreadsoutand vanishesondistan es omparable

tothediameteronwhi hmagnetsare distributed. The a tual

∆B

isredu ed whi h lowers the apture velo ity and thus the beam ux. We made additional se tions

0 296 592 888 1184 100 300 500 B ( G ) z (mm)

Figure 1.17: Ideal (red) and al ulated proles, without (bla k) and with (blue) end

aps.

improvement on the eld prole at both ends (`end aps'). As seen on Fig. 1.17,

mat hing tothe ideal proleisenhan ed, espe iallyatthe higheld side where the

ideal prole exhibits amarked in rease.

1.6 Me hani s and eld measurements

1.6.1 Me hani al design

The Zeeman slower onsists in 9 mounts supporting 8 U-shaped aluminum proles

(see Fig. 1.18). The U-shaped proles go through the mounts by means of square

holes evenly spa ed on a ir le whose diameter de reases from mount to mount

a ordingtoEq. (1.4)andEq.(1.15). Magnetsarethen insertedone aftertheother

in the U-shaped prolesand lamped by a smallplasti wedge. End aps are lled

with the suitableblo k magnets and s rewed together with their spa er inthe rst

and lastmount. The wholesetup isthen veryrigid and allpartstightly positioned.

Indeed, as said before, al ulations are very reliable and Zeeman slower operation

is known to be robust so there is no need for adjustment. This setup an even

be simplied by removing some mounts: the Zeeman slower we use now has only

three mounts. Mountsare made of twoparts s rewed together. The Zeemanslower

an then be assembled around the CF-16 pipe without va uum breaking e.g. after

baking out the UHV setup.

1.6.2 Shielding

The 8-pole ongurationprodu es very littleeld outside(seeFig. 1.20(a)), ex ept

of ourse, at both ends. However, to lower stray elds even further, we have made

123

143

Figure 1.18: (a) Pi ture of the Zeeman slower: [M℄ mounts, [EC℄ output end ap

s rewed in last mount, [U℄ U-shaped proles, [S℄ half part of the shield, [sp℄

5 mm

spa er betweenend apand shieldside. (b)Individualmount; [T℄ threading tos rew

the two parts of the mount together, [P℄ entral square milling in whi h CF-16 pipe

goesthrough. ( )detailofa squareholetoshowU-shapedprolesinsertion,magnets

[m℄ and plasti wedge [W℄. Dimensions in mm.

the mounts. In this way, me hani al properties and prote tion are improved. As

seen on Fig. 1.19 (a), the inner eld is almost unae ted. On the ontrary, the

outer magneti eld falls down mu hqui ker allthe more sin e the plateau around

0.5 G

inFig.1.19 (b) isprobablyanartifa tasso iatedwith the probe. Inpra ti e,

(b) (a) 1150 1200 1250 1300 1350 1 10 200 400 B ( G ) 1184 z (mm) 0 296 592 888 1184 1 10 200 400 B ( G ) z (mm)

Figure1.19: Cal ulated (red) andmeasured (bla k) magneti eld proles. (a) S an

along the beam axis. (b)Close up of theoutput region. In the al ulation the shield

is not taken into a ount. Dotted and dashed lines indi ate the Zeeman slower and

the shield physi al ends. Log s ale below break.

1.6.3 Field parameters and measurement

The length of the Zeemanslower is

ℓ = 1184 mm

orresponding toeightse tions of

148 mm

-longmagnets. The apture velo ityisthen

v

c

≈ 450 m/s

and

∆B = 388 G

.

A bias eld

B

bias

= 200 G

isadded to avoid low-eld level rossingsaround

120 G

.

These eld parameters together with the magnet size and properties determine the

distan e and angle to the axis of the magnets. In our ase, the best hoi e was

a slope of

−15.9 mm/m

orresponding to

d(0) = 49.5 mm

and

d(ℓ) = 30.7 mm

. Entran e and outputend aps are both made of

10 mm

-side ubi magnetsof N35 grade (

B

R

= 11.7 kG

). They are lo ated on ir les whose diameters are

94.0 mm

and

66.0 mm

respe tively.

Figure1.19 displays alongitudinal s an of the magneti eld on the axis of the

Zeeman slower with end aps and shield. It an berst noti edthat the

longitudi-nal prole is intrinsi ally very smooth as the magnets make a uniform magnetized

medium throughout the Zeeman slower. After alibration of the magneti

mate-rial a tualremanent eld, deviationsfromthe al ulatedprole are less than afew

Gauss. Besides, one usually observes only lo alized mismat hes at the jun tion

be-tweentwomagnetsattributedtothedispersioninthemagnetizationofthemagnets.

The shield input and output sides atten the inner eld at both ends. Of ourse

the ee t de reases when they get further apart but the Zeeman slower should not

be lengthened too mu h. A

5 mm

spa er (tag [sp℄ in Fig. 1.18) is a good trade o. Then, the a tual magneti eld measured parameters are

B

bias

= 200 G

and

∆B = 350 G

onlyslightly smallerthan the al ulated value.

Figure1.20 depi tsatransverse utofthe magneti eld. Itisrealizedalongthe

(a) -100 -50 0 50 100 1 10 100 200 300 B ( G ) u (mm) -8 0 8 268 270 272 B ( G ) u (mm) (b)

Figure 1.20: (a) Measured magneti eld without shield a ross the beam axis at

z ∼ 460 mm

along the

u

-dire tion of Fig. 1.16. (b) Close up of the entral region.

Dashed lines indi ate the atom beam extension and a

1 G

magneti eld span. Log s ale below break. The shield was removed to allow the probe to go through. With

the shield, the inner eld is almost unae ted and the outer eld is below the probe

sensitivity.

features: (i) little outer eld (ii) highly homogeneous inner eld. In the vi inity of

the axis, the measured prole is however less at than expe ted. This is mainly

due to the nite size of the probe. Anyway, magneti eld deviations stay within a

Gauss orsointhe regionof interest. Withthe shield,the outer eldis belowprobe

sensitivity.

1.7 Performan es

The laser system for the slowing and dete tion is the same as for the dipole

on-guration. The velo ity distribution presents here a simple peak (no splitting)(see

Fig.1.21), whi his onsistent with the sub-Gauss stray eld. We alsomeasure the

velo ity distribution in the absen e of slowing light to observe the thermal

distri-bution. By integrating the thermaland slowed distribution, we determine that the

apture rate is

75%

. When we vary the detuning, the output velo ity vary linearly with

δ

0

(see Fig. 1.22 (a)). We nd in the same way as inthe dipole onguration that the ux is onstant for output velo ities over 40 m/s (see Fig. 1.22 (b)). We

do not observe a broader peak at large output velo ity (see Fig. 1.23). This is

0 200 400 600 800 0 5 100 200 0.6 0.4 0.2 0.0 0 200 400 600 800 F l u o . ( m V ) A b s . ( % ) v (m/s) 70 110 150 190 0 2 4 F l u x ( 1 0 1 0 a t o m s/ s) T (°C)

Figure 1.21: Red: thermal beam uores en e signal. Bla k: absorption and

uo-res en e signals of the slowed beam. Inset: temperature dependen e of the atom

ux. (b) (a) 700 800 900 0 100 200 Fi n a l v e l o c i t y ( m / s ) Frequency (MHz) 10 25 100 0 1 2 3 F l u x ( 1 0 1 0 a t o m s/ s) Final velocity (m/s)

Figure1.22: (a) Outputvelo ityas afun tionof thedetuning

δ

0

, weobserve alinear relation with a slope

α = 0, 95 m.s

−1

_/MHz

. (b) Atomi ux as a fun tion of the

output velo ity.

1.7.1 Needed laser powers

Be ause we use a tapered amplier we have a lot of power available in the y ling

and repumper beams. Figure 1.24 depi ts the output ux as a fun tion of y ling

and repumping power. Unexpe tedly, it turnedout that several tenth of milliwatts

of repumping light are ne essary to obtain the maximum ux. A total power of

100 mW

of y lingandrepumperlightisne essarytoget anon- riti aloperationof

the Zeemansloweratitsbest uxand anal velo ityof

30 m/s

. The orresponding intensity,

24 mW/cm

2