A longitudinal Stern-Gerlach interferometer : the “beaded” atom

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Submitted on 1 Jan 1991

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A longitudinal Stern-Gerlach interferometer : the

“beaded” atom

Ch. Miniatura, F. Perales, G. Vassilev, J. Reinhardt, J. Robert, J. Baudon

To cite this version:

Ch. Miniatura, F. Perales, G. Vassilev, J. Reinhardt, J. Robert, et al.. A longitudinal Stern-Gerlach interferometer : the “beaded” atom. Journal de Physique II, EDP Sciences, 1991, 1 (4), pp.425-436.

�10.1051/jp2:1991177�. �jpa-00247527�

Classification

Physics

Abstracts

32 60 32.90

A longitudinal Stern-Gerlach interferometer

_:

the

_«

beaded

_»

atom

Ch.

Miniatura,

F. Perales

(*),

G.

Vassilev,

J.

Reinhardt,

J Robert and J. Baudon Laboratoire de

Physique

des Lasers

(**),

Universitk Pans-Nord, Avenue J. B. Cldment, 93430 Villetaneuse, France

(Received

I June 1990, revised 21 November 1990,

accepted14

January 1991)

Rksumk.- On donne le pnncipe d'un interfdromdtre atornlque ^dans lequel est utilisd l'effet Stem-Gerlach

longitudinal

Des rdahsations

possibles

d'un tel mterfdromdtre fonctionnant _avec

des atomes mdtastables de gaz rares ou

d'hydrogdne

sont ddcntes Quelques exemples d'objets de

phase

_sont discutds, et

quelques applications

sont

suggdrdes

A leur sortie de l'interfdromdtre, les atomes

possddent

des

propndtds

inhabituelles, telles qu'une locahsation

multiple

fi

l'dgard

des variables extemes

(atomes

_{« en}

chapelet »)

Abstract. The

principle

of _an atomic interferometer based _on the

longitudinal

Stem-Gerlach effect is _given Possible realizations using beams of metastable _rare gas atoms or metastable

hydrogen

atoms are described Some

examples

of

phase-objects

_are discussed and _some

possible applications

_are

suggested

The atoms

coming

out of the interferometer exhibit uncommon

properties,

particularly

_a permanent

multiple

locahsation

regarding

the extemal atomic vanables (« beaded _»

atoms)

1. Inwoduction.

In

spite

^{of the} fundamental role

played by

the

phase

in

Quantum Mechanics, only

_very few

experiments dealing

with atomic

physics

and atomic collisions have been camed

through

to

study

its properties. ^It is well known that _a

specific

state

la>

of a quantum system ^{is defined}

by

_a ket

ii

_{up to an overall}

_phase

_{factor :}

[a>

=

exp(ia) jai

It _means that all _states defined

by

the _same ket and different

phase

factors must'be

considered _{as a}

single physical

state of the system ^{This axiomatic}

property

^{confers the}

peculiar

mathematical structure of fibered space ^to the Hilbert space of state vectors

ill.

As the overall

phase

factor _exp

(i

_a

)

is not measurable

~postulate

of the

measure), nothing

_can be

(*)

Present address

Department

of

Physics,

New York

University,

New York 10003, U-S-A-

(**)

Laboratoire associd au CNRS URA n° 282.

said about _a. The

only

_way to get some information about the

phase

_is to prepare the system in _a linear

superposition

^{of several states}

(completely

coherent pure

state),

such _as

[q2»

= aj

exp(iaj) I)

₊

a~exp(ia2) (2)

In this _case, if

nothing

can be said about _a

j or a~, the

phase

difference

(

_a

j a

2)

is, at least

in

pnnciple,

measurable In the atomic _{case, in} order to acltleve the preparation ^{of such} a

superposition

and, eventually,

to

modify

and control the

phase difference,

it _is

obviously

necessary ^to prepare, and

eventually modify

_{a pure} atomic state further To describe these

operations properly,

_one must consider the

complete

atomic state, _i.e., at the _same time, the internal state,

depending

_on internal

variables,

and the extemal state

depending

_on external

vanables,

_e.g, those variables

descnbing

the motion of the atomic center of _{mass, in a} fixed frame For sake of

simplicity,

let _us consider the _{case in} which the

(free)

external state has been

prepared

_{as a}

plane

_wave

(ideally

monokJnetic

beam).

In such

conditions,

the initial atomic state _can be written _{as :}

[a»

₌

^exp(ia) exp(IKZ) jai

where Z _{is a} fixed _axis, K the atomic

wavenumber, jai

_a

specific

internal _state and _{a an} overall unknown

phase.

As K has the _same value and _as the internal state

a)

_is ^the same for all the atoms, the

complete

states of the different atoms are all described

by

^the _previous

expression in which the

phase

_a is

randomly

distributed.

Any evolution,

induced for instance

by imposed

extemal

fields,

will

generally

concern both internal and extemal states. An

enlightening example Qf

such _an evolution is given

by

the famous Stern-Gerlach

expenment.

^In this

experiment,

internal states

corresponding

to different values of _jy

(component

of the intemal

angular

momentum _on axis Y

perpendicular

to

2~

_are

spatially separated

from each other

by

_a

_magnetic

field

gradient

VB

parallel

to Y. In the

original

version of the

expenment,

^the atomic beam _is not

polarized,

the coherence is _zero and there _{is no} definite

phase relationship

between internal states In such

conditions,

the

experiment

is

usually

described

by

_saying that _some of the atoms are deflected _{in one}

direction,

the other _{ones m} the other direction. On the contrary, ^{if the beam} is

polarized,

_i-e-

if all atoms are in the

sami

_{internal state, which}

is a linear

superpos1tlon

^of eigenstates ^of j _y, then for _{one given} atom there exJsts definite

phase

differences between the

components

^of the intemal state

(total coherence).

As _a consequence, the

spatial separation

^induced

by

VB

now concerns each atoms

ind1vldually

_: if_j

y is

double-valued,

the extemal state will consist of two waves,

travelling along

axJs Yin opposite directions. It should be

possible,

in

pnnciple, by

using a convement

magnetic

^field

configuration,

with

opposite gradients,

to recover an atomic state s1mllar to the in1tlal _one, except ^that different

phase-shifts

_{are now} added to the different internal _state components ^{As far} as we

know,

such _an

expenmental

device has not been set up so far. It would constitute _a realization of an atomic interferometer and

Schwinger

et al.

[2],

who

analyzed

it

theoretically

_{in great}

detail, clearly

demonstrated its

feasibility.

From _a

practical

point of view

however,

great difficu1tles _are

expected

if _{a transverse} magnetic field

gradient

is

used, especially

_concerning the _« recombination _» of the

sphtted

atoms into _a

single

beam. Another

configuration, suggested

and

bnefly

discussed in reference

[2],

consists of _a

longitudinal gradient)

i,e, a

gradient parallel

to the direction Z of the atomic beam Phase

shift

effects,

and

correlatively

the

spatial

separation, now occur

solely along

the _axis Z As _a consequence, the initial

plane

_wave ^remains _a

plane

_wave

along

the whole device. As it will be

seen

further,

the recombination becomes much

easier,

_{even in a} realistic

expenment

in which neither VB _nor the atomic

velocity

_are

strictly parallel

to Z.

The _mm of the

present

paper is to

suggest

an atomic

interferometry

method based _upon the

longitudinal

Stern-Gerlach

effect,

to examine its

feasability

and _to

investigate

_some of its

fundamental consequences. In part

2,

the

pnnciple

of the

suggested _expenment

_is

detailed,

and

possible

realizations

using

beams of _{rare gas or}

hydrogen

metastable _{atoms are} discussed.

In

part 3,

two

particularly important examples

of

phase

^effects _are described _:

(i) topological Berry's phases [3]

which _can be used _{as a} test,

(it)

the

phase-shift

induced

by

a

phase- object

_{», consisting} of _a

simple profile

of magnetic field the

gradient

of wltlch _is

longitudJnal

Some

possible applications

of the method _are

suggested.

Part IV _is devoted to a

study

of the atoms coming ^out of the interferometer the

complete

coherence exJsting ^{witltln their internal} state also _appears wltltln their external state _{as a} kind of _« delocahzation _»

2. The

longitudinal

Stern-Gerlach interferometer.

2 PRINCIPLE. The

pnnciple

of _a

longitudinal

Stem-Gerlach interferometer is based _on

the Zeeman state

preparation

of the atoms, I-e- _on the

polarization

of the atomic beam.

From this point of _view, it _is

quite

s1mllar to that of interferences obtained when _a

crystal plate,

cut

parallel

to the

optical axis,

_is traversed

by

_a

parallel light beam,

the

polarization

of which _is tilted with respect to the two

polarization eigenvectors

^{of the}

plate.

Within the

plate,

the two

components

^{of the}

polarization

accumulate different

phases,

but _no interference

effects _can be

directly

_{seen in a}

detector,

since the two beams have

orthogonal polarizations.

However,

after _a

polarizer

tilted

with,respect

to the

plate

eigenvectors, ^the

resulting amplitude

_{is a} linear

superposition

of the two

amplitudes

_coming out of the

plate,

and the

interference effects become observable.

The

pnnciple

of the

suggested

expenment is shown _in

figure

I

together

with its

optical

counterpart. ^A

non-polarized

beam of atoms, of internal

angular

momentum

(«spin»)

j # 0 =

I _in

Fig. I)

is first

passed through

_a

polarizer

P which selects _one Zeeman state

~,mo).

Then _{a «mixer» or}

spin-flipper M,

descnbed

below, produces

_a coherent

superposition

of Zeeman states. This atomic

(intemal)

state enters an interaction zone in

which each Zeeman

component

accumulates its _own

phase

In this region, ^it is assumed that the extemal fields do not _cause any transfer of momenta to the extemal atomic motion

(except longitudinally)

Then _a second

spin-flipper

M' builds _{a new} coherent

superposition

in

which the

amplitudes

_are linear combinations of the

amplitudes

_conning out of the interaction

zone. An

analyzer

A selects _one Zeeman state, and the final

intensity

^{is measured in} a

detector D.

atoms

P M M' A D

unpolarized ,

~ ^,

~~~~'

P A D

') _n~,n~ ~

=. .

light _i ,,

interaction

' zone

Fig. ^I -Scheme of the Stem-Gerlach interferometer

(upper part)

P _{is a}

polarizer,

M and

M'are

spin-flippers

A _{is an}

analyzer

and D _a detector The lower part of the

figure

shows the

optical

counterpart, m which the interaction _zone consists of _a

birefnngent

medium (optical index n,_~

correspond

to polarization eigenvectors 1, 2).

For several

expenmental

_reasons, metastable atoms of _{rare gases}

(He*(~Sj), Ne*(~P~), )

and

hydrogen (H

^*

(2sj

j~, F

= I

) )

_are

good

candidates for this

expenment

:

(i)

their spin is _non-zero and -their

hyperfine

structure is

generally much simpler

than that of alkali atoms,

(ii) they

_are

easily produced,

and detected with _a

high efficiency (10-30 9b), (iii) they

_can be ekcited to upper radiative levels

by

standard low power

(10-20mW) CW-dye

lasers.

2.2 POLARIzERS In

pnnciple,

_any transverse

gradient

of

magnetic

field

(Stern-Gerlach expenment, hexapole

_magnet,

...)

is able _to

polarize

^the atomic beam.

However,

there exists other efficient methods which _{are more} convenient _to handle.

(i) By

means of _a

circularly («* polarized laser,

tuned _{on a} transition from the metastable level to _a radiative level allowed to cascade down to the

groundstate,

it is

possible

to prepare

the metastable atoms in a pure Zeeman state mo = ±j. For

instance,

_in the _case of

Ne*(~P~),

_a total

polarization (mo

= ±

2)

has been achieved

by using

the

~P~-~D~

^transition

(A

=

614.3 _nm wlth'a laser _power of about 15-20

mW,

the

intensity

of the

polarized

atomic beam

being

about 50 9b of the initial

Ne*(~P~) intensity [4].

The _same method _can be used with

H*(2sjj~) ^htoms,

^with a

CW-dye

laser locked _in

frequency

_on the Balrhefa line

(A

= 556.3 _nm

)

(ii)

In the

specific

_case of

hydrogen,

_a

partial polarization

is achieved

by passing

the atomic beam

through

_a

magnetic

^fidld B of about 600 G

(Lamb

and Retherford's method

[5])

:

hyperfine

Zeeman levels

2sjj~,

^F =

I, M~

= I and

2sjj~

^F =

0, M~

= 0 _are

degenerated

with the radiative levels

2pjj~,

^F =

I, M~

= 0 and

2Pjj~,

^F =

I, M~

=

I,

at B

= 530 G and

597 G

respectively.

The

coupling

induced

by

the

motionjl qlectric

field

E~

= v x B

(v

^{is the}

atomic

velocity)

results into a

complete quenching

of both previous

2sjj~ Zeemai

_{states. As}

a

consequence, an'incoherent mlxtilre of

hyperfine

states

2sjj~,

^F =

I, M~'= 1,0

_is obtained The

only disadvantage

of such _a

partial polarization

_{is a} loss of contrast _in the interference

pattem observed _on the

M~

= -I

component.

2 3 SPIN-FLIPPERS. The

problem

is now to prepare a linear

superposition

of Zeeman

states. If the first method

(i)

of 2.2 _is used to

polarize

the beam into _mo

= ±

j,

it _is rather easy to

modify

it _in order to

get

^certain linear combination of

[j,m)

states: if the

«* laser excitation takes

place

within _a

sufficiently

low

magnetic

^{field b} non collinear to the laser beam _so that the induced _Zeeman

splitting

is

negligibly

small with respect ^to the laser line

width,

and the Larmor

penod

_is

large compared

to the

time-of-flight through

the laser

beam,

then a definite

superposition

of

[j,m)

~states referred _{to axJs} b _is

prepared.

This combination _can be modified _{in a} controlled way

by changing

the

angle

between b and the laser beam.

A _more

general

method has been used _{some years ago in} neutron

[6]

and atom expenments

[7].

The basic idea _is to

produce

_a non-adiabatic evolution of the spin,

by sending

the atoms

through

_a

special configuration

of magnetic ^{field B} : two

oppositely

wound solenoids

produce

a

B~ component

^which reverses its sign

abruptly

_in the middle of the device. As the beam direction is

parallel

but not identical _to the solenoid

axis,

the atoms _{see a} transverse

component B~

^{which has} a

Gaussian-type profile.

Since the necessary fields _are small

(B

few 10 mG

),

the

system

^needs to be

carefully

shielded. For _a

typical length

of few _cm, the

dynamical phases

accumulated

by

each Zeeman _state

along

its

path through

the spin-

flipper

_are

negl1glbly

small

(~10~~ _rd).

This _means that the extemal atomic vanables are

almost unaffected. For _an incoming ^{atomic state} exp

(iKZ)

_j,

mo),

the

spin-flipper produces

the

outgoing

state _:

exp

(iKZ) £ _a~~~

_{j, m}

)

m

The coefficients

a~~~

^{are controlled}

by

the _{current i} in the solenoids. As mentioned _in

[7],

one of the difficulties _{is to ensure} that the coefficients

a~~

_~ ^{are almost identical for all atoms in} the beam It _{is a} matter of

collimation,

the

key _parameter being

the ratio

d/L,

where d _is the beam

diameter,

L the

length

of the system. ^Such a device has been constructed and tested _on

H*(2s) _[8]

_{in an}

expenment

^similar to that of

Hight

et al

[7].

^In our expenment, ^the Lamb-

Retherford method has been used to

polarize

^the

incoming

beam and _to

analyze

the

outgoing

one The

outcomlng H*(2s)

flux has been measured _{as a} function of I, for

given

atomic velocities selected

by

the

time-of-flight technique (Fig. 2).

It is worthwhile

noticing that, actually,

such _a

complex

device _{is not} needed _to

produce

a

superposition

of Zeeman states. It has been verified

that, by simply

_passing the atoms

through

_a short _zero field

chamber,

such _a

superposition

was

produced [8] However,

_in this _{case, no} control

parameter,

except the atomic

velocity,

is available.

N(AAB.

_U

~

. o

D

. .

o .

. ~

~ , a

2 lo ~

° ~

i

A D

@o o_o -

o o

~ 6

' °

~ j o

a a ,

o 1.5 ; ~

Fig 2

current i the spin-flipper The selected

atomic velocity is ^{. (D)}

v =10km/s, (.) v

_z~

v 4 3

km/s,

(o) v =

5.8

km/s

3.

Experimental

tests.

3 TOPOLOGICAL _PHASES.

Topological phases (also

known _as

Berry's phases) developed

by

_a

quantum

system

govemed by

_an Hamlltonian

depending

on at least two

parameters

which

undergoes

_a

cyclic

adiabatic

evolution,

have been

widely theoretically

studied

[3].

Experimental

measurements of such

phases

have

already

been

performed

_on

photons _[9],

neutrons

[10]

and nuclei

[I Ii.

This

type

^of measurements is _a

good

test for _an atomic

interferometer insofar _{as :}

(i) topological phases

_are

predictable, (ii)

the measurements

imply

an ehmmatlon of the

dynaqJical part

^{of the}

phase,

I _e. that

part

coming from Zeeman

splittings

inside the interaction zone. Let _us consider two

possible types

^of interaction :

(i)

A conical magnetic ^field B

= B

(Z) Q(Z),

where

Q(Z)

_{is an} unit vector

descnbmg

_a

cone : ..

This _is the

typical configuration

used in neutron expenments

[10].

In this _case, the

topological phase

_is

simply ma,

where D _is the solid

angle

sustained

by

the _cone

[12]

(ii)

A

loop skirting

round _a conical intersection

point

_:

Magnetic

B and electric E fields _{are two}

parameters

in the Zeeman-Stark Hamiltonian. In the _case of

H*(n=2),

the adiabatic _energy

surfaces, corresponding

to

2sjj~,

^F =

I, M~

=

I and

2pij

^F

=

I, M~

= 0 levels at _zero

fields,

exhibit _a conical intersection

point

I at B

= 597 G and E

=

0

(Fig. 3)

The field

profiles

B

(Z)

and

E(Z)

_can be chosen _so that the atom

representative

point follows _a closed

path

e in the B-E

plane.

If point ^I lies inside

e,

then the

predicted geometncal phase

_{is w,} otherwise it _{is zero}

[12]. Actually

these

predictions

can be made

only

for stable states. In the present case

however,

the

2pj/~

^{state has} a short radiative

lifetime,

_{i-e- a} width r

large compared

to the Stark

splitting

and of the _same order of

magnitude

_as the

2s-2p splitting.

It has been shown

recently [13]

that _m such _{a case,} for

special cycles,

the

Berry phase

_can take _an intermediate value

n/2 together

with _an

exchange

of the

internal _states.

2s~,F-i,m~--1

2p~

,

Fai,

m~_o

/2

,

/~ _Cl

B

c~

E

Fig

3 -1 conical intersection point.

Cycle Cj (full line)

_gives rise to a Berry's

phase equal

to w

whereas

cycle

_C~

(broken line)

_{gives nse} to _a

Berry's phase equal

to 0

3 2 A SIMPLE PHASE-OBJECT Let _us consider _a magnetic field

profile

such _as

(i)

the

gradient

VB _is collinear to axis

Z,

I-e to the atomic

velocity (for

_an ideal

atomic,beam coinciding

with axis

Z~, (it)

B=0 _on both sides of the

profile,

_i-e- for Z<0 and

Z>L. The tridimensional extemal motion of _an atom of

spin j =1/2

_m such _a field _is described

by

two equations ^which are

decoupled

_m the adiabatic

approximation

_:

T~

_{+ e~}

(z) F~ (z)

=

EF~ (z)

TR is the kinetic _energy of the extemal

motion, F=

are the

amplitudes corresponding

to the two intemal states w=

(for

which _m

=

±1/2 respectively),

_e=

(Z)

= ±

e(Z)

_are the local

Zeeman energies in

B(Z).

It _{is seen} that the local internal energies

play

the role of

potentials

for the extemal motion.

Owing

to the

separation

^{of variables}

X,

_Y,

Z,

the solutions take the form _:

F~

_{= exp}

ii (Kx

x ₊

Ky

Y~

f~ (z) f=

_are solutions of the equation

1-)$+

E(Z)) ^f=

⁼

(E-)(K]+K()) f=

where M _is the atomic mass. As

e(Z)

_{is a very}

slowly

_varying function _at the atomic-

wavelength scale,

the JWKB approximation is

largely justified

The local wavenumbers _are defined _{as :}

K_ (Z)

=

[Kj± U(Z)]

^~/~ where U

(Z)

=

~

) _e(Z)

h

fi2 K~ is such that E

=

(Kj

₊

K)

₊

Kj)

2M

It is

readily

verified that the JIVKB solutions which behave _as

exp(IK~Z)

for Z < 0 _are given

by

_;

f

_z

f_ (Z)

=

0 exp I K_

(Z')

^dZ'

K=

o

For Z _>

L, they

become

z

f_ (Z)

= exp i

K= (Z') dZ')

o

As the Zeeman energy

splitting

is small

compared

to E

(typically e=10~~eV

_for

B

=

200

G,

E

= 0. I

ev~,

the

approximation

K_=K~(1± _2K~ ~~)

is

justified

In such

conditions,

^for Z

>

L,

_one has

f=(Z)=exp[iK~(Z±@ )j

with

L L

AZ

=

Kj

^~

U(Z)

dZ

=

Mg»~ Kj

^~

B(Z)

dZ

o o

where g is the Landk factor and _{»~ is} the Bohr magneton. ^As an

example,

for _a

magnetic

field of 200G _over L=

lsmm,

and _a kJnetic _energy

(along Z~

of

0.lev,

_one

gets:

AZ = 800

A.

The solutions

F_ finally

consist of two

plane

_waves, shifted

by

AZ with

respect

to each other _:

F=(Z>L) =exp(iK. (R± _@Q~) ).

As _a consequence, to _an

incoming (Z

_<

0) complete

state _:

[a~ [w~)

_{+a_}

[w_)]exp(iK.R) corresponds

the

outgoing (Z >L)

state

a~ w~

)

_exp iK R ~~

Q~ + a w exp iK

(R

^{+ ~~} Q~

2 2

This

analysis

_can be

generalized

to any

type

^{of extemal motion For}

example,

^let _us

consider _an almost monochromatic

wavepacket,

the

spectral density

of

which,

_p

(K),

_is

sharply peaked

at _a value

Ko

=

Ko

_Q~ If the internal _{state is w_}

),

the JIVKB solutions _{are :}

F_

=

d~K

p

(K)

_exp

Ii ^K~

^{X +}

^K~

^{Y +}

j~ ^{K_ (Z')} _dZ'j

^{~ t}

R3 _o h

By expanding

the

phase

term up ^to the first order _{m w}

= K K

o, one gets.

1.

^.z

^E~

F=

₌ exp i

Ko~

X ₊

Ko

~

Y ₊

K(

_dZ'

~

^{t x}

0

x

d~w

_{p ~lLo + w}

)

_exp

iw _V~ K~

X ₊

K~

^Y +

j~

^{K_ dZ' ~ t}

R~ o h

Ko

~ ~ _~~

h~

K(

~~~~~

~* ~~°

~ ~

~~~~~

~~~

~°

2 M ForZ<0:

F=

_=exP

i(Ko.R-~tj Eo dWP~Ko+W)exPliw. (R-vat)j

R~

The first

exponential

factor _is the carder-wave whereas the

integral

_{over w} is the

slowly

varying

envelop

of the

packet,

which _{is a} function

L(R

_v

o

t)

where vo =

AK

o/M

is the group

velocity.

It is

readily

verified that for Z

> L the

wavepacket

becomes _:

AZO Eo _AZO

F_ =exp(i(Ko. (R±jQ~) --tj) xL(R±-Q~-vat)

h 2

where

AZO

is the value of

AZ,

defined

before,

for

K~

₌

Ko.

As _a consequence, ^to the

incoming complete

state _:

ia+lw+) +a-lw-)iexp

I

Ko.R- ~°t) ^.L(R-vat)

corresponds

the

outgoing

one :

[P~ a~[w~)

+P _a

[w )],exp

_i

Ko.R-~°t

h

~

where _:

AZO AZO

~*

_{~~~ ~}

~~°

2 ~

~~

^~ 2 ~~ ~°

Therefore,

_as far _as its extemal motion _is

concemed,

the atom appears

sphtted

into two coherent

parts, corresponding respectively

to the

orthogonal

mtemal states w~

).

This _{is a} standard result of _a

longitudinal

Stern-Gerlach

experiment operating

with

polarized

atoms More

generally,

if the total atomic momentum is

j,

then

(2

_{j +} I

wavepackets

_are obtained

for the extemal

motion,

each of them

corresponding

to _one of the

(2j +1) mutually

orthogonal

internal _states

[j,

_m

).

The major interest of such _a

longitudinal

Stern-Gerlach

expenment

is that the

spatial separation

^induced

by

the

magnetic gradient

^remains

finite,

whereas in _a transversal Stern-Gerlach

expenment

^this

separation

becomes

asymptotically

_a

linearly

_increasing function of Z.

Obviously,

the

previous

calculation has been canned out in

an idealized situation, since _a

strictly longitudinal gradient

has been assumed. Cumbersome effects induced in the extemal motion

by

transfers of transversal momenta will be examined _m the _next

paragraph.

3.3 APPLICATIONS. Since the interaction between

polarized

atoms A

(spin j)

and

sphencally symmetric

atoms B _is

anisotropic,

the _« elastic _» scattenng is

actually

descnbed

by

a set of Zeeman transition

amplitudes f~~~(#),

where

[j, mo)

is the initial Zeeman state,

[j,

_m

)

the final _one and

fi

_the

_scattering

_direction.

_{Anisotropy (or}

«

polarization »)

effects have been

already

observed _m collisions between oriented excited alkali atoms and

heavy

_rare

gas ^atoms

[14],

_{or m} collisions between oriented metastable Neon atoms

Ne*(~P~,

_m

= ± 2

and _rare gas ^atoms or molecules

[15].

In these expenments, ^{the difference A of} the differential

cross sections

corresponding

to

opposite

initial onentations

(mo=±3/2

for

alkalis,

mo= ±2 for

Ne*)

is measured. In absence of fine _structure transitions

(eg.

_m the

Ne*-Ne

collision)

_one has _:

A

=

£ ((f+2,m(Jl)(~ (f-2,m(#)(~)

Using

_a laser-induced fluorescence

detector,

_operating within _a

magnetic

field of few

10~G,

_{it is}

_possible

_to

improve

_our information

by individually measuring

the

squared amplitudes [f=~,~[~.

Further

information,

about the relative

phases

between these

amplitudes

_can be obtained

by

_using the interference method

by _inserting

_a

spin-flipper

between the collision volume and the

detector,

_one is able _to transform the

onginal outcoming

state

£ _fmom ^(k) lj,

_m ^~

i

m

into the _new

superposition

of Zeeman states _:

£

A

,

f~ ~(fi)[j, _m')

^~

o

j/

m,m'

Then, by

_measunng Zeeman states

populations

1_

²

£A~,f~~~(R)

m

one gets interference _terms such _as

(f~~~, ft~~~+c,c.) containing

the relative

phase

between the two

amplitudes.

Another type ^of

possible application

deals with the

phase-imaging,

_{m a} bulk _or at _a surface The basic

pnnciple

_is s1mllar to that of the

phase-imaging

of _an

accidentally birefnngent

medium

(e,g,

_a

mechanically

constrained

isotropic medium)

_{using a} wide

light

beam. In the

present

_case,

by

_use of _a wide atomic

beam, together

with _a

position-sensitive detector,

it would be

possible

_to

_get

_a two-dimensional picture ^{of the}

phase

accumulated

along

each

trajectory,

^{either in} a bulk

(magnetic agreggates),

_or ^after a reflection _{on a}

surface,

_even if the

interaction

responsible

for the

phase

shifts is _so small that _no

significant

effect is observable

on the atomic trajectories

Experiments

of this kind have

already

been realized with electrons

(e.g.

impinging on a

ferro-magnetic surface) [16]. Owing

to the shortness of the atomic

wavelength,

_a

high

sensitivity is

expected

In

addition,

the _use of beaded atoms

(described

m

part 4)

_is

expected

to

provide original

information about

crystal

surfaces.

4. Beaded atoms.

4.I PREPARATION. As mentioned _m

3.2,

within the

interferometer, just

after _a

phase-

object

_consisting of _a

longitudinal

_magnetic field

gradient,

the

complete

atomic state

~

=

l/2

for sake of

simplicity)

has the

general

form

(apart

a common

phase factor)

a+

iw+ P+

+ a~

iw-

P~

Therefore the atomic state consists of two

wavepackets depending

_on the extemal vanables R, ^shifted

by AZ,

each of them

being

associated to one of the two

orthogonal

internal states

w_ Now if

such

a state is sent

through

the second

spin-flipper

charactenzed

by (almost)

real coefficients _«~

~

(i,

_j

= ±

)

it becomes _:

~(+ ) _lW+

₊

~(~ lW- )

where :

£(+ )

# a,

~ ~~ P

~ + a~ ~_ P_

Hence,

for each internal state

[w~),

the extemal motion _is described

by

_a double

wavepacket,

with the

spatial separation

AZ. As _a consequence, ^at the

output

^{of the}

interferometer,

_i-e- after the

analyzer,

_one

_gets

beaded atoms, ^{for which each mtemal}

state is combined with _a double

(or

_more

generally

_a

(2

_{j +} I

)-fold) wavepacket,

charac-

terized

by

the interval AZ

along

the atomic

velocity Obviously,

_{m a} real

experiment,

because of the field

configuration

itself and because of the

angular

_aperture of the atomic

beam,

VB

cannot be considered _as

stnctly longitudinal.

Then the different components of the atomic

state will

undergo

different transversal momentum transfers. This results into _a

angular

separation ^{of the}

wavepackets,

I-e-

finally

into _{an «}

explosion

of the beaded _atom.

Actually

one has to answer the

following question

how

long

_m time _{or m} distance is a beaded atom

observable? In other

words,

how

long

does the transversal

separation by

remain small

enough compared

to the

longitudinal

_one AZ? Let _p be the

order'of _magnitude

_of

d~

Bid

y B and 3t _a reasonable value of

AZ/A

Y. It is

easily

shown that the observation

length L,

within which

AZ/A

Y _m

3t,

_is

given by

L

=

fp _/3t,

where

f

is the width of the

magnetic

^field

profile

_m the

phase-object. Finally,

this shows that the condition to be

imposed

to the

gradient

m order to

get

a value of L

comparable

to that of

f (few cm)

_is not drastic.

4 2 DETECTION _OF BEADED ATOMS In order _to detect the beaded _atoms

produced by

^the

longitudinal

Stern-Gerlach

interferometer,

it is necessary ^to induce a transition from the intemal metastable level to _an internal radiative level. This _can be

easily

achieved

by

_means of

a laser

optical

excitation

(e,g,

^for rare gas metastable

atoms)

or

by

a static extemal field

(like

m the Lamb and Retherford's method used for metastable

hydrogen

atoms

[5]).

In the

following,

_we will focus _our attention on this latter _{case as an}

example.

The external static

magnetic

field _{gives nse} to _a motional electric field _m the reference frame of the atom which induces _a transition from the

2si/2, 1/2 )

state to the

2pim

+

1/2 )

state

(m

a fine _structure

description).

This transition _is followed

by

the radiative

decay

of the

[2pj/~

+

l/2)

state

resulting

_m the _emission of _a

Lyman

«

photon. Therefore, by detecting

the emitted

photon,

one measures the

population

of the

)2sjj-1/2)

state. In other

words,

the detected

fluorescence

signal

3 is

proportional

to

[[£_

_[[~ =

d~R _{£~_} ~(R),

i e..

3 _cc

d~K30(K)jl

₊ C

(K )cos 1l~~K)j

with _:

=

«'-+

_a+

, » = « a-,

Jo(K)

=

jp(K)j2 _jjA

_{j2 +}

_{j» j2j}

~~~~

"

IA12+ 1»12

In this

expression,

the atomic

phase 1l'(K)

stands for _:

'l'(K)

"

i'~K)

₊

~fiat(K)

Mg»~ IL B(Z')

dZ'

where, w(K)

=

Arg (» IA )

and

~fi~~(K)

=

Kz AZ(K)

=

° Note that _w

Kz

depends

bn K _since the

mixing

coefficients of the

spin-flipper depend

_on K.

The _expression

given

for 3 shows

clearly

that the fluorescence

signal

_is the _sum of the

various atomic interference pattems

(intensity 30~lL)

and contrast

C(K

associated with the

various atomic wavenumbers

describing

the ingoing

state)

If _one

excepts

^{the slow} K-

dependence

of the contrast, one has here the

equivalent

of the interference pattern ^obtained

m an

optical

interferometer _{using a} non-monochromatic

light

_source. As it _is well known m

such _{a case,}

only

few

fringes

around the

bnght

central

fnnge

_are observable. It will be the

same for the atomic interference

pattern provided AZ(K )

is of the order of

magnitude

of the

atomic

wavelength

_Ao

= 2

w/Ko (where Ko

^is the wavenumber for which 30~lL

)

_is

maximum).

As the

typical

values for

o are of the order of few

Angstrom,

it _means that the

magnetic

field

responsible

for the

longitudinal

Stem-Gerlach effect has to be small

(typically

_a few

mG).

But it _is not _a necessary cond1tlon to observe structures in the fluorescence

signal

for indeed it may

exist several wavenumbers _K~ for which

[VK1l'(K)]~,

_~ ^0.

This _means that the wavenumbers _{K~ are} stationary

phase _points.

In this _case, the

fluorescence

signal

exhibits slow variations around these points, even if

AZ(K)

_is

large compared

to

Ao. Unfortunately,

the

problem

of the existence of such

stationary phase

points

is rather difficult to solve because it

depends

_on the kind of

spin-flipper

_{one uses} and because it relies _on the exact

knowledge

of the

expression

of the

phase

of the

mixing

coefficients 5. Conclusion.

The

interferometry

method described _m this paper is

actually

_{a new}

development

of ancient basic

ideas, onginated

at the very

beginning

of quantum

physics by Stern, Gerlach, Rabi, Schwmger

and others and

mainly developed

_m neutron expenments. ^{The central}

problem

here _is the atomic

phase,

and the

only

_purpose of all devices descnbed _m this paper is to

measure and to

study

the charactenstics of this

phase

_{m vanous} situations

fjoq_

the

viewpoint

of the

phase,

these

problems

_are

very

close to those encountered _m

optics.

This

analogy explains

the

necessity

of using

well-collimated,

and

polanzed

atomic beams In such

conditions,

it is

obviously

_an

advantage

to preserve

(as

far _as

possible)

the

longitudinal

geometry

of the

beam,

all

along

the interferometer. The easiest way ^to fulfill this condition _is to _{use a}

longitudinal

Stem-Gerlach

configuration.

Since the atomic

wavelength

_{is very} short

(sac),

at least at thermal _energy

(few 10~~eV),

the

phases developed through

usual

magnetic

^fields

(0,I-few l0~tJauss)

_are

generally

_very

large. Nevertheless,

small

phases,

of the order of _{w, are} sufficient _to

change

the interference

pattem significantly

Such small

phases

_are

accompanied by negligibly

small

(on

a

macroscopic scale)

effects _on the extemal motion.

Therefore,

the atomic

mterferometry

_can be

applied

to the

analysis

of small

perturbations,

which would be

completely

invisible _on

atomic

trajectories.

One of the most

promising

consequences of the _use of _a

longitudinal

Stern-Gerlach interferometer _is the

production

of _« beaded atoms. Further

investigations

of the

properties

of these very

special

_{species are}

certainly needed, particularly

_m the

following

directions

(i) Complete

treatment of the interaction with radiation

(spontaneous emission,

laser

fields, squeezed fields, microcavities,

beaded

planetary Rydberg atoms,...).

(ii)

Collisional properties : the standard collisional treatments have to

be'modified

since

the incident _wave is _{now a}

(2 j

+ I

)-uple plane

_wave this _is

expected

to

deeply modify

the effect of resonances, atomic

exchange, itc...

The present

method,

which

simply

needs

polarized

neutral

beams,

_can be extended to other species and would be of

great

^interest m the _case of cold atoms where the atomic

wavelength

can be

Very large.

Acknowledgments.

The authors wish to thank Pr. J.

Bellissard,

Dr. V. Lorent and Dr M.

Ducloy

for

stimulating

discussions. The authors also thank the Referee for his criticisms which have led to _a clarified version of

part

^4.

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