Angular correlation measurements in a thermal beam of H* (2s) atoms using a Stern-Gerlach atomic axicon

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Angular correlation measurements in a thermal beam of H* (2s) atoms using a Stern-Gerlach atomic axicon

J. Robert, Ch. Miniatura, S. Nic Chormaic, J. Lawson-Daku, O. Gorceix, F.

Perales, J. Baudon

To cite this version:

J. Robert, Ch. Miniatura, S. Nic Chormaic, J. Lawson-Daku, O. Gorceix, et al.. Angular correlation

measurements in a thermal beam of H* (2s) atoms using a Stern-Gerlach atomic axicon. Journal de

Physique II, EDP Sciences, 1994, 4 (11), pp.2061-2071. �10.1051/jp2:1994247�. �jpa-00248109�

Cla,,iiication Pfiv.~« _; Al?~iiii.>,

12,61) 07.60L 03,65

Angular correlation measurements in

_a

thermal beam of

H* (2s) atoms using

_a

Stern-Gerlach atomic axicon

J. Robert

('),

ch. Miniatura

('),

S. Nic Chormaic

(2),

J. Lawson-Daku

(')~

O, Gorceix

').

F. Perales

('

and J. Baudon ^'

l'l Laboratoire de Phy,ique de, LJ,er,(~l. Univer,itd Pari~-Nord. Avenue J-B- Cldnient.

9343(1 Villetaneu,e. France

(2) Penuanent addre,, Dept of Experimental Phy,ic~. St Patrick', College. Maynooth. Ireland

(Ret <>11-«<' 6 Mai /994, Jai-J,e</ 7 Septemb<>1 /994, _at apt<><' 9 .ieptem/><>1 ^/994)

Rdsumd. L'effet de gradient, magndtique, tran~ver,e, en interfdromdtrie atomique ^de type

Stern-Gerlach e,t de faire interfdrer de, onde~ plane, _ayant initialeiuent de, _vecteur, d'onde diffdrant par leurs directions. II _en rdsulte que, dans le signal d'interfdrence~ induit par le

gradient

longitudinal, le contra,te e~t attdnud par une fonction d'autocorr61ation angulaire. Cet eflet e~t dtudid expdnmentalement,ur _{un jet} thermique d'atome~ mdta~table, H (?, ), don, le _ca~ d'un gradient tran,ver,e radial (« axicon,> atomique).

Abstract. The effect of tran,ver~e magnetic

gradient~

in Stern-Gerlach atom interferometry i, to make interfere plane _~ave, the _momenta of which differ _in their direction~ A, _a re~ult the contra,t of the interference pattern

produced

by the

longitudinal gradient

_i, attenuated by _an angular auto-

correlation function _in the momentum ,pace. ^Thi, effect i,,tudied experimeiitally on a thermal beam of meta~table H I (2,) atom,, with _a radml tran,ver,e

gradient

(atomic « axicon »).

1. Introduction.

An atomic beam emitted

by

_any ^kind of _hource (an oven for alkali _{atom;, an} electron

boinbardtnent

region

for ineta;table atomh, etc. j, can be characterized if _one con~iders the

external motion

by

_a

den;ity

matrix in momentum ,pace, or

equivalently by

_a ;tati~tical

ensemble of

wavepackels~

_or

by

_a

« minimal

»

wavepacket

in the sense

given by

Gabor ]. In order to

investigate experimentally

the

angular

coherence~ in _a beam, _one need~ _a

~ignal

in which

off-diagonal

terms of the

density

matrix, p (k, k'l, with ⁱ k i

iik'ii,

_are involved.

Obviously,

_an atomic interferometer in which incident

plane

waves

having originally

different _momenta k, k' _are allowed to

finally

interfere in the detected

signal,

i~ _a device

adapted

_to the measurement of such

quantities.

If two ~uch _{waves are} made to interfere it

nece~~arily implies

that _one of them (at least) experiences a transverse momentum tran~fer,

1') A~~ocid _au CNRS. URA 282.

such that both final wavevectors be

equal.

Stern-Gerlach _or

polarization

interferomet- ry

[2]

is _one of the

Simplest

methods that can be used for this purpo~e. Let _us

briefly explain

the

principle

of this type ^of interferometer

(see Fig,

lj : first the atomic beam is

spin-polarized (polarizer

P), then _a linear

superposition

of Zeeman _states

M)

i~ built

by

a

Majorana spin- flipper (F)

the atoms then pass

through

a

magnetic

field

region (B)

where each _state

M)

accumulates its _own

phaseshift

a second

spin-flipper

IF'j builds another

superposition

of Zeeman _{states, one} of these

being finally

^filtered

by

an

analyzer (A)

_; the

emerging

atomic flux is measured

by

a detector

ID

). In this

device,

Zeeman

energie~

_F~

=

Mgp

B

B,

where g is the Landd factor and _p

~ the Bohr magneton,

play

the role of

potentials

for the external motion.

If B is

time-independent

the incident atomic _wave is

elastically

scattered

by

these

potentials

in

a way

specific

to each M value. When there _are transverse

gradient~

of B, momentum transfers k

- k'

= k ₊ 6k with k

=

k', will occur. This is

just

the desired effect and it is the

general

principle

nf the method used here. It is also worthwhile

making

~ome

practical

considerations,

particularly concerning

the orders of

magnitude

of

energies,

momentum transfers and various

sizes

iii

with the low fields used here IS G I the Zeeman

energies

l~ nev _are very small

compared

to the kinetic energy 11 0.5eV in the present

experiment using

metastable

hydrogen

atoms) _; deflection

angles

^induced

by

transverse

gradients

_are also

quite

small

(6k/k

₅

10~~

_rd)

iii)

the de

Broglie wavelength

is very ^{small at} the

macroscopic

scale of the

magnetic

field

profile

(here A

=

0.41

_{whereas B is}

_varying

_at

a mm

scale) (iit)

it is assumed that the

diaphragms provide

_a

good

collimation of the beam (the

angular

aperture ^is less than 2°

FWHM)

this allows _us to _use a

paraxial approximation,

H2

~

p

A

o

Fig. I. Experimental _{~etup :} electron gun G, polarizer P and analyzer A, spin-flipper~ F. F', magnetic

~hielding _M, detector D B I, the region where the _variou, magnetic gradient, acting _on the external

motion of H ^~ atom~ _are pre,ent.

2. General form of the

signal.

Suppose

that _a

wavepacket (produced by

the _source and all

diaphragms preceeding

the interferometer itself) is incident _on the interferometer

*,1)

= dK

i,

C iK<,

K<,)

⁽¹⁾

where

K,,)

is _a

plane

_wave of _momentum

K,,.

Becau~e of the presence of

diaphragms

within

the

interferometer,

this

wavepacket

is _not allowed to reach the detector, _even if this is very

wide. If _no

magnetic

^field is

applied

then the incident _wave (I)

produces

an

outgoing

wavepacket

V/,,~) =

dK,,

dk C

(Kj,

D

(K,,,

k

k)

₁₂₎

where the D-s _are characteristic of the inner

diaphragms.

In the present ca~e the size of these

diaphragms

is

macroscopic

(~mm) and the Fresnel diffraction at their

edges

can be

completely ignored.

Nevertheless

they

will

play

a

key

^role here.

simply

because

they

truncate the wavefronts incident _on them. AS the deflection caused

by

the

magnetic

field is

extremely

small, the effect of these

diaphragms

remains the _wme in the presence of B.

Consequently

V/,,~)

(multiplied by

the linear combination of M-states

produced by spin-flipper

F) _can be

regarded

_as the

incoming

state in the collision with the

magnetic potentials,

Conditions

(ii

and

(it) being

fulfilled, the Glauber

approximation

[31 i~

widely justified

in the treatment of the collision itself. In the _case of _a

plane

_wave

k)

incident _on the limited range

B-profile,

and for

a

specific

M level

(referred

to

quantization

axi~

fi),

_the

_phase

_at

a

point

_r

beyond

the

profile

is

readily

calculated _{as an}

integral along

the rectilinear ray

parallel

to k

passing through

this

point

~P k _{r +} M~ (k, r j (3)

where

gp~

,,rJ

~ (k, r)

=

B d-I.

(41

h~

i,J,

Here _v fit/ui where _iii i~ the atomic _{ma;~. .~} i; the absci;ha

along

the _ray. Therefore for _a

plane

_wave

[k) passing through

the entire interferometer (I,e. P. F. B, F'. A), the

outgoing

external motion ha~ the form

jj Au

_exp ii (k _{r +} M~ lk. _r

,1

where A,w are coefficient~

characterizing

F and F'. Then for the

incoming

_~tate

(2),

_{one gets} the

outgoing

state

P,~~,, =

~

_AJ~

dK,,

dk C

(K,,

D

(K,,,

k exp ii (k _r +

M~

(k, _r

))]

(5

u ~

In such conditions, _a wide detector (I.e. not

participating

in the beam collimation) measures a

time

independent

flux

proportional

to

~'oUl

=

jj Au

AIi

dK,, dK,(

dk dk' C

(K,,

D

(K,j,

k C ~

(K,[

D *

(K,[,

k' _x

,i,,i

x dr exp ii ((k k'j _{r +}

M~

(k, _r M' _~ (k', r)

j] (6j

In this very

general

form of the

signal,

it may be

already

noticed that ~ince the

integral~

over r

lead _{to a set} oi functions

f'u,,j.(k, k'),

different _{wave vector;} will be allowed to

participate

in the interference

;ignal.

A SIMPLE EXAMPLE. In order to di~cuh; _more

specifically

the role of transver;e

gradients,

let

u; con~ider the case of _a unidimensional

gradient

(I,e, the ;tandard Stem-Gerlach

configur-

ation) _:

B=(B,,+Gi)I,

for =e

(0,L(

= o elsewhere

(7)

J<JUR~AL DE PHYS'QUE II T 4 N' '' NO'EMBER lL<u4 77

where k i~

orthogonal

to the axis I of the beam. Thi~

simple

form of B is

actually

an

approximate

_one,

only

valid in the

vicinity

of the I axis. If one assumes a k _vector

slightly

inclined with respect to

I,

then the

phase

can be

explicitely

calculated irom

equations

13 and (4)

4l=

(k+M6k).r+M~P,, (81

where

This _means that the

emerging

wave is _a

plane

_wave, the momentum of which i~

k ₊ M 6k

(in optics

thi~ would be the effect of _a

prism), phaseshifted by M~Po, by

^the

longitudinal

Stern-Gerlach effect due to

B,,.

In this

special

_case the

signal given by equation (6)

can be

simplified

to

yield

( v'~~~^(~ ₌

jj

_AJ~

A(,

du

e'~'~

^{~'~'~ ~~"~~} _{r (u M 6k (u}

))

r * (u M' 6k (u )) (9)

M,w

where

r ju

=

dK,,

c

(K,,)

D

jK,,,

u

lo)

Here r represents a momentum distribution

combining

that of the wave incident _on the interferometer with the collimation effect due _to the inner

diaphragms.

It is

clearly

_seen in

(9)

that the

longitudinal

Stem-Gerlach interference pattern,

governed by

the

phaseshift

(M M')

4lj,,

^{will be}

damped by

an

an,qulai

auto-c.r~iielatir~n

jiifictir)fi.

In

principle

it is

possible

to

change

either C

(Kj,), by acting

on the _{source, or} (more

easily) D(K,j,

u),

by changing

the inner

diaphragms.

Notice that if _a ~tatistical ensemble of

wavepackets

is considered, _one has to

replace

rr* in

(9) by

the average

@ (off-diagonal

element of the

density

matrix).

An

expression

of the

outgoing

state

(Eq.

(5)) in the

~pecial

case of _a unidimensional

gradient

was

given

_a

long

time ago

by

Bloom and Erdman

[4].

The

expression they

^obtained for the

signal

is identical to that

given

ⁱⁿ

equation

(9), except ^that interferential crossed terms are

missing simply

because these authors

only

considered the effect of the

magnetic

field and _not of _an interferometer.

REMARK ABOUT THE CALCULATION usiNG cLAssicAL TRAJECTORIES. Let _us consider _a

monokinetic beam

consisting

of

independent

atoms, each of them

following

_a classical

trajectory (approximately

_a

straight line)

at a given

velocity

_vi,.

Actually

^the external motion is seuii-classical since, for each M, value _a

phase

is accumulated

along

the

trajectory.

The

phase

difference

(M

M') _~ can be calculated as before

by

_use of the Glauber

approximation.

For

the sake of

~implicity

let _{us assume} that _a

single

inner

diaphragm

ID is present. The

signal

takes the form

S

=

jj

_AJ~J~. dS _exp [I (M M') _~ (l

,i,,i

D

where _p

=

4l~

+ 6k.x. One may notice that for Ml M' the

integral

is similar to the

Fraunhofer diffraction

auiplitude

in the direction k'= k+ 6k.

Using

the definition of

D(Kj,,

k)

(Eq.

(2) and

Eq. (8)),

^{it is}

easily

verified that

e'"~

^{~'~ '~"}

ldu

^D

^(Kj,,

u M 6k D *

(K,,,

u M' &k j ₌

=

ids e'~'~

^~'~ "~" _~ ~~ ''

=

dS

e'~~

^~'~ ~ l

2)

D

D

This _means that the

signal given by equation

is identical _to that

given by equation (9)

for

an incident

plane

wave of momentum

K,j uivj,/h.

The

physical meaning

of this result is the

following (cf.

the treatment of

impact

parameters ⁱⁿ semi-classical

collisions)

the size of the

diaphragm

^is so

large

that it _can be divided into domains

A,, ~~,

...,

A,,, ^{the sizes of which}

are

large compared

to A, but also small

enough

to

give

_an almost _constant value

~,, of _~ for all the rays

starting

from any

point

in

~,,.

^The intensities

corresponding

to all these domains have _to be added _to get ^the

signal.

It is then

easily

under~tood that the distribution of initial

phases

among the different domains A,,

plays

_no role in the final

intensity, giving

the

same result for _an

incoming plane

wave

[K,,)

and for _a random distribution of the initial

phases.

This property can be

generalized

to any

configuration

of B and _any

configuration

of

inner

diaphragms, provided they

_are wide _{at a} A-scale. The fact that in this _{case a} treatment

using quasi-geometrical optics

is valid does not at all invalidate the _more

general

result~

derived

previously (e.g. Eqs.

(6) and (9)), which involve

angular

correlation functions in

momentum space.

A CYLINDRICALLY SYMMETRIC GRADIENT. The _use of _a Cartesian-unidimensional

gradient

is

experimentally

difficult because it cannot be

produced independently

of _a uniform field

Bj, which in addition needs to be

sufficiently large (B,,»

G i-vi ). As _a consequence the

interference order for the

longitudinal

Stem-Gerlach effect is

necessarily high

(m 6) which

cau~es a severe loss of contrast, even if _a

velocity

selection is made, because of unavoidable

instrumental

imperfections

_(«

Humpty-Dumpty

_» ^effect

[5().

Most of the

experiments

de-

scribed later use a

magnetic quadrupole.

This consists of two identical and

orthogonal pairs

of

oppositely

wound

rectangular

Helmholtz coils. In the

vicinity

of the I axis the field _can be

approximated by

B

= G (- ii ₊

_vj),

for _{= e} [0. L

= o elsewhere II ~)

Within the interval [0, L], the field lines _are

hyperbolae

_{(i_>.} C~t. and the

magnitude

of the field is B

=

Gi~,

where i~ is the distance _to the I axis. This lead~ to a radial

gradient

of B. For each M value this

configuration

behaves _{as a}

parallel plate,

the index of which is

proportional

to i~, I-e- _an

optical

_« axicon

». For _an incident

plane

wave

k),

the momentum of which lies

in direction

I(0,

_p,

y ) with p « y w I,

equation

(4)

give~

qJ (k, r)1 ^{~~ ~} G

~'~

dir

.v~

+ »

~

= +

pit j

^~'~

'4) fir

,, y

For _a

specific

M value, the total

phase

at a

point

_r

beyond

the B

region

(= m

L)

is then

~P

=

I.(P-v

₊ Y=) ₊

M~. jis)

A _{wavevector can} be defined _at this

point by

K(rj

=

V,4l

k+M 6k(r) (16)

6k(rj

has the

following

components

&l~

j~~

G

~

(Ar;h )

_Ar~h

)

ii P ' '

61,

=

j~~

G

) ^{(,fi ,fi)}

_'7)

iv

61- ~

fit,

Y

where l'j = v

~

= ><, ₌ i'-

~

IL

₌). It may be noticed that k &k

=

0 (elastic collision)

Y Y

and that

61=[

«

61,.

If p

=

0, _one get~

&k1 ~~ _~

GLi~

(18)

In other word; the outgoing ^waveiront (for each M ;tate) _i, a cone, the normal oi which make;

an

angle ~~~

^{GL with I, where E is the kinetic} energy. ^It can be shown that thi~ re~ult E

remains

approximately

true for

p

# 0 but ~mall and in the

vicinity

of the output

plane

= = L, where 6k has _an

expre;sion

^similar to (18),

i~ being replaced by

another radial unit

vector I

[

centered at the point (.I

=

0. _>'= ~

L).

2

In order to calculate the

;ignal

obtained in this _{case more}

explicitely

we can u~e a « coaoe

graining

_» method I-e- divide the detector

plane (assumed

to be _{not too} far

beyond

the output

plane

₌

=

L) into domain~

(di)

^centered at

points

i-I, the ~ize; oi which _are at the _same time

large compared

to and ~mall

enough

to con;ider that

6k=61i~

is uniiorm _over

di.

Then starting ^irom equation (6) and after _~ome calculations _one

finally

gets

i P~,~,,11^~ = Cst,

jj

AJ~J~ du r(u ) r ~[u ₊ (M M') 61(ii _FL (19)

~iu

an

expre~sion

_very ~imilar to (9) (with _4l,>

= 0), except ^{that the direction oi the} momentum transier is _no

longer

fixed but I; radial. In thi~ _case the

longitudinal

Stem-Gerlach interference will be

damped by

a « radial

»

angular

autocorrelation function.

3.

Experiment

and discussion.

The

general principle

of the

experiment

ha; been

already explained

in section I, and _a detailed

description

of the technical a,pect; ^ha~ been given

previously

elsewhere [6].

Only

the main features will be

briefly reported

here. A beam of H~ (?s) atom~ is

produced by

electronic

bombardment _{or a} thermal beam oi

Hi

molecules. The main contribution to the

resulting

normalized time of

flight

distribution is well fitted

by

f'(ii

=

12.5 _ii~ ^~ exp (- ?.5 ii~ ^~

(20)

where _ii

=

tit,,,

t,,

being

the mo~t

probable

time of

flight,

which

corresponds

to a

velocity

of 10km/~ (i.e.

A10.41).

_{The bedim is}

_{partially polarized}

_{in the}

_hyperfine

_~tate;

_?sj~~,

F

=

I.

M~

=

0,

by

_passage

through

a transverse field

Bp

of 600 G (Lamb and Retherford's

method

[7]j.

The

Majorana spin flipper

F operate~

by

a non-adiabatic passage from the

attenuated

Bp

field to the iield

existing

in the B

region.

In the low fields used here,

H'~ _atoms behave _ah

spin particles,

hence the

operation

_can be de~cribed

by

_a

Wigner

rotation matrix D

(p

=

D~

'(0, p, 0).

In the

following

treatment both

velocity

and field

dependences

oi the Euler

angle p

will be

ignored.

The spin

flipper

F' and the

analyzer

A work in the _same

manner a; F and P

respectively. Finally

the radiative

decay

of

emerging

atom~ i~ induced

by

~tatic Stark

quenching

and the

re~ulting Lyman

_cv

photons

are detected

through

_a

Mgfj

window

by

a channel electron

multiplier.

In

region

B (~ee

Fig.

2) the

magnetic

field is

produced by

_a

magnetic quadrupole (ci.

Sect.

?)

in addition to two

parallel

horizontal wires which

produce

a uniform field

B,, j

in the

vicinity

of

the I axis. The total iield _can be

approximated by

Bi[-Gii+(B,,+G>.)I(,

ior _=e

(0,L(

~ ° el~ewhere

(?

This iield has

properties quite

,imilar to those of _a

quadrupole

iield

(Eq.

l~)) except ^that now

i~ I; centered in the (.i, _i')

plane

on the

point

(0,

Bjj/G).

z

2L

x 2d

x

io _io

~i

^y z

' 2h

Fig. ?.- The magnetic

contigurition

_u;ed

in region ^B con,i,t~ of _two pa>r, of orthogonal anti- Helmholtz coil~

~upplied

^with current ii and of _a ,traight ^frame ;upplied ^with current in (L

=

4? _mm,

</=6 _mm, h =8 mm). Such _a configuration tran,form, _a plane _~ave

(K,j)

with internal ,tote

M)

into _a conical _wave.

In the first ~et of

experiments B,,

i~ scanned ior _a fixed value of G

(Fig.

3). Here the inner

diaphragmq

^consist of two holes of radiu~ 2 _mm

separated by

15 cm. It i~ worthwhile

noticing

that

only

the~e hole; deiine the beam collimation. When G

=

0 _one finds the standard

longitudinal

Stern-Gerlach pattern ^with

only

a few

fringes

^visible since the atom~ are not

velocity

~elected. The contrast around the central

bright fringe

^is about 30 %. A, G is increased

the overall contrast

continuously

decreases and _{no more}

fringes

are ob~erved _once

G ~ ??5 mG/cm.

In the ~econd set of

experiment,

_we have studied, with the _same set of collimation holes, the attenuation oi the central

fringe _(B,j

= 0), oi the first minimum

(B,,

=

3~ mG and of the iir;t lateral _maximum

(B,,

=

78 mG ), as a function of G

Fig.

4).

Finally

in the third ~et oi

experiments

^{the effect} of the inner collimation _on the central

fringe

attenuation ha~ been

investigated (Fig.

5). When the radius of both hole~ is reduced

by

_a factor of ?, it is _~een that, apart ^from an obvious reduction of the atomic flux, the _new pattern ^{is twice}

a~ wide _as the fir~t _one. When the two hole~ _are

replaced by

two annular apertures (internal

a

b

c

d

e

-0.15 0

B~(Gj

^0.15

F>g. 3.- Interference pattern; ^obtained a, a function of Bu, ^for variou~ fixed value~ of G:

G 0 (a) _; 55 mG/cm (bj ; 110 mG/cm (c) 167 mG/cm (dj 2?0 mG/cm (e). The light line~ _are

calculation~.

6

a

i i

iGimGicm))

Fig. 4.- Interference pattern~ ^obtained as functions of G for _variou, fixed value~ of Bu;

B~j 0 mG (a 32 mG (b) 78 mG (c). The light lines _are calculation~.

radius 1.5 _mm, external radius 2 mm) _a

clearly oscillatory

~tructure with _a better _contrast i~

observed.

Ajj these

experimental

features are in

qualitative

agreement ^with the theoretical

predictions

oj~ section 2. A

quantitative interpretation

needs _a

slightly

more elaborate _treatment than that

u,ed in the _case of a radial

gradient, firstly,

because the initial

spin polarization

is _not

perfect

a

~

Z

b

c

O.5 0 _o.5

G(mG/cm)

Fig. 5. Interference pattern~ obtained _a~ function~ of G with _two collimation hole~ of diameter

4 _mm (a) 2 _mm (hi and with two annuli (internal diameter 3 _mm, external diameter 4 mm) (c). The

light lines _are calculations.

and

secondly,

because _a uniform field is added to the

quadrupole

^field. Nevertheless the _same basic

principles

and

approximations

will be u~ed here in the framework of the vectorial model for _a

spin

F

=

I. Let

f

be _a column _vector the components ^{of which} are

v'~,

the extemal

motion a~~ociated with the Zeeman _~tate M

)

referred _to the

j

axis. For _an incident

plane

wave

of momentum k,

slightly

inclined with re~pect to the I axis, let _us ~et

flf

=

exp(il=),

4l j22)

As the elements of ~P _are

slowly varying

functions _{one can}

neglect

their second derivative~ (cf.

the Glauber

approximation),

which allows _us to determine the evolution operator

connecting

# (,t, I', Z < 0 tO ~fi _I, V, Z ~ L)

gp~ La U

= exp i [- G-t-F ₊

(B,j

+ G»

IF,

123)

hvi>

where

u/v,,

= Iv, v hl./ui. Now

taking

into account the presence of _a

single

inner

diaphragm

D of _area S

together

with the

partial

initial

polarization

of the beam and the time of

flight

distribution

given

in

equation

(20), one can calculate the normalized

signal

N

=

j

da

f(ui

^~~

i-

i o+

(p

^uD

(p )j 1)

_j2

(24)

D S

Using polar

coordinates _p, in the

plane

_{t., >.,} ^{with the}

origin

at

point (0, Bj,/G

_{I, one} gets

N

=

j~

^da

^j

^{(u ~}

^~~

^{~~ [a cos (2 cup + b cos (cup + c. (25)}

i> D .~

where _:

gv _~ LGU

~

'

cos4

_(o

p

. h '

1' ~'~~

_~°

~

_~~

cv

~~~, 16 ~

t =

-a-b.

The numerical value p ^{=60° is derived from the}

experimental

contrast observed when

G

=

0

(Fig. 3a).

The

regular

full lines in

figures

3, 4, 5 repreqent the calculated patterns obtained in the various situations considered here. The agreement with the

experimental

results is

quite satisiactory

_even

though

^{there is} a small

discrepancy

for the annulus

(Fig.

5c) in the

matching

of the horizontal scale, which is

probably

due to the

approximate expre~sion

for B

(Eq. (21)) being

less valid in this _case.

In order to get a better

physical understanding

^{of the result} (25) and make _a connection with the

general

formulation of section 2, let _us assume that (I)

Bj,

0 it) the incident beam on

the interferometer is characterized

by

_a

diagonal den~ity

matrix

C(K,,)C*(K,(j=

F(K,j) 6(K,[ Kj,),

^{where F is}

strongly peaked

in the I direction. In such conditions in

equation

(19) one has :

rju

r~(u

₊

(M

M~ 51(ii

i~

= dK,~ ^F

(K,,)

o

jK,,,

u o

*(K,,,

_u +

jM

M~ j

&ii~

_j

Thi~ _means that

only

the inner

diaphragm;

are involved here. With _a

~ingle diaphragm,

D(K,,,

u _m 5)(u) is

simply

the ?D-Fourier transform oi _a square function R( _t, >')

=

in;ide the

diaphragm,

₌ 0 outside, I-e- the Fraunhofer difiraction

amplitude

of the

diaphragm.

For _a

given

value of _K,> the

integral

_{over u} in

equation

(19) _can be written

Ida, ^da,

^{51(u 51* (u +}

^(M

^M')

^61i~

⁾ =

at-

d>'R~(

_t,

>. exp

ii

(M M') 6k i~ =

=

dS exp [I (M M') fit i~

v

Finally

the

signal given by equation (19)

takes the

following

form

jj

_A,w,,i.

j~ dKjj fi(K,j)

dS exp ii (M M') 61 i~ (26)

jj~t iJ D

where

fi(K,j)

is the

K,,

distribution. Let _{us now} turn to

equation

(25). In the present case p = i~

,

cv =

61. The

integral~

_over

only

contain the three coefficients _a, b, _c. and

give fjnally

three constants

characterizing

the

spin flipper efficiency.

In conclusion~ apart ^from

some

normalizing

constant, the re;ult

given by equation (25)

is identical to that

given by equation

(191.

4. Conclusion.

We have shown that a Stem-Gerlach atomic interferometer

using time-independent

transverse

gradients

^of

magnetic

fields

provides

_{on a} wide detector _a

signal

which contains (as attenuation

factors in the contrast oi the

fringes) angular

correlation functions oi the beam in the

momentum space. The~e functions _are

generally

determined

by

the atomic source,

by

the

collimating diaphragm~ preceeding

the interferometer _a; ~i,ell as

by

the

diaphragm~

_present

within the interferometer. The basic _proces; here is the elastic transverse momentum transfer

(M

6k ) induced

by

the transverse

gradient,

which make interfere

plane

waves which have initial momenta

differing

in their directions (but not in

magnitude).

As _a consequence there is _a strong

similarity

between these attenuation factors and the Fraunhofer diffraction

amplitude

at an

angle

the order of

magnitude

of which is &I/k. In the present conditions this

angle

is

extremely

small

(5 10~~

_rd)

: this is the _reason

why

the attenuation patterns are so

clearly

visible when

diaphragms

the size of which is very

large compared

to are u~ed. It may be

noticed that in such _a situation the

signal

_can be calculated _as well

by quasi-geometrical optics,

I-e-

by averaging

the

inhomogeneous longitudinal-gradient

induced interference patterns over

all rectilinear

trajectorie~ passing through

the

diaphragms.

In the

experiments reported

here the whole

generality

of the method is

obviously

not

exploited

since the _source itself is

highly

incoherent and the beam collimation is

entirely

made

by

the inner

diaphragms.

Nevertheless these

experiments quantitatively

demon;trate the

validity

of the theoretical

predictions

and show the

capability

of the method in

measuring angular

correlations in _an atomic beam.

Variou~ further

developments

of this

technique

can be

imagined.

Let _us summarize very

briefly

some of them _:

(I)

mea~urement of the

signal

^received on a detector

having

_a small size (at the

macroscopic scale),

the

question

addressed here

being

how is the

signal

distributed in the detection

plane

'? (in

spite

of its

difficulty

the _use of _a uni-dimen~ional

gradient

would allow the easiest

interpretation

of the results)

(it)

construction of _a

(slightly) separated

beam

interferometer,

by

use of four successive

gradients

(+ G, G, G, ₊ G) (iiij effect of _a

put.led

transverse

gradient:

in this _case the _momentum tran~fer is _no

longer dispersive (thereiore

an easier

operation

of _a

separated

beam interferometer

using

four

gradient pulses

can

be

expected).

Acknowledgments.

The authors wish _to thank K. Rubin

(City College,

New

York)

for his enthousiasm in the work.

S-N-C- wishes to thank the Commis~ion of the

European

Communitie~ for the

provision

of a

SCIENCE

bursary (BS/SCI*/915186).

J. L. D. thanks the ALBAR fundation for grant.

References

Ii Gabor D., Re,,. Mod Ph_v.I 28 (1956) 260 _;

The fact that time-independent >nterferometric experiment~ cannot dist>ngui~h between the coherent wavepacket ~tructure and the incoherent m>xture of ~uch packets has been pointed out by Mopurgo G., Awl Ph>'.I (N Y 97 (19761 519 _:

~ee also Bernste>n H. J. and Low F. E., Ph_i'i Rei'. Lent. 59 (19871951.

[2] Miniatura Ch., Perale~ F., Vassilev G., Reinhardt J., Robert J. and Baudon J., / Phv.< II Fiaiite 1 (1991) 425 ;

For other atomic

>nterferometry

techniques. ~ee App/ Ph_;I B 54(51 (1992), special _{i~~ue «} Optic~

and Interferometry ^{with Atom~} _» ^{and reference; therein.}

[3] Glauber R. J., _in Lecture~ _in Theoretical

Phy~ic~,

vol. I, W. E. Brittin and L. Dunham Ed~.

(Inter~cience Pub(., New York, 1959) _p. 135.

[4] Bloom M. and Erdman K., Can ./ Ph_i'.I ⁴⁰ (1962) 179

See al~o Bloom M., Enga E. and Low H., Can../ Phi'.I. 45 (1967) 1481.

[5]

Englert

B. G., Schwinger J. and Scully M. O., Ffiwi</ Flit-( 18 (1988) 1045 Schwinger J., Scully M. O and Englert B. G., z Ph_;~ D lo (1988) 135.

j6( Robert J., Miniatura Ch., Le Boiieux S.. Reinhardt J.. Bocvar,Li V. and Baudon J., Ew.I>phi _I. Lent.

16 (199 II 29.

[7( Lamb W. E. and Retherford R C. Phv~ Rei' 79 (1950) 549.

JOURNAL DE PHhSIOUE H T 4 N II NUVLMBER 1~~4 78