Interferometry With Metastable Rare Gas Atoms

7/28/2019 Interferometry With Metastable Rare Gas Atoms

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ATOM INTERFEROMETRY

I N T E R F E R OM E T R Y WI T HM E T A S T A B L E R A R E G A S A T O M SF U J I O S H I M I Z UDepartment o f Applied Physics, U niversity of Tokyo, Bunkyo-ku, Tokyo

I. In t roduc t ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153II. Atomic B e a m Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

III. Young ' s D oub le -S l i t E xp e r ime n t . . . . . . . . . . . . . . . . . . . . . . . . . 158IV. Holograph i c Ma nipu l a t i on o f At oms . . . . . . . . . . . . . . . . . . . . . . . 161V. T w o - A t o m Cor rel a t ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

I . I n t r o d u c t i o nMe t a s t a b l e r a r e g a s a t o m s o f f e r a t t r a c t i v e f e a t u re s fo r t h e s t u d y o f i n t e r f e ro m e t -r i c p h e n o m e n a o f a t o m i c wa v e s . Me t a s t a b l e r a r e g a s a t o m s , wh o s e e x c i t e d s t a t ee n e rg i e s e x c e e d s e v e ra l e l e c t ro n v o l t s , c a n b e d e t e c t e d w i t h h i g h q u a n t u m e f f i -c i e n c y b y c o n v e n t i o n a l p a r t i c l e d e t e c t o r s . Th e y c a n b e c o o l e d t o a n u l t r a c o l dt e m p e ra t u re b y l a s e r f i e l d s (S h i m i z u et a l . , 1 9 8 9 ) . Th e l o n g d e B ro g l i e wa v e -l e n g t h o f c o o l e d a t o m s i s a n a d v a n t a g e wh e n d e s i g n i n g i n t e r f e ro m e t r i c m e a s u re -men ts . In add i t ion , the lowes t exc i ted s ta te o f ra re gases has two metas tab le l ev -e l s , o n e o f wh i c h i s t h e z e ro a n g u l a r m o m e n t u m l e v e l , J = 0 . Th e J = 0 a t o m sh a v e n o m a g n e t i c l e v e l d e g e n e ra c y a n d a r e n e a r l y f r e e f ro m e l e c t ro m a g n e t i c i n -te rac t ions . Th is a l so s impl i f i es the behav io r o f a toms .

We describe, in th is chapter , three experiments with a laser-cooled metas tableneon a tom ic beam in a J = 0 s ta te . W e f irs t describe the configur at ion of our a tomicsource . Then , we show the resu lt o f the Youn g ' s doub le -s l i t in te r fe rometer , wh ich i sthe s imples t example o f a tomic in te r fe romet ry (Sh imizu et a l . , 1992a). In the fourthsec t ion , we d i scuss a more soph is t i ca ted a tom in te r fe romet r ic app l ica t ion , the ma-n ipu la t ion o f an a tomic beam by a computer -genera ted ho logram (Fu j i t a et a l . ,1996). We also describe, in the las t sect ion , an in terferometr ic effect concerningmore than one a tom, which has no t p rev ious ly been demons t ra ted exper imen ta l ly .

I I. A t o m i c B e a m S o u r c eP h o t o n s d o n o t i n t e r a c t w i t h e a c h o t h e r . Th e m a x i m u m i n t e n s i t y o f a n o p t i c a lb e a m , t h e re fo re , i s l i m i t e d o n l y b y t h e d a m a g e t h r e s h o l d o f o p t i c a l c o m p o n e n t s .In a n a t o m i c b e a m , t h e i n t e r a c t i o n a m o n g a t o m s c a n n o t b e n e g l e c t e d a n d d e t e r -

15 3 Copyright 9 1997 by Academic Press, Inc.All rights of reproduction in any form reserved.ISBN 0-12-092460-9


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154 F. S h i m i z u

m i n e s t h e m a x i m u m i n t e n s i t y . A s a r e s u l t , t h e f l u x o f a n a t o m i c b e a m i s m a n yo r d e r s o f m a g n i t u d e l e s s th a n t h a t o f a n o p ti c a l b e a m . W h e n d e s i g n i n g t h e i n t e r-f e r o m e t r i c a p p a r a t u s , i t i s i m p o r t a n t t o d e s i g n t h e b e a m s o u r c e t o m a x i m i z e t h es p a t i a l l y c o h e r e n t a to m ic f l u x .

W e c o n s i d e r r e l e a s i n g a t o m s f r o m a r e s e r v o i r t h r o u g h a h o l e h a v i n g d i a m e t e rD . O n a v e r a g e , a t o m s i n th e r e s e r v o i r c o l l i d e w i t h e a c h o t h e r i n a t i m ercoI = 1 / ( a n ) , w h e r e c~ i s th e b in a r y c o l l i s i o n a l r a t e c o n s t a n t a n d n i s t h e a to m icd e n s i t y . T h e r e l e a s e d a t o m s m u s t m o v e w i t h o u t c o l l i s i o n o v e r a m i n i m u m d i s -t a n c e , D , i f t h e y a r e to f o r m a n a t o m i c b e a m . T h e r e f o r e , rco1 -----D / v s, where vs i st h e a v e r a g e a t o m i c v e l o c i t y i n t h e s o u r c e . T h i s g i v e s t h e m a x i m u m a t o m i c f l u xF :

n v "a" D 2 _ "rrvZDF = ----~ - - . (1 )4 4 16 aT o o b t a in i n t e r f e r e n c e t h e a t o m i c w a v e h a s t o b e s p a t i a l ly c o h e r e n t , a n d o n l y t h ea t o m s e m i t t e d w i t h i n t h e d i f fr a c t i o n a n g l e

AdB __ h0 d i f = D - m v s D

c a n b e u s e d . T h e r e f o r e , t h e u s u a l f l u x F coh i s 7rh 2Fc~ = F02if - - 16am2D" (2 )

A t l o w t e m p e r a t u r e s , t h e c o l l i s i o n a l r a t e c o n s t a n t a i s n e a r l y c o n s t a n t a n d t h ec o h e r e n t a t o m i c f l u x Fdi f d o e s n o t d e p e n d o n t h e a t o m i c v e l o c i t y . G r o u n d s t a t ea t o m s c a n c o l l i d e a n y n u m b e r o f t i m e s i n t h e r e s e r v o i r . H o w e v e r , a s i n g l e c o l l i -s i o n c h a n g e s t h e i n t e r n a l s t a t e o f t h e m e ta s t a b l e a to m s . T h i s s e t s t h e l im i t t o t h ep r o d u c t o f th e d i a m e t e r a n d t h e a t o m i c d e n s i t y o f t h e t ra p . T o o b ta i n a l a r g e d e n -s it y, th e s i z e o f th e r e s e r v o i r m u s t b e a s s m a l l a s p o s s i b l e , a n d t h e m a x i m u m f lu xF i s o b t a i n e d w h e n t h e r e s e r v o i r d i a m e t e r is a p p r o x i m a t e l y e q u a l t o D . T h eJ = 0 m e t a s t a b l e r a r e g as b e a m d e s c r i b e d b e l o w a u t o m a t i c a l l y s a t is f ie s t h is c o n -d i ti o n . T h e a t o m s a r e c o l l e c t e d fr o m 4 7 r d i r e c t io n s , a r e c o o l e d , a n d i m m e d i a t e l yr e l e a s e d a f t e r t h e y h a v e b e e n c o l l e c t e d i n t h e t r a p.

I n m a n y i n t e r f e r o m e t r i c a p p l i c a t i o n s , o n l y o n e - d i m e n s i o n a l c o h e r e n c e i s r e -q u i r e d . I n s u c h a c a s e t h e f l u x is g iv e n b y

" B ' h V sF1D = F0co h = 16cema n d t h e t h e r m a l s o u r c e p r o d u c e s m u c h h i g h e r f l u x t h a n t h e l a s e r - c o o l e d s o u r c e .

F i g u r e 1 s h o w s t h e e n e r g y d i a g r a m o f n e o n r e l e v a n t t o th e a t o m i c s o u r c e .O t h e r r a r e g a s a t o m s h a v e s i m i l a r le v e l s t ru c t u re . T h e l o w e s t e x c i t e d s t a t e m u l t i -p l e x I s h a s f o u r f i n e s t r u c t u r e l e v e l s , o f w h i c h t w o a r e m e t a s t a b l e . T h e y h a v e a


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INTERFEROM ETRY WITH METASTABLE RARE GAS ATOMS 155

t r , , 2 p s ( J = 1 1 5 0 7 7 4 c m 1

2p , , 2 P g ( J = 3 ) 1 4 9 6 5 9 c m ~=...11)r

p-

(: b, " 1 s 2 ( J = 1 )

l s ~ " " l S 3 ( J = O ) 134820cm11s4(J--1)

Ee-

G ~ ~ 0 c m lGround stateF I o . 1 . T h e e n e r g y d i a g r a m o f n e o n r e l e v a n t to t he g e n e r a t i o n o f t h e u l t r a c o ld m e t a t a b l e a t o m i cb e a m . S p a c i n g s b e t w e e n t h e g ro u n d , l s , a n d 2 p s t a e s a r e n o t to s c a l e .

l i f e t im e o f o r d e r 1 0 2 s e c a n d c a n b e c o n s i d e r e d t o b e s t a b l e f o r th e p r e s e n t a p p l i-c a t i o n s . T h e m e t a s t a b l e s t a t e s h a v e a n e n e r g y o f a p p r o x i m a t e l y 1 7 e V , a n d t h ea t o m s a r e e a s i l y d e t e c t e d w i t h h i g h q u a n t u m e f f i c i e n c y b y u s i n g a c o n v e n t i o n a lc h a r g e d - p a r t i c l e d e t e c t o r s u c h a s a n e l e c t r o n m u l t i p l i e r , c h a n n e l t r o n o r m i -c r o c h a n n e l p l a t e d e t e c t o r (M C P ) . O n e o f t he m e t a s t a b l e l e v e ls h a s a n g u l a r m o -m e n tu m J = 2 a n d is u s e d t o c o o l a n d t ra p t h e a to m s . T h e c o o l in g i s o n t h e t ra n -s i t i o n b e tw e e n th e J = 2 l e v e l a n d a J = 3 l e v e l i n t h e u p p e r e x c i t e d s t a t em u l t i p l e t 2 p . T h e o t h e r m e t a s t a b l e l e v e l h a s z e r o a n g u l a r m o m e n t u m , a n d t h e r ei s n o e f f ic i e n t m e t h o d f o r c o o l i n g t h e b e a m u s i n g t h i s t r a ns i t io n . W e u s e o p t i c a lp u m p i n g f r o m t h e c o o l e d J = 2 m e t a s t a b l e s t a t e t o g e n e r a t e t h e J = 0 le v e l. F o u rJ = 1 l e v e l s in t h e 2p s t a t e a r e o p t i c a l l y c o n n e c t e d t o th e J = 2 m e ta s t a b l e l e v e l .T h e a to m s in th e J = 1 l e v e l c a n d e c a y t o a n y o n e o f t h e f o u r l e v e l s in l s b ye m i t t i n g a s p o n t a n e o u s p h o t o n . I f a n a t o m d e c a y s t o o n e o f t h e tw o J = 1 l e v e ls ,i t c a s c a d e s t o t h e g r o u n d s t a t e b y e m i t t i n g a 7 4 n m V U V p h o t o n . F o r a n a p p r o -p r i a te e x c i t a ti o n s c h e m e , a p p r o x i m a t e l y a h a l f o f th e J = 2 m e t a s t a b l e a t o m s a r et r a n s f e rr e d t o th e J = 0 m e t a s t a b l e l e v e l u s i n g o p t i c a l p u m p i n g . A l l c o o l i n g a n do p t i c a l p u m p in g t r a n s i t i o n s f o r N e , A r , a n d K r a r e e i t h e r i n t h e r e d o r n e a r -i n f ra r e d r e g i o n . T h e r e f o r e , t h o s e p r o c e s s e s c a n b e d r i v e n w i t h d i o d e o r a r g o n - i o n


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156 F. Shimizu

l a s e r pum pe d dye l a ser s . T he t r a ns fe r p r oc e s s i nvo lve s on ly a s ingl e spon ta ne ouse mis s ion p r oc e s s , a nd the he a t ing i s on the o r de r o f 1 /xK .

Both the J = 2 and J = 0 a toms a re good candida tes for an u l t r aco ld a tomicbe a m. T he y a r e ge ne r a t e d f r om the g r ound s t a t e ga s e i t he r by e l e c t r on bomba r d -m e n t o r s imp ly by d i s c ha r ge. I n op t im iz e d c ond i t i ons , a be a m f lux o f o r de r 1015sec -]1~-~ i s poss ib le . T he f ir s t s tage of the cool ing i s to send a counte r p ro paga t -ing l a se r be a m r e sona n t w i th t he l s ( J = 2) to 2 p ( J = 3) t r ans i t ion . The ve loc i tysp r e a d c a n be r e duc e d by a dd ing a c op r opa ga t ing l a se r be a m tha t ha s a s l i gh t lylow e r f r e que nc y a nd p r oduc e s a mov ing s t a nd ing w a ve w i th ve loc i ty de s igne df o r t he a tomic be a m ( F a u l s t i c h et al . , 1992) . In pr inc ip le , th i s cool ing schemec ompr e s se s t he ve loc i ty sp r e a d to t he l imi t i ng ve loc i ty o f dopp le r c oo l ing . T hee ne r gy sp r e a d m v A v , how e ve r , i s h ighe r t ha n the dopp le r - l imi t e d t e m pe r a tu r e . T oob ta in f u r the r c oo l ing , one ha s t o t r a p the a toms a nd r e l e a se t he m. W he n f r e e df rom the op t ica l and e lec t romagne t ic force , the a toms a re acce le ra ted by grav i ty .S ince the ene rgy spread remains cons tan t , the ve loc i ty spread i s r educed as thea tom s a re acce le ra ted . Fo r the J = 2 m e tas tab le a tom s, the gene ra ted b eam is in -ev i tab ly in te rmi t ten t , because the t r apping and re leas ing of a toms has to be a l te r -na t e d . T he J = 0 me ta s t a b l e a tomic be a m c a n be ma de c on t inuous , be c a use t hea toms a re no t a f fec ted by the t r apping l igh t and the quadrupole magne t ic f ie ld ofthe t rap.

F igu r e 2 show s the c on f igu r a t ion o f t he N e a tomic sou r c e w e u se d in ou r e x -pe r ime n t s . T he me ta s t a b l e a toms w e r e ge ne r a t e d u s ing a w e a k d i s c ha r ge th r ougha g l a s s c a p i l l a r y ha v ing a d i a me te r o f a pp r ox ima te ly 0 . 5 mm. T he c a thode w a sins ide the g la ss tube , and a grounded me ta l d i sk ou ts ide the cap i l la ry se rved asthe a node . T he c u r r e n t t h r ough the c a p i l l a r y w a s t yp i c a l ly 20 mA . T he sou r c epa r t w a s su r r ounde d by a l i qu id n i t r oge n sh r oud to r e duc e the a ve r a ge ve loc i tyo f t he be a m. A typ ic a l t e mpe r a tu r e o f t he be a m w a s 300 K . T he be a m pa s se d

N ed i s c h a r g et u b e

. , , 5 9 8 n m6 4 0 n m la s e r \ \ " ~ T r a n s f e r la s erC o l l im a t o r Z e e m a n t u n i n g s o l e n o i d \ \ I /d e f l e c t o r , ~ i i i i ~ ............... ~ . _ ~ IJ(2 d im e n si o n a l) iNi:,:,:,::::::::::i::::qi!::iiiiiiiiiiiiiii~i::i::ii::i~,~.. II

i s > . . . . < - . . . . . . . . . -M - L M . . I 5. .............................................................................. s 5 T r a p ~ 64 0 nm~ - - " ~ I i i i i i i i ii i i i i i i i i~ii i!! i i i i i i i i i ii i~ii":!! i i i i i i i i i ii i i i i!!!!!! i ii~~ ' ~ " = . . .. . .. . .. . .. C o o l i n g a n dt r a p p i n g l a s e r

T r a p p i n g la s e r /I l S 3 n e o n b e a m( 3 b e a m s )

/ /FIG. 2. A schematic diagram of the J = 0 metastable atomic source.


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I N T E R F E R O M E T R Y W I T H M E T A S T A B L E R A R E G A S A T O M S 1 5 7

t h r o u g h a tw o - d i m e n s i o n a l b e a m c o l l im a t o r , w h i c h c o n s i s t e d o f t w o s et s o f ac o n c a v e a n d c o n v e x m i r r o r s w h o s e c e n t e r s o f c u r v a t u r e w e r e p l a c e d a t t h e s a m ep o i n t . A r e s o n a n t l a s e r b e a m w a s s e n t p e r p e n d i c u l a r t o t h e a t o m i c b e a m , a n d i tz i g z a g g e d b e t w e e n t w o m i r r o r s a s s h o w n i n t h e f i g u r e . T h i s b e n t a n d s i m u l t a n e -o u s l y c o l l i m a t e d t h e a t o m i c b e a m t o a d i r e c t i o n p a r a l l e l t o t h e w a v e f r o n t o f t h el a s e r at t h e e x i t o f t h e m i r r o r s e t . T h i s i n c r e a s e d t h e J = 2 m e t a s t a b l e a t o m i c i n -t e n si t y b y a f a c t o r o f 3 0 a n d r e d u c e d t h e n u m b e r o f g r o u n d s ta t e a t o m s e n t e r i n gt h e t r a p p i n g r e g i o n . T h e a t o m i c b e a m w a s l e d i n t o a r e g i o n w h e r e t h e r e w a s a na x i a l m a g n e t i c f i e l d g r a d i e n t a n d s l o w e d b y a c o u n t e r p r o p a g a t i n g l a s e r , u s i n gs t a n d a r d D o p p l e r t u n i n g t e c h n i q u e s . A t th e e n d o f t h i s s l o w i n g s t a g e , t h e d ir e c -t io n o f t h e m a g n e t i c f ie ld w a s r e v e r se d , f o r m i n g a n a x i al l y s y m m e t r i c q u a d r u -p o l e f i el d. A t o m s w e r e t r a p p e d a t t h e c e n t e r o f t h e q u a d r u p o l e f i el d b y f o u r l a s e rb e a m s d i r e c t e d t o w a r d t h e c e n te r . T h e s i z e o f t h e a t o m i c c l o u d a n d i t s t e m p e r a -t u r e w e r e a d j u s t e d b y c h a n g i n g t h e m a g n e t i c f i e l d g r a d i e n t , l a s e r i n t e n s i t y , a n dd e t u n i n g f r o m r e s o n a n c e . T h e m i n i m u m v e l o c i t y w e o b t a i n e d f o r t h e n e o n t r a pc o r r e s p o n d e d t o a t e m p e r a t u re o f a p p r o x im a t e l y 8 0 / x K .

T h e J = 0 m e t a s t a b l e a t o m s w e r e g e n e r a t e d b y s e n d i n g t h e f o c u s e d l a s erb e a m i n t o t h e t r a p . T h e l a s e r f i e l d p u m p e d t h e l s ( J = 2) a toms to the 2 p ( J = 1 )l e v e l , w h i c h t h e n d e c a y e d t o t h e l s ( J = 0 ) l e v e l ( S h i m i z u et al . , 1 9 9 2 a ) . T h eJ = 0 a t o m s w e r e f r e e d f r o m t h e t r a p p i n g f o r c e s a n d d r o p p e d v e r t i c a l l y u n d e rt h e a c t i o n o f g r a v it y . T h e i n t e r f e r o m e t r i c c o m p o n e n t s w e r e p l a c e d n e a r l y v e r t i c a lf r o m t h e b e a m s o u r c e .

T h e c o l d a t o m i c b e a m g e n e r a t e d b y t h i s p r o c e d u r e h a s m a n y e x c e l l e n t f e a -tu res . F ir s t, i t i s a con t inu ou s a to mic be am wi th the cha rac te r i s t i c s o f a s ca la r f i e ldt h a t i s i n f l u e n c e d v e r y s l i g h t l y b y o p t i c a l a n d e l e c t r o m a g n e t i c p e r t u r b a t i o n s . S e c -o n d , i t s t e m p e r a t u r e i s e x t r e m e l y l o w . T h e b e a m w a s c o n s t a n t l y a c c e l e r a t e d b yg r a v i t y , a n d t h e v e l o c i t y s p r e a d r a p i d l y d e c r e a s e s a s t h e a t o m s d r o p p e d . A t a v e r -t i ca l d i s t ance o f 1 m, the typ ica l ve loc i ty sp re ad Av /v ~ V2s/(2v 2) i s o f o r d e r 1 0 - 3 .T h i r d , t h e s i z e o f t h e s o u r c e c a n b e a d j u s t e d b y v a r y i n g t h e s i z e o f t h e f o c u s e dp u m p i n g l a s e r a n d t h e t r a p s i z e .

T h e c o l l i s i o n r a t e a o f n e o n a t o m s i n t h e t ra p i s o f o r d e r 1 0 - 9 c m - 3 s e c - l .T h i s g i v e s a m a x i m u m d e n s i t y o f n ~ 1 0 ~2 c m - 3 f o r t h e t r a p d i a m e t e r D = 0 . 2m m a n d v s = 2 0 c m / s e c . T h e c o h e r e n t f l u x a c h i e v a b l e f r o m t h i s tr a p i sFco h ~ 103 sec - l a t the t rap den s i ty n ~ 1012 cm -3 . To keep th i s den s i ty , we nee da n i n c o m i n g a t o m i c f lu x o f N = ( ' r r /6 )D3n/ ' rcol - -7rv2D/(6ce ) ~ 4 109 sec -1.T h e p r e s e n t s o u r c e a n d s l o w i n g s t a g e c a n s u p p l y a fl u x t h a t i s a t le a s t 1 02 t i m e st h i s a m o u n t . T h e r e f o r e , i t w i l l b e p o s s i b l e t o f u r t h e r i n c r e a s e t h e c o h e r e n t f l u xF co h b y i n d u c i n g t h e s t i m u l a t e d p r o c e s s w h e n c o n v e r t i n g t h e a t o m i c s t a te f r o ml s 5 t o l s 3 b y o p t i ca l p u m p i n g . T h e o p t i c al p u m p i n g p r o c e s s c a n d i s s ip a t e m o -m e n t u m u p t o h/A, w h e r e A i s t h e w a v e l e n g t h o f t h e s c a t te r e d p h o t o n . I f o n e c a ns q u e e z e t h e l s 3 a t o m s i n t o a s i n g l e m o d e o f th e t r ap , t h e c o h e r e n t f l u x s h o u l d i n -c rease by a fac to r o f (D/A) 2.


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I I I . Y o u n g ' s D o u b l e - S l i t E x p e r i m e n t

T h e Y o u n g ' s d o u b l e - s l it i n t e r f e r o m e t e r i s t h e s im p l e s t a t o m i n t e r f e r o m e t e r a n dw a s f i r s t d e m o n s t r a t e d b y C a r n a l a n d M l y n e k ( 1 9 9 1 ) u s i n g a t h e r m a l b e a m .S i n c e t h e d e B r o g l i e w a v e l e n g t h o f a t h e r m a l a t o m i s o n t h e o r d e r o f i t s c l a s s i c a ls i ze , t h e s p a c i n g o f t h e i n t e r f e r e n c e p a t t e r n i s u s u a l l y e x t r e m e l y s m a l l . W h e n t h ed o u b l e - s l i t i n t e r f e r o m e t e r i s o p e r a t e d w i t h l a s e r - c o o l e d a t o m s , t h e i n t e r f e r e n c ep a t t e r n b e c o m e s s u f f i c i e n t l y l a r g e t o b e r e s o l v e d b y a c o m m e r c i a l l y a v a i l a b l ep o s i t i o n - s e n s i t i v e d e t e c t o r . A s i s s h o w n i n F i g . 3 , w e c a n p r o d u c e a n i m a g e o ft h e i n t e r f e r e n c e p a t t e r n s i m i l a r t o t h e a n a l o g o u s o p t i c a l , d o u b l e - s l i t i n t e r f e r e n c ep a t t er n s c o m m o n l y d i s p l a y e d i n o p ti c s t ex t s. B e c a u s e o f th e s m a l l k i n et i c e n -e r g y , t h e c o l d - a t o m i n t e r f e r o m e t e r i s v e r y s e n s i t i v e t o p e r t u r b i n g p o t e n t i a l s . T h ea c c e l e r a ti o n o f a to m s b y g r a v i ty i n t h e in t e r f e r o m e t e r m a y c h a n g e t h e a t o m i c v e -l o c i t y b y a n o r d e r o f m a g n i t u d e , a n d t h e a t o m i c t r a j e c t o r y i s n o t s t r a i g h t b u t p a r -abo l ic .

W e c o n s i d e r a d o u b l e - s l i t i n t e r f e r o m e t e r t h a t i s a l i g n e d p a r a l l e l t o t h e d i r e c -t i o n o f g r av i ty . T h e p h a s e d i f f e re n c e o f th e a t o m i c w a v e b e t w e e n t w o p o i n t s , r ]

. _ . " . - . . . . * . ~ . , . . .9 9 . . . " - . . j . . ' - . " . , . . . : . - . .9 . . ~ - r . . . . h a . . , . . . . 9 . .. "9 " . " - . " " " ! - - . . " : . o- . ~ . a . . . r : " " . 9 . 9. . ,:.~:.~,: : ~ . . : - ! . 9

9 " " " " . , . . . ; . : ' j '. - ' . ~ , . - A ~ r . - ~ ' . . .C , .; - , ~ ,. . ~ . - - " . .: . ; - . : . . . - . . . 9 " " . . " " .. . . ~ . . . : . . . ~ . . . ~ , . .. " " ~ - J r : " ~ i ~ : ~ 9 " ~ . ' 9 9 : " . t . . . .9 . - " . . . . - ' : : " ' " i i i i : ' : ' ~ ' ' ' ' ' ~' ~ . ' . ~ :, ~ . ~ - "" , " . ~ : ' : ' . : . ~ : : ' . : : .. . . - . . . ,

9 . :.~':~. :~: .~..-~ . , :; ~-:.-:;-...:-'. . . ., :~ ? : ~ ; . . . . .. . . . : . f ~ . ~ . , . . ' . ' . , . ' . , : . . . .. . . . . ' . " . . . . ~ . . . . - ~ ' ~ - ~ , . . ' ~ , ~ . - " ~ , - " . . - . . .9 . ' . . : - - ' - ~ " . . ' ; . ~ . ~ ' : " ' . . .9 " " " i " ,~ " - . ' .~ F ; ' . ? : _, . _ ~ , _: '~ '~ * . .. . ~ ; " " " ~ 9 "" ~ 9 " " ". . : ' . .

~ ' ~ < ~ : . ~ - . " , . -~ - , . .. = . - . . . . . : : : . ' : ."9 -:.. ? :;:.-~-~:.'.!,~e~~ ; - - ~ ; : - ; . - . . 9i : i i . ;9 ' ~ .' K _ ~ i ' . - r , ~ _ _ ~ . _ ' r ' : l ~ : , ~ . . . . , " . , ~ . ~ - . : 9 .,9 '~'~.-.. ,'.~m'r .r 9 ', . ~ . . . . . 9 -9 . , . o l e . . " . d ' x 2 " ' 9 : 1 . "" . ' "9 " " " ~" ~c ~- ' ~' ~' ~" ~ W " " . . . . . . . . . : '~ " " " "9 .: 9 : , , f~ : , ? , ~ 3 r ~ ; ~ , 7 ; ~ : i ~ . ~ , , . '. . : ' : : .: . ' . ' : ' . . . " ." ' . " ; ' ~ : : ~ ' ~" .g~ lr . " ~ . t . .= " . . : ~ : " ~ "" . ; ? ~ .. . - . 9 ~ . . . " . " : , : ~ ; " ~ . -! , , ,~ . . . . ' . , ~ ' ~ t - . . , ' i ' ~ . . ' " - ~ . . . . . . . 9. . - " " . . " . ~ " " : " ~ : :' . ~t ' -" . " ' ~' " '. " " " : ' " .

" . i ! i i ' . . - ' . . " , " :: :' ~' ~, "- '~ '~ -; *~ ,' '~ . ~: ,~ [ ~,, , ~ ! . ~ : ~ ' ; ' ~ ' ; " . . . . .. . i . : .i ' : . "~ . . . . ~ - , . , . , ~ . " , ~ x ~ : ~ = . : ~ . : 9 9 :" .9 " . . . . ~ . ' . . " . . , ' ~ '_ !" ~ o ' ~ : ~ .~ . : ' ~ 'r " ' .~ . .. " 9 . "" "9 " " . . L . . ~ . . . v . ~ . ~ t ; " , 9 " . . . , : " " - ,L ' . " " . " . , : .9 . . . . . . , ~ : ~ -~ . . . . - _ ., ,~ : - . . . : . . . .9 , " . ~ ' . ' . ' : . . : ~ - . " ~ i ~ t ~ . . . ~ , K . ~ . . ~ : % " . : ~ " . : " " .9 . . . . " . . : :.~: ; ~ . ~ ~ : ~ : ~ ; -~ . ~ . . ' . .~ : : : . _ ~ , ~ , . . ~

9 .." 9 ' .~ " ~t ,~.~ "~ .' - ? .t~: . . . . ". . , ~ : , - : . :~ : , : ~ , ,~ ' ~ .: , - ~ . _ . . . . . .9 . . . " . 9 . . ~ . , . . ~ , . ~ ; . . ~ ; ~ . : ~ r " : . . . . . . . .. . . . . , , " . " ' . . . " . . ' . : : . ! . " . . ' .- '~ : - i : , ; ,. , J ' . . . . . 9 . . , . . . . 9 .. . ' - " . ~ , . . ~ . ~ ' . . ' ~ - - ~ . . . . . ~ . . - - " . . . . .

. . " . - ' , ' = ' ~ ' - , ' a -~ .~ '? : ~ . . . - - . ~ . : . . . 9 . . . . . 9 . - . . - :. . . . : ~ . ; ' t '. - - ,- . " ~ : ~ - . , , . ~ - : , ~ . : " . , ' . . . : ' : ' : .- " ~., 9 . . . . . . :9 9 ~ l r . , . , - .~ , ~ . . : . . . . . , . . : . , . .

9 ~ ' : " , 1 - . . , , 9 ~ . ~14 ' 11 9 , , 1 , : . . . . . ~ , . . . . .. ' . - . ' . . . . . 9 ~ _ ~ . o , ~ 1 . ~ . ~ . . 9 . . . . . . . . . - . . . . .9 . " : . ' - . . . q - . , . . r ~ ' . Z ~ r ' . ' . . , . ~ . :. . ., : , . , . .. .- ~ - . . . . . .... . . . " " : . ' - ' , ' . ~ '. ~ :~ ., ~ -: , " . ~ . ~ . ~ ' 2 ~ : ~ ' , " - " . 9 " . . . . " ,9 . . " : " " - . " . , : .~ .~ : t " ' , : - ' i ' , ,~ -~ . ' ~ -~ . . ' . . . : 9 - . : . . 9 9

. . . 9 . . . , . . . . ~ - ~ - . : . . - , . : , . " . . ~ " " ~ . " - . . ' _9 - . 9 . - . .. , '- .' :. " . . . . ~ ' . ~ ~ .. . , '. . ,

F I G . 3 . T h e d o u b l e - s l i t i n t e r f e r e n c e p a t t e r n .2 r a m


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INTERFEROMETRY WITH METASTABLE RARE GAS ATOMS 159

a n d r 2 , i s o b t a i n e d i n s e m i c l a s s i c a l a p p r o x i m a t i o n b y i n t e g r a t i n g t h e w a v e v e c t o ra long the c l a ss i ca l a tomic pa th

wh ere Vg i s the g rou p v e loc i ty o f the a tom . F or an a to m tha t f a l ls ve r t i ca l ly by ad i s t ance l, the ph ase 4~ as a func t ion of the dev ia t ion ~ f rom the ve r t i ca l l i ne i s, i f~: < < l (see Fig. 4),

~ 2 m V O + v+ = c o n s t a n t + - - + ( O x 4)21 h 2w h e r e v0 and v are the ini t ia l and f ina l ve loci ty , respect ive ly. T he pha se di f ferenceArk between a tomic waves f rom two sl i t s tha t a re separa ted by d therefore , i s ,

x d m v o + VA + - - - - -l h 2

Doubleslit

> xScreen(detector)FIG. 4 . The t ra jec tory of fa l l ing a toms in the double -s l i t in te rfe rome te r .


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a n d t h e i n t e r f e r e n c e f r i n g e s e p a ra t i o n A x isl h 2a x = ( 3 )d m v o + V "

Th i s fo rm i s e x a c t l y t h e s a m e a s t h a t o f an o p t i c a l i n t e r f e ro m e t e r , w h e re t h e o p ti -c a l wa v e l e n g t h i s r e p l a c e d b y t h e a v e ra g e d e B ro g l i e w a v e l e n g t h .

In the exper imen t shown in th i s sec t ion , the a tomic source , doub le s l i t , andt h e d e t e c t o r we re p l a c e d a l o n g a v e r t i c a l l i n e . Th e d i s t a n c e b e t we e n t h e s o u rc eand the d etector , 10 + 1, w as 82 cm . Tw o paral le l s l i ts we re cu t in gol d fo i l . Eac hs li t h a d a w i d t h o f 2 / x m , a l e n g t h o f 2 m m , a n d t h e s l it s we re s e p a ra t e d b y d = 6/ x m . An a t o m t h a t p a s s e d t h ro u g h t h e d o u b l e s l i t wa s d e t e c t e d b y a t wo - s t a g em i c ro c h a n n e l p l a t e d e t e c t o r (MC P ) e q u i p p e d w i t h a f l u o re s c e n t p l a t e . Th e i m -a g e s o f i n d i v id u a l a t o m s we re r e c o rd e d o n v i d e o t a p e . Th e i r p o s i t io n s we re r e a db y a n i m a g e p ro c e s s o r f ro m t h e t a p e , a n d t h e i n t e r f e ro m e t r i c p a t t e rn wa s r e c o n -s t ruc ted f rom the pos i t ion da ta . F igure 3 shows the pa t te rn when the doub le s l i twa s p l a c e d 3 5 m m b e l o w t h e s o u rc e , a n d t h e i n i t i a l a t o m i c v e l o c i t y wa s n e a r l y0 . Th e f igu re con ta in s app rox im ate ly 8 x 103 po in t s , and the e f fec t ive accum ula-t io n t im e wa s a p p ro x i m a t e l y 1 0 m i n .

An a t o m t a k e s a p p ro x i m a t e l y a f r a c t i o n o f a s e c o n d t o r e a c h t h e MC P f ro mthe source . Therefo re , the var ia t ion o f the in te r fe rence pa t te rn wi th a tomic deB ro g l i e wa v e l e n g t h c a n b e i n v e s t i g a t e d b y p u l s i n g t h e o p t i c a l p u m p i n g l a s e rs y n c h ro n o u s l y w i t h t h e v i d e o f r a m e . F i g u re 5 s h o ws t h e v a r i a t io n o f t h e i n te r f e r -ence pa t te rn as a func t ion o f the t rans i t t ime o f the a tom f rom the source to theMC P . Th e t o p c u rv e s h o ws t h e i n t e r f e r e n c e p a t t e rn p ro d u c e d b y a t o m s wh o s ei n i t i a l v e l o c i t y wa s a p p ro x i m a t e l y 0 . Th e m i d d l e a n d b o t t o m c u rv e s a r e t h e p a t -t e rns t aken 50 and 100 msec , respec t ive ly , befo re the top cu rve . The a tomic ve-loc i ty a t the source , v0 , in those cu rves i s 50 and 100 cm/sec , respec t ive ly . Thevar ia t ion o f the f r inge spac ing wi th a tomic ve loc i ty i s c lea r ly seen in thosec u rv e s . Th e s h i f t o f t h e p e a k p o s i t i o n wa s c a u s e d b y t h e h o r i z o n t a l d i s p l a c e m e n to f the t rap and the do ub le s li t.

T h e n u m b e r o f f r i n g e s i n t h e d o u b l e s li t i n t e r f e r o m e t e r d e p e n d s o n t he r a -t i o b e t we e n t h e s l i t w i d t h d w a n d s e p a r a t i o n d ; i t i s a p p r o x i m a t e l y d / d w w h e nd > > d w . I n a n a t o m i n t e r f e r o m e t e r i t i s r a t h e r d i f f i c u l t t o i n c r e a s e t h i s n u m -b e r o w i n g t o th e l i m i t e d b e a m i n t e ns i ty . T h e l e n g t h d i f f e r e n c e o f t w o a t o m i cp a t h s i n t h e c e n t r a l p e a k i s 0 a n d a t m o s t s e v e r a l d e B r o g l i e w a v e l e n g t h s ,e v e n i n t h e o u t e r m o s t p e a k . T h e r e f o r e , t h e d o u b l e - s l i t i n t e r f e r o m e t e r i s n o t ad e v i c e t o m e a s u r e a c c u r a t e l y t h e w a v e l e n g t h o f a t o m s . H o w e v e r , i t i s e x -t r e m e l y s e n s i t iv e t o t h e d i f f e r e n c e o f p e r t u r b i n g p o t e n t i a l s b e t w e e n t w op a t h s . W e h a v e d e m o n s t r a t e d t h e p h a s e s h i f t c a u s e d b y t h e g r a d i e n t o f t h eS t a r k p o t e n t i a l ( S h i m i z u e t a l . , 1 9 9 2b ) . T h e p h a s e s h if t a c c o m p a n y i n g e l a st i cc o l l i s i o n s w a s m e a s u r e d b y S c h m i e d m a y e r e t a l . ( 1 9 9 5 ) u s i n g a t h e r m a l N ab e a m .


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INTERFEROMETRY WITH METASTA BLE RARE GAS ATOMS 161

f / )

= .

0 2 4 ~ 8 10Leng th (mm)

F IG. 5 . Va r i a t i o n o f t h e i n t e r f e r e n c e p a t t e rn w i t h t h e t r a n s i t ti m e o f th e a t o m s . T h e t r a n s i t t i m e so f a to m s f r o m t h e s o u r c e t o th e d e t e c t o r a r e 4 0 0 m s e c ( t o p c u r v e ) , 3 5 0 m s e c ( m i d d l e c u r v e ) , a n d 3 0 0m s e c ( l o w e r c u r ve ) .

IV . H o l o g r a p h i c M a n i p u l a t io n o f A t o m sTh e d o u b l e s li t i n t e rf e ro me t e r may b e co n s i d e r ed a s i mp l e fo rm o f a t o mi c b eammanipu la t ion . More e l abora te s t ruc tu res , such as a g ra t ing (Kei th et al., 1988,1991) and Fresne l zone p la te (Carna l et al., 1991) , were used to def l ec t and fo -cu s an a t o mi c b eam b y i n t e r f e ro me t r i c mean s . I n o p t i c s , h o l o g rap h i c me t h o d scan be used to man ipu la te an op t i ca l beam to an a rb i t ra ry shape by pass ing i tth rough a ho logram tha t i s a semi t ransparen t f i lm wi th an in te r fe romet r i c pa t -t e rn . For an a tomic beam, i t i s no t poss ib le to make a semi t ransparen t f i lm, andthe ho logram fo r a toms has to have a b inary pa t t e rn composed o f e i ther to ta l lyt ransparen t o r to ta l ly opaque a reas . The t echn ique to approx imate con t inuouschange o f t ransmiss ion by a b inary pa t t e rn was a l so deve loped in op t i ca l ho lo -g raphy to genera te a ho logram by computer ; th i s i s ca l l ed b i n a r y h o l o gr a p h y(Lohmsann and Par i s , 1967) . We descr ibe , in th i s sec t ion , the s imples t exampleo f a co mp u t e r -g en e ra t ed a t o m i c b i n a ry h o l o g ram t h a t g en e ra te s a Fo u r i e r t ran s -fo rmed wave f ron t o f the ob jec t wave . By t ransmi t t ing a p lane wave th rough the


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162 F. Shimizuh o l o g ra m , t h e p l a n e wa v e f ro n t i s c o n v e r t e d t o a wa v e f ro n t t h a t g e n e ra t e s t h epa t te rn o f the ob jec t a t in f in i t e d i s tance .

Th e s c h e m e t o c o n s t ru c t t h e b i n a ry F o u r i e r h o l o g ra m i s i l l u s t r a t e d i n F i g . 6 .W e d iv ide the ob jec t pa t t e rn w e wa n t to reco ns t ruc t in to N N square ce l l s ,where N i s an in teg ra l power o f 2 . We ass ign a va lue to each ce l l tha t i s equa l tothe average t ransmiss ion ins ide the ce l l . Each va lue i s then mul t ip l i ed by a ran -d o m p h a s e f a c t o r , a p ro c e d u re t h a t i n c re a s e s t h e s t a b i l i t y o f t h e r e c o n s t ru c t e dp a t t e rn a g a i n s t d e fe c t s o f t h e h o l o g ra m . Th e n , w e c a l c u l a t e t h e F o u r i e r t ra n s fo rmo f th e o b j e c t p a t t e rn b y a F F T a l g o r i th m . T h e r e s u l t in g N N c o m p l e x n u m b e r sAnn exp(i~bnm), w here n , m = 1 . . . . N , rep re sen t the t ransm i t tance and pha sec h a n g e o f c e l ls o f t h e F o u r i e r h o l o g ra m . To e x p re s s a c o m p l e x n u m b e r , i t is n e c -e s s a ry t o e n c o d e t h e p h a s e a s we l l a s a m p l i t u d e i n fo rm a t i o n . To a c c o m p l i s h t h i stask , we cu t a s l i t pa ra l l e l to a s ide o f the ce l l whose l eng th i s p ropor t iona l toAnn, at a po si t io n Ax = (~bnm/27r)Al fro m the edg e of the cel l , w he re AI is thel e n g t h o f t h e c e l l . W h e n t h e h o l o g ra m i s i l l u m i n a t e d b y a p l a n e wa v e , t h e c o m -p l e x a m p l i t u d e o f t h e wa v e t r a n s m i t t e d t h ro u g h t h e s l i t a t t h e r e f e r e n c e p l a n et h a t h a s a n a n g l e 0 = h d a /A l is p ro p o r ti o n a l t o Ann exp(iq~nm . Th i s r e f e r e n c ep lane co r responds to tha t o f the f i r s t o rder d i f f rac ted wave f rom the g ra t ing tha thas p i t ch AI . In the ho logram tha t we used in the exper imen t , a ce l l was d iv idedin to 77 77 subce l l s , and the com plex am pl i tude i s app rox im ated by the num bera n d p o s i t io n s o f t r a n s p a re n t s u b c e ll s . Th e e x a m p l e i n F ig . 6 s h o ws t h e v a l u e- 1 + 3 i when 77 = 4 . The t ransm i t ted w ave con s i s t s o f d i f f rac ted wa ves o f d if -f e r e n t o rd e r s w i t h a n a v e ra g e d i f f r a c ti o n a n g l e

O ( n , m ) = (nhaB/l,mhda/l),

O b j e c t ~ H o l o g r a m

I F ' " . . .. . 1o u r i e r " , " . .. .t t r a n s f o r m | t ! ! " ,, . .. . .. .J. - - ' " : - ' , " '~ ' :, ' " ' ~ ' ~ x. . . . . . . / a '. ,, " . . .

Z / ' / / " , " " " ' , " " ' ,. . . . . . . . . . ... " , , , S t r u c t u r e o fT . . . . . . . , . . . . ' . a c e i lJ " ' ' , , ' - - ' . - T ' , ~ x " , " .. .J. l / i H o,ogr=. .. .. j p l a n e

W a v e f r o n to f t h e l s t o r d e r o ~ = 3 r~ 2d i f f r a c t e d w a v e ( 3 i - 1 )FIG. 6. Coding of the ph ase and am plitude on a binary Fourier hologram.


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w h e r e n i n d i c a t e s t h e d i f f r a c t i o n o r d e r p e r p e n d i c u l a r t o t h e s l i t d i r e c t i o n , a n d mi s t h e o r d e r p a r a l l e l t o th e s l it . T h e i n t e n s i t y o f t h e w a v e w i t h n o r m l a r g e r t h a nr / d e c r e a s e s r a p i d l y w i t h i n c r e as i n g o r d e r , b e c a u s e t h e a m p l i t u d e o f t h e w a v e isc a n c e l e d o u t o w i n g t o p h a s e v a r i a t i o n s i n s i d e a s u b c e l l . T h e d i f f r a c t e d w a v e s o fo r d e r ( 1 , m ) p r o d u c e a p a t t e r n c o r r e s p o n d i n g t o a n o b j e c t a t i n f i n i t e d i s t a n c e .T h e w a v e s w i t h ( r / n - 1 , m ) h a v e t h e c o n j u g a t e p a t t e r n . A l l w a v e s w i t h ( r/ n , m )d o n o t c a r r y p h a s e i n f o r m a t i o n , a n d t h e i m a g e i s a s p o t .

T h e h o l o g r a m w a s f a b r i c a t e d o n a S i N f i l m h a v i n g a t h i c k n e s s o f 1 0 0 n m b yu s i n g t h e p r e c e d i n g p r o c e d u r e . T h e n u m b e r o f ce ll d i v is i o n s N w a s 1 2 8, a n d th es iz e o f t h e s u b c e l l / / r / = l/4 w a s 0 . 3 / z m . F i g u r e 7 s h o w s t h e p a t t e r n o f t h e r e c o n -s t r u c t e d i m a g e , w h e n t h e h o l o g r a m w a s p l a c e d 4 5 c m a b o v e t h e M C P .

T h e r e l a ti v e r e s o lu t i o n o f a b i n a r y h o l o g r a p h y , d e t e r m i n e d b y t h e n u m b e r o fc e l l s , i s a p p r o x i m a t e l y 1 / N , p r o v i d e d t h a t t h e a t o m i c b e a m i s m o n o c h r o m a t i c .T h e s i z e o f N i s li m i t e d o n l y b y t h e c a p a c i t y o f t h e c o m p u t e r . T h e m o n o c h r o -m a t i c i t y d e p e n d s o n t h e v e l o c i t y s p r e a d a t t h e a t o m i c s o u r c e . F o r t h e b e a m i nS e c t i o n I I, t h e v e l o c i t y s p r e a d a t t h e s o u r c e i s a p p r o x i m a t e l y 2 0 c m / s e c , a n dAv/v i s r e d u c e d t o 1 0 - 3 a t a v e rt i c a l d i s t a n c e o f l m . T h e r e f o r e , a p a t t e r n h a v i n g ar e s o l u t i o n o f 1 03 c a n b e e x p e c t e d . I n t h e e x a m p l e j u s t d e s c r i b e d , w e d i d n o t u s ea f o c u s i n g d e v i c e , a n d t h e r e s o l u ti o n o f t h e r e c o n s t r u c t e d i m a g e w a s l i m i t e d b yt h e si z e o f th e d i a p h r a g m p l a c e d n e a r t h e h o l o g r a m . T h o u g h i t i s d i f f ic u l t to d e -s i gn t w o - d i m e n s i o n a l i m a g i n g o p t i c s o f a t o m s u s i n g e l e c t r o m a g n e t i c p o t e n t i a l s ,it is re l a ti v e l y e a s y t o i m p l e m e n t t h e f o c u s i n g e f f e c t i n to t h e h o l o g r a m . W h e n t h e

(a). r : . " , ~ . : . : - : ! ': : : : . i' ~ , : ' : , , .9 ~ . . . ' . : : 9 " ~ ' . ( ' . ! . : . . , . . . . ..:."...:- '.':-V":~':"-~.~ :... :-~.G...::,:'~.:,; " ):::;"." !.:4-" ~..f~-..-~;.~, "" ....~-" .... " ' 9

::" ~" "-.,'-'~"':~:".'.T~-~i;::~~":""i;'~";~:";""""" '" ...' ~ : ' . . : . . ~ . , . . ': ~ , . . . . . ' , ; , ~ , ~ ' : , ~ ' . . . . ' . ~~. -": .... . .' .. '. .~ , 6 . ~ : ~ _ . , ~ : ~ ; ' . ~ : . ~ . ~ ~ 5 ~ . " i , ;. .. .. " , .. , % ~ ~ , , ~ . ~ . . . . .. : - ~ . ~ ~ : , ~ , ~ : . ~ ; : % , ~ .. ! . - ,, . . " .. " , , . ; : , . .. ': ~ . , ' , ~ ~ . ' X , ' q l9. ~ : - . : . ~ # . . : ; ~ '.~ : ~ . . - t . . ~ i ~ ; ~ : ~ . ~ ~ , '- .. '. % : -. :: .: .. :~ ! ~ . . : : ~ ~ ~ ~ I " - " ~ , ~ . . . .

" : . - " , " ~ 5 ~ - ? ~ - : , : ~ . i l ~~ . . , , , . . . . . . u ~ - ~ . ~ . . ~ L ~ , ~ , / ~. . . . . . , :~" . ...~...-.~.,.',..~.~ a ..: ~, ~ . ; ~ ".- r : " . 9

9 " . _ ; ' , 5 . . ' ~ ; " : ~ ~ . . ~ . ~ . . . . . - : . "9 . . . . : ~ . '

i i

5 m mFIG. 7 . Reco ns t ruc ted a tom ic pa t te rn of the hologra m: (a ) the ent ire v iew, (b) an expa nde d v iew

of the (1 ,0) d i f f rac t ion pa t tem. (F ig . 7 cont inues next page . )


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164 F . S h i m i z u

(b )i " ' " --''- ':''''~ ':--" . . . . . ' . . . . . . 1: - . . . . . - " - : , : - ' ~ . L ? ' . ~. ~ . 9 . : " . . . . . . . . . . . . . . . ~ . . . - . . -. . . - . . , : : . : : . : . : , . . . : : : . . . , " . . . . : : . . : . .. -9 : , : . : . . . . . . . . 9 . . - . . . - - . . . 9 9 .

L ' . " ' " Y" " " . . " 2 . . " . . 9 9 " . - . . : ' : " " . . . . ' , " . : : ' . 9 . .- ' . ".[ '~.i, ~" ." : " - : , " ~ ~ ; . ~ . -: : . : ,! ~ . ~ ; ." ~ . : -. & ~ .b . . : " . . " . . ' " : .F'.." ":'".-." ::~": ".~ ,'~'g ~" t~'~ :~'/' .'.~_ !~:e - ". '. "..I : ' ~ " ; : ' " " " i . ~ ~ ~ " . ~ , " : : ' ~ " ' - . : .

. . . . e , ~ . 1 , 9 : . s l . ' . . r r . . ~ . . . . 9 , , -" . . -. " .Z . . . " : : . . - ~. ~ :" "~ " '~" ' . - .~F '. ' - . ' - . . . " ' . : "I , .. u ' . ; . - : - , ~ : a ~ . . ~ ~ Q a 9 -" "." ".- .. ""9 . ~ . . : , . , . ' ,, , , ; . . , , ; . . , , . . . 9 . . . . ,! ~ , . , , - : , . - ; - ~ , ~ ~ , . - , , . . ; : . . .

"...' ,:-: ~'-. "r~ : ' # , ~ ' , r . ' 7 ~ . . , ..., - ...: 9 :..9 - . 9 : " ~ ~ . ~ . ~ . ~ . . ~ : . ~ ' . : ' .~ " . "" " "" "" "9 . . . . . - . . . . . : . . . , . , ; : - . . : . ' . , . ~ " : ' . . ~ . . 9 .." " " " " - " " ~ . " ~ % ~ , ~ . T ' ; . : - " , ' " ' - . . . . " "' """" 9 ~ .~ .~ ~ ". .: L .." ".~ " '" "'- ........ '. , . ' . : : ." , ~ : ~ . l ~ - ~ . j . , . : . - . . - ~ " : . . . . - , : .'. "." "" . . ' : A . ~ - . ~ - . . ' : - 9 . . . ' e ," , ' . . '?. . . . : 9 9 ~ . . ' ~ . . d , ' A ~ - ' , t " ,, ,; . 9 9 . . . ~ , 9 . " 9 . - . , , . ". .~ , ; . : . . . ; " . , . ~ r , . . . . . . . . ' . . " . - . . . .

I. 9 " . . . . . . . . . 9 . ' . . ~ . 9 " . . " 9 : 9 " " 9 " . o . " . ~ . " .9 . - - . . . , . . . . . . . r , . , . . . . . . . : - . ~ " . , . . ' . . "

1 m mF I ~ . 7 . ( c o n t i n u e d )

f o c u s i n g i s a p p r o x i m a t e d b y a p a r a b o l i c p h a s e s h i f t , t h e c o r r e c t i o n t o t h e F o u r i e rh o l o g r a m is a c c o m p l i s h e d b y i m p o s i n g a p h a s e s h if t o n t h e o b j e c t p a tt e r n ( M o r i -n a g a , 1 9 9 6 ) .

V . T w o - A t o m C o r r e l a t i o nS o f a r w e h a v e d i s c u s s e d i n t e r f e r o m e t r i c e f f e c t s o r i g i n a t i n g f r o m t h e q u a n t u mm e c h a n i c a l w a v e n a t u r e o f a s i n g l e a t o m . I n o p t i c s , v a r io u s h i g h e r o r d e r i n t e r f e r-o m e t r i c e f f e c t s h a v e b e e n d i s c u s s e d a n d e x p e r i m e n t a l l y d e m o n s t r a t e d . I n c o n -t r a s t m u l t i p a r t i c l e i n t e r f e r o m e t r i c e f f e c t s h a v e n e v e r b e e n e x p e r i m e n t a l l y t e s t e df o r a p a r t i c l e b e a m w i t h a f i n i t e m a s s . T h e m a j o r o b s t a c l e w a s t h e b e a m i n t e n -s i t y . T o o b s e r v e t w o - a t o m i n t e r f e r o m e t r i c e f f e c t s , t w o a t o m s h a v e t o b e f o u n d i na s in g l e e x t e r n a l q u a n t u m s ta te 9 T h i s p r o b a b i l i t y w a s c o m p l e t e l y n e g l i g i b l e w i t ha c o n v e n t i o n a l a t o m i c b e a m . T h e r e f o r e , n o m u l t i a t o m q u a n t u m e f f e c t s c o u l d b ee x p e c t e d . T h i s s i t u a t i o n h a s c h a n g e d o w i n g t o a d v a n c e s i n l a s e r c o o l i n g t e c h -n i q u e s . I n t h i s s e c t i o n , w e d e s c r i b e a n e x p e r i m e n t i n v o l v i n g a l a s e r - c o o l e d n e o nb e a m , w h i c h is t h e a t o m i c a n a l o g o f t h e H a n b u r y - B r o w n a n d T w i ss i n t e n s it yc o r r e la t i o n e x p e r i m e n t ( H a n b u r y - B r o w n a n d T w i s s, 1 9 5 7) . T o o u r k n o w l e d g e ,t h i s i s t h e f i r s t e x p e r i m e n t t o b e c a r r i e d o u t t h a t m e a s u r e s m a t t e r w a v e c o r r e l a -t io n s b e t w e e n t w o a t o m s .

T h e j o i n t p r o b a b i l i t y t o f in d a n a t o m a t ( r , t ) a n d t h e n a t ( r ' , t ' ) i s ( M a n d e l , 1 9 8 3 )e ( r , t ; r ' , t ' ) = < q ~ l S t ( r , t ) r t ( r ' , t ' ) 8 ( r ' t ' ) 6 ( r , t ) l ~ > (4)


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w h e r e 6 t ( r , t ) a n d 6 ( r , t ) a r e a t o m i c c r e a t i o n a n d a n n i h i l a t i o n o p e r a t o r s . T h ea t o m i c w a v e f u n c t i o n m a y b e w r i t t e n

1~ = E c ( n l ' n 2 . . . . ) V / n ln l , n 2 . .. ! n 2 . . . . ( a ~ ) n ' ( a t 2 ) n 2 . . . l O >

t i s t h e c r e a t i o n o p e r a t o r o f t h e a t o m t h a t is in t h e s i n g l e - a t o m e i g e n s t a t eh e r e a iq J i ( r ) e x p ( - i t o i t ) . F o r b o s o n - l i k e a t o m s , n ; c a n t a k e a n y p o s i t i v e i n t e g e r v a l u e ,w h i l e f o r f e r m i o n - l i k e a t o m s r / i = 1 o r 0 . I n s e r t i o n o f t h is e x p r e s s i o n i n t o E q . ( 4 )g i v e s t e r m s p r o p o r t i o n a l t o q J i ( r ) ~ ( r ' ) ~ ( r ) d / ~ ( r ' ) . I f t h e a t o m s a r e i n a c h a o t i cs ta te , a l l t e rm s o th e r tha n i = k , j = 1 and i = 1, j = k van i sh on a ve ra ge . Th e re -f o r e ,P ( r , t ; r ' , t ' ) = E

all combination of( i , j ) n i n j l c ( n 1 . . . . n j . . . . . ) [ e

[ l~ P i r ) 1 2 1 q , j ( r ') 1 2 + l ~ ( r ) ] 2 ] q J i ( r ') [ 2 + f 3 1 q J ~ ( r ) ~ b ~ ( r ' ) ~ ( r ) $ i ( r ' )e i ( t . o i _ o j j ) ( t - t ,) .a t_ ~ t } ( r ) ~ b ~ ( r , ) q t i ( r ) ~ . ( r , ) e - i ( o ) i _ t o j ) ( t - t ' ) } ]

w h e r e / 3 = 1 f o r b o s o n - l i k e a t o m s a n d / 3 = - 1 f o r f e r m i o n - l i k e a t o m s . W e a s -s u m e t h a t t h e e i g e n f u n c t i o n s d / i ( r ) a r e n e a r l y p l a n e w a v e s a t t h e d e t e c t o r a n d t h a tt h e d e t e c t o r s u r f a c e i s m a t c h e d t o t h e w a v e f r o n t o f t h e e i g e n f u n c t i o n s . T h e n , w em a y d r o p t h e r d e p e n d e n c e o f qt, a n d r e p l a c e t h e s u m m a t i o n

2 Eall combination of ( i , j )b y t h e i n t e g r a t i o n

n i n j l c ( n ~ . . . . n i . . . . n j . . . . ) 1 2 1 i 1 = 1 1 2

f w ( o . ) i ) W ( t . o j ) d o J t d o ) jw h e r e W ( t o ) i s the k ine t i c ene rgy d i s t r ibu t ion o f the a toms in the source . Th i s g ives

I"2(g ) = e ( r , t ; r ' , t + r ) = f W ( t . o i ) W ( ( o j ) [ 1 + 13 co s { ( ( oi - t .o j) '/ '} ]dcotdto .O u r t i m e - o f -f l ig h t m e a s u r e m e n t s h o w s t h a t t h e v e lo c i t y d i s tr i b u ti o n o f th e a t o m si s n e a r l y G a u s s i a n , w i t h a w i d t h c o r r e s p o n d i n g t o t h e t h e o r e t i c a l l im i t o f D o p p l e rc o o l i n g :

W ( v z ) = ~ e x p \ v 2 /w i t h v o = V ' h y ~ ( 2 m ) , w h e r e y is t h e n a t u r a l w i d t h o f t h e c o o l i n g t r a n s i t i o n a n d mi s t h e m a s s o f t h e a t o m . U s i n g to = m v ~ / ( 2 h ) , w e o b t a i n

/ 3F2('r) = 1 + V'I + Ato2"fl (5)


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166 F. Shimizu

wh e re A t o =m v~/ (2h ) . Th e re fo re , we e x p e c t a p e a k a t z = 0 w i t h w i d t h a p p ro x i -m a t e l y e q u a l t o t h e e n e rg y s p re a d o f t h e a t o m s i n t h e s o u rc e a n d w i t h h e i g h te q u a l t o t h e b a c k g ro u n d c o u n t .

Th e b a s i c e x p e r i m e n t a l s e t up o f t h e t wo -a t o m c o r r e l a t i o n m e a s u re m e n t i ss i m p l e . W e d e t e c t t h e 1 s 3 n e o n b e a m a t s o m e d i st a n c e f r o m t h e s o ur c e a n d m e a -su re the d i s t r ibu t ion o f the t ime in te rva l be tween two success ive pu l ses h i t t ingt h e d e te c t or . Th e d e t e c t o r s u r f a c e h a s t o m a t c h t h e wa v e f ro n t o f th e a t o m i c w a v ewi t h i n t h e a c c u ra c y o f th e l o n g i t u d i n a l c o h e re n c e l e n g t h o f t h e a t o m i c w a v e s .S i n c e a t o m s m o v e v e ry s l o wl y c o m p a re d t o p h o t o n s , t h e c o h e re n c e l e n g t h i su s u a l l y s h o r t e r t h a n t h e d e p t h o f a c o m m e rc i a l p a r t i c l e d e t e c to r . I n o u r c a s e , t h ea v e ra g e v e l o c i t y o f t h e a t o m a t th e d e t e c t o r wa s v = 4 m / s e c , a n d t h e e n e rg ys p re a d Ato/2,n'= 2 MHz . Th e re fo re , t h e c o h e re n c e l e n g t h wa s a p p ro x i m a t e l yv /(TrAto) = 0 .6 /xm. To sa t i s fy th is co nd i t ion , we used a go ld - coa ted con cav em i r ro r a s t h e m e t a s t a b l e a t o m d e t e c t o r . An e l e c t ro n wa s e m i t t e d wh e n am e t a s t a b l e a t o m h i t t h e m i r ro r s u r f a c e a n d wa s d e t e c t e d b y a m i c ro c h a n n e l p l a t ed e t e c t o r . Th e e l e c t ro n p u l s e s i g n a l f ro m t h e MC P wa s f e d i n t o a t i m e i n t e rv a lc o u n t e r a n d t h e n p ro c e s s e d t o g i v e t h e i n t e rv a l d i s t r i b u t i o n o f m e t a s t a b l e a t o m s .Th i s m e a s u re m e n t g a v e t h e t i m e - i n t e rv a l d i s t r i b u t i o n . Ho we v e r , t h e d e v i a t i o nf ro m t h e s e c o n d o rd e r c o r r e l a t i o n fu n c t i o n wa s n e g l i g i b l e i n t h e r a n g ez - 1 /(Ato ), becau se , in ou r expe r ime n t , the p robab i l i ty to de tec t mo re than onea tom in th i s per iod was very smal l .

I n t h e a c t u a l e x p e r i m e n t , a c o m p l i c a t e d p ro c e d u re wa s n e c e s s a ry t o d i s c r i m i -na te the s igna l f rom the t rans ien t e f fec t o f the e lec t ron ics . The la rges t spur ioust r a n s i e n t s i g n a l wa s c a u s e d b y t h e MC P . Th e m i c ro c h a n n e l p l a t e d e t e c t o r a p p a r -e n t l y h a d a s m a l l p ro b a b i l i ty t o p ro d u c e a n o is e p u l s e w i t h i n 1 ~ s e c wh e n i t wa sh i t by an e lec t ron . Al tho ugh th i s p roba b i l i ty was o f o rder 10 -3 o r l es s , the no i sep u l s e s c o m p l e t e l y m a s k e d t h e s i g n a l . To e l i m i n a t e t h e n o i s e s i g n a l , we d i v i d e dt h e m i r ro r s u r f a c e i n t o fo u r q u a d ra n t s , a n d e l e c t ro n s t h a t we re e m i t t e d f ro m d i f -f e r e n t q u a d ra n t s we re d e t e c t e d b y s e p a ra t e MC P s . F u r t h e rm o re , t o e n s u re t h a tt h e o b s e rv e d c o r r e l a t i o n s p e c t ru m wa s o f a q u a n t u m s t a t i s t i c a l n a t u re , we r e -p e a t e d t h e e x p e r i m e n t w i t h t wo d i f f e r e n t c o n f i g u ra t i o n s , f ro m wh i c h d i f f e r e n ts p e c t ra l s h a p e s w e re e x p e c t e d .

F i g u re 8 s h o ws t h e s e t u p fo r t h e c o r r e l a t io n m e a s u re m e n t . Th e 1 s 3 n e o na t o m i c s o u rc e , e l e c t ro s t a t i c l e n s e s , a n d t h e d e t e c t o r m i r ro r we re a l i g n e d v e r t i -c a l l y w i t h i n a n a c c u ra c y o f 3 x 1 0 - 4 . T h e m i r r o r w a s p l a c e d 8 2 c m b e l o w t h ea t o m i c s o u rc e a n d t h e e l e c t ro st a t ic l e n s wa s 3 3 . 6 c m a b o v e t h e m i r ro r. Th e l e n s ,w h i c h h a d a n a p e r t u r e o f d i a m e t e r 2 . 3 m m , e x p a n d e d t h e a t o m i c b e a m c o h e r -e n t l y t o c o v e r t h e e n t i r e m i r ro r s u r f a c e . W h e n a v o l t a g e o f 6 k V wa s a p p l i e d t ot h e le n s , t h e a n g u l a r d i v e rg e n c e o f th e a t o m s i n c re a s e d a p p ro x i m a t e l y b y a f a c t o ro f 4 0 , a n d o n l y th o s e a t o m s t h a t p a s s e d t h ro u g h t h e c e n t r a l 0 . 4 m m d i a m e t e r h i tt h e m i r ro r . W h e n t h e v o l t a g e wa s o f f , a l l a t o m s t h a t p a s s e d t h ro u g h t h e a p e r t u reh i t t h e m i r ro r . I n t h e fo rm e r c a s e , t h e a t o m i c b e a m wa s n e a r l y c o h e re n t . I n t h e


15/17

I N T E R F E R O M E T R Y W I T H M E T A S T A B L E R A R E G A S A T O M S 167

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ov oconcave ens

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c o n c a v e lens

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FIG. 8. The exper imen tal setup for the correlat ion me asure me nt.

l a t t e r c a s e , t h e p h a s e v a r i a t i o n o f t h e a t o m s e m i t t e d f ro m d i f f e r e n t p a r t s o f t h es o u rc e w a s m u c h l a rg e r t h a n 2 7 r a n d t h e b e a m c o u l d b e c o n s i d e re d s p a t i a ll y i n -c o h e re n t . Th e o n e -d i m e n s i o n a l d e fo c u s i n g l e n s p l a c e d 1 5 c m b e l o w t h e s o u rc ewa s u s e d t o e q u a l iz e t h e a v e ra g e a t o m i c c o u n t b e t w e e n t h e t wo c a s e s .

F i g u re 9 s h o ws t h e s p e c t ru m i n t wo c a s e s . Eq u a t i o n (5 ) s h o ws t h a t t h e s p e c -t ru m s h o u l d h a v e a p e a k a ro u n d ~" = 0 w i t h a w i d t h o f a p p ro x i m a t e l y 2 / A to wh e nt h e tr a n s v e r s e s p a t i a l c o h e re n c e o f th e a t o m i c b e a m i s p e r f e c t. I f t h e a t o m i cb e a m i s p a r t i a ll y c o h e re n t , t h e h e i g h t o f t h e p e a k w i l l d e c re a s e , b u t i ts w i d t h w i l lremain approx imate ly cons tan t . Th is fea tu re i s c lea r ly seen in the f igu re . By as -sum ing A to = 2 x 106 sec -1 , wh ich w as de te rm ined f rom a t ime o f f l igh t me a-s u re m e n t o f t h e v e l o c i ty s p re a d , o n e c a n f i t t h e e x p e r i m e n t a l d a t a u s i n g Eq . (5 )to obt ai n / 3 = 1 .0 __+ 0 .31 for the c oh ere nt cas e in Fig . 9(a) and f l = 0 .27 _+ 0 .22


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168 F. Shimizu

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