Magnetic coherences in atom interferometry

(1)

HAL Id: jpa-00248107

https://hal.archives-ouvertes.fr/jpa-00248107

Submitted on 1 Jan 1994

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Magnetic coherences in atom interferometry

Jörg Schmiedmayer, Christopher Ekstrom, Michael Chapman, Troy Hammond, David Pritchard

To cite this version:

Jörg Schmiedmayer, Christopher Ekstrom, Michael Chapman, Troy Hammond, David Pritchard.

Magnetic coherences in atom interferometry. Journal de Physique II, EDP Sciences, 1994, 4 (11),

pp.2029-2042. �10.1051/jp2:1994245�. �jpa-00248107�

(2)

Classification

Physic-s

Absfi.acts

07.60L 42.10M 35.10D

Magnetic coherences in _atom interferometry

Jbrg Schmiedmayer('. 2), Christopher

^R. ^Ekstrom('.

~),

Michael S.

Chapman(').

Troy

D. Hammond

(')

and David E. Pritchard

(')

(') Department of Physics and Research Laboratory of Electronic~, Ma~sachu~etts In~titute of

Technology, Cambridge,

Massachusets 02139, U-S-A-

(2) institut fur

Experimentalphysik,

Universitht Innsbruck, 6020 Innsbruck Austria

(Receiied 2 Maj' 1994, ace.eptefi _m final ffiim 22./~ilv 1994)

Abstract. We have u,ed _a separated beam atom interferometer to investigate magnetically- induced phaw shifts of the different magnetic sub~tates of the sodium ground state. We have observed periodic rephasing of the 8 independent magnetic substates present ⁱⁿ an unpolarized sodium beam _a~ the differential

magnetic

field is increased _on

opposite

sides of the interferometer.

We have also demonstrated that two phase ^~hifts with similar velocity dependences _can be uwd to cancel each other, _creating _an interference pattern ^from one magnetic ^sub~tate although _an

unpolarized

beam

is sent into the interferometer. Finally _we discuss _some

application~

of these techniques.

1. Introduction.

interferometric measurements have _a

long

tradition that includes the beautiful

experiments using light by

^Fizeau

[I],

Michel~on

[2]

and

Rayleigh [3]

_as well _as _matter _wave interference

devices such _as the electron interferometer

[4. 5]

and the

perfect crystal

neutron interferometer

[6],

both

being

well-known for their contribution to

precision experiments

and fundamental

physics [7]. Recently,

there have been several atomic interference

experiments culminating

in

the demonstration of atom interferometers

[8-15]

that are now

beginning

to be used _as

experimental

tools~

especially

in the field of atomic

physics [15-20].

We have

recently

demonstrated the first atom interferometer in which the two

interfering

components ôf êach âtom's wave

(henceforth

called

beams)

are

physically

isolated,

permitting

the controlled

application

of different interactions _to the two beams. In

previous experiments

with this device _we studied differential interactions

(electric

fields and collisions) ^that shifted all

magnetic

substates of the

ground

manifold

identically

so that all atoms in _our

unpolarized

beam

experienced

the _same

phase

shift

[18~ 19].

j*) Current address. Fakult>t fur Physik, Universitit Konstanz, Konstanz, Germany.

(3)

2030 JOURNAL DE

PHYSIQUE

II N° li

In the

experiment

described here~ _we

apply magnetic

fields _to the _two beams in _our

interferometer. These beams _are

unpolarized,

and the various

magnetic

^substates have

different interaction

energies

with the field. More

importantly,

the differential

phase

shift

depends

_on the

magnetic

moment of any

particular

substate, _so that the interference patterns of the various substates _are shifted

by differing

_amounts. Here, the

phase

and contrast of the resultant interference pattern ^observed under these conditions is the

subject

of

study.

We show that the interference

pattern

has

interesting

discontinuous

phase changes

and that the _contrast

exhibits many

sharp rephasings.

We also show how this differential interaction,can be canceled for

only

_one substate

by

the

application

of another differential interaction whose

strength

is

independent

of the substate with the resultant interference pattern

arising

from

just

_one of the substates.

2. The MIT atom interferometer.

The MIT _atom interferometer

8]

has _a Mach-Zender geometry

employing

three 200 _nm

period

nanofabricated transmission

gratings [2

Ii mounted _on separate translation stages ^inside the

vacuum chamber ~p ² x 10~^~ torr ). The

path length

difference for all velocities in the central

fringe

is _zero,

thereby producing

_a robust white

fringe [22].

A detailed schematic of _our _atom interferometer is shown in

figure

I.

We _use _a sodium seeded argon driven

supersonic

beam with _a _mean

velocity

of

v ₌ 000 m/s which

corresponds

to a de

Broglie wavelength of1~~

m 0.17

I. _By varying

the

gas pressure in the _source _we _can

change

^the FWHM of the

velocity

distribution from

AVIV

= 70 %

(no

seed

gas)

to AVIV

< 8 % _at 3 bar of Ar pressure. The Na beam is collimated

translation

stages

^reference

P

~°"~(~~"" P~°~°~"~~ photodiode

~ ~~~

signal

/~i

hot wire detector

Na/Ar

(~

_0.6

m o,6 _m

~(

~°~~~~

10 micron copper foil H~~N~

interaction

region

laser

Fig. I. A ~chematic, not to scale, of _our _atom interferometer (thick lines _are _atom beams). The oth and lst order beams from the first

grating

strike the middle grating where

they

_are diffracted _in the lst and St orders _so that they ^form _an interference pattern ⁱⁿ ^the plane of the third grating. which acts _as _a

mask to sample this pattern. ^The ^detector ^located beyond the third grating, records the flux transmitted by the third grating. ^The ^lo cm

long

interaction region with the lo _~m thick copper foil between the _two

arms of the interferometer is positioned ^behind ^the ^2nd

grating.

An

optical

interferometer (thin lines _are laser beams) _measures the relative position of the 200 _nm period atom gratings (which _are indicated by

vertical dashed lines).

(4)

by

two 20 _~m wide slits

separated by

90 cm. The atoms are detected with a Re hot wire detector, ^located in _a separate vacuum chamber ~p 2 _x 10~ ^~ _torr ), ^behind the interferometer.

The detector has

high efficiency,

fast time response l ms) ^with a

background

_as low

as

I0cps.

The detected _source

brightness

of the sodium beam is _as

large

_as

5 _x

10'~

_atoms _str~ ^' _s~

'.

At the

position

of the second

grating~

the centers of the

interfering

beams _are

separated by

55 _~m, which is _more than the beam width, a

trapezoid

with 20 _~m flat top. ³⁰ ~m FWHM

and 40 _~m width _on the bottom. At this

position, immediately

behind the 2nd

grating,

the two

interfering

beams _are

physically

isolated

by

_an interaction

region consisting

^of a stretched metal foil

jseptuml positioned symmetrically

between _two side

plates,

each

spaced

2 _mm from this septum. ^The atom wave,

separated

in the two arms of the

interferometer,

_passes _on

opposite

sides of the foil. This septum is

only

10 _~m thick and up ^to 10 _cm

long.

The shadow of the stretched foil _on the detector is

typically

20-30 _~m wide, caused

by slight

deviations

from

perfect

flatness and

imperfections

at the ends. The observed interference

amplitude,

with

the septum ^between the _two _arms of the interferometer, is _up _to _a few thousand counts per

second,

allowing

_us to determine the

phase

to better than 10 mrad in _a minute measurement.

This

conducting, physical

barrier between the

separated

beam allows _us _to

apply

different interactions to the

portion~

of the atom wave on each side of the interferometer and to measure

the

corresponding phase

shifts. These

experiments

have _a

sensitivity

set

by

the inverse of the interaction time. An intrinsic line width of 10 kHz can be obtained with _a 1000 ms-' fast beam and _a 10 _cm

long

interaction

region.

In other word~. _a

potential

of 4 _x 10-' eV

gives

a

phase

shift of 2~ radian.

Neglecting systematic

_errors, ^the

fringe

_can be

split by

_a factor

proportional

to

,iC

_where _C

=

~' '~

is the _contrast of the interference pattern and

~max ~ ~min

N is the total number of _atoms counted in the measurement. For _our interferometer _a

typical

one minute measurement

give~

_a

sensitivity

better than 10 mrad which

corresponds

to an

energy

sen~itivity

of

roughly

^6.6 _x 10~

'~>eV/,"min.

or about 16 Hz/ ,"min.

3.

Magnetic phase

shift.

The

experiment pre~ented

here _concern~ the interaction of the

ground

state of the Na atom with

a

magnetic

field.

Speciiically,

we will de~cribe

experiment~

that demonstrate the

interplay

between the interference patterns

produced by

the 8 different

magnetic ground

state~ of the

sodium atom. Before _we describe the

experiment

in

detail,

let _us consider the

phase

shift

produced by

a

magnetic

field

applied

to one arm of the interferometer for _an atom in _a

specific magnetic

~ubstate.

if _we

apply

_a

magnetic

field B to one part of the _wave function, _we introduce _a

potential

U(v) ₌ _~ B= g~

m~m~B,

where g~ m~ is the

magnetic

moment and the m~

gives

the

projection

of the

magnetic

moment on the direction of the

magnetic

field. The

potential changes

the

unperturbed

(-vector ko =

,fi

_to _II-1)

=

,/2

m (E~,,~ g~ m_~ m~ ^B and

fi fi

introduces _a

phase

^shift _~g

given by

~g

(lu

=

~

g~ m_~ m~ B (.i h

= g~ m_~ m~ B r) iv

h-

lo

^hV

In the limit that the

potential

_~ B is much smaller than the kinetic energy of the _atom

E~,,~, we can

integrate

_over the unpertu

fifiath.

The action of the

potential

_can also be _view ed

as an refractive index _n

=

~~= jl _-)m -(

_The

pha~e

^shift is then

given by

E

JCL RNAL DE PHi ~'QUE T 4 N' 'I ~U"EMBER lW4 j

(5)

2032 JOURNAL DE PHYSIQUE Ii N° I1

Table I.

Maqnefic

moment.<

(if

(lie 8

chfft>lent n>aqJ>etic.

.<iib~<late.<

of'(lie Na,qioiinc/

-<(ate

mF=-2 mF"-I mF"0 mF"1 mF=2

F=2 -pB

-(~

₀

_(~

_p~

~/_~ ^~B ~ ^_~B

2 2

~g(I) I

Ill

_{fill ))} _di. _in

our interferometer with its 10 _cm

long

interaction

region,

a

phase

shift of rad

corre~ponds

to U

= 6.6 _x 10~ ^'~ eV which translates into _>1[

=

?.7 _x 10~

'.

_For

a 000 m/~ fast Na atom in a F

= 2, _ni~ _± 2 ~tate in _a

magnetic

field of G

we find _a

phase

^~hift of 880 rad

corresponding

to refractive index of (I n)12.5 _x

10~~

With

typical performance

of _our interferometer _we achieve _a

sensitivity

of _x 10~ ^~ G/

, min.

Because _our atomic beam i~

unpolarized,

_an

applied magnetic

field B results in _a different Zeenian

pha~e

shift for each of the different

magnetic

moment

projection~.

The

ground

state of

~odium consists of _two

hyperfine ground

state~

having

total

angular

momentum F of 2 and

giving

_a total of 8 substates. Because the two

hyperfine

_state~ have

g-factor~

that _are the _same in

magnitude,

but have

opposite sign.

^these

eight

_states five

only

5 different

magnetic

_moment

projections (Tab.

I). There will be one ;tote (F

=

~, _m~ ? j with _one Bohr magneton ^of

magnetic

moment

projection along

the z-axis~ two with half _a Bohr magneton

(F

=

2,

_m~

= I

and F

=

I, _m

~ = ), two with _no

projection

(F

=

? and I, _m~

=

0), two with minus _one half

(F=?,m~=-I

and F=I.

m~=I)

and _one with minus _one Bohr magneton

jF

_2~ _mr

= 2). This

give

_us~ in the low field

regime~

8 states with 5 different

magnetic

moments.

Throughout,

_we have

neglected

the _curvature in the Zeeman manifolds. Thi~ I; _a

valid

approximation

in _our

experiments

~ince _we operate ^at total

magnetic

fields of

approximately

5 G, while the

decoupling

of the Zeeman _~tates _occur~ at several hundred gauss.

The total interference pattern ^will ^be ^the ^incoherent sum of the patterns

produced by

the 5

individual

magnetic

moment

projections, weighted appropriately.

In the _case of _a _mono-

chromatic atomic beam the contrast of the interference pattern as a function of the

magnetic phase

shift _~g~ of the F

=

2, _m~ 2 ~tate (the

« stretched state ») is

C

-

I C°~

~~9i)

₊ 2 _C°S

l~

+

ii

^C ^C°S~

i~

^cos

i~

⁽²⁾

Co

i~ the initial contrast. The main feature in e~uation (2) is _a

rapid

decrease of the _contrast with

rising phase

shift _~g~ and later revivals of the full _contrast at

specific

value~ for

~g~. The n-th revival i~ _at _a

point

where the

phase

shifts _are [~g~[ ^=4n~ ^for ^the

(m~[

= 2 states, _~o

=

2 _nw for the

[m~[

₌ I states, and 0 for the m~ = 0 states. At these

points

the interference patterns

iej?fia.<e.

The behavior of the total observed

pha~e

shift is best _seen

by looking

at the _sum of the

different

phase

vectors

(Fig.

?). ln _an

unpolarized

beam the

magnetic phase

shift for the

+ m~ and _m~

magnetic

sub~tates cancel, and therefore the total

phase

shift will be either 0 _or

qr. At the

rephasing peak~ (4

_ii _qr

< ~gj < (4 _n ₊ _qr, the total

pha~e

^shift ^is 0, but jnbetween the

rephasing peaks,

for

phases

(4 _fi ₊ ) _w _< _~g~

< (4 _n ₊ 3 _w _we find

regions

where the observed total

phase

^shift is _w.

The situation is

complicated by

the

velocity

distribution of _our beam. As _one _can _see from

equation

(I ). the

magnetic phase

^shift ^is

dispersive,

that is~ it

depends

_on the _wave _vector

(6)

0 n/2 _n 3n/2

~2

_4n

7n/2

3n 5n/2 ~^~

l~

li

contrast

100§b 60§b 0§b 10§l 0§b

phase

0 0 _n

Fig. ?. Pha,e vector, for [he 5 different independent magnetic moment projection, and the re,ulting

contra,t and pha,e, for _vJriou, applied pha,e ,hilt, jij ol the ,ireiched _;tote. The length of [he pha,e

vector, give, the probability to find the ,tote in an

unpolarized

beam. It _1, 2 for [he [m~ (black

vector,) and _Jiij o (white vector,) ,tJte, and for the [ni~ ^? ~tates (gray vector,). The _~um of the pha,e vector,, i, ,hown below. It, length _gi~e, the relative contra,t, it, direction the total pha,e ot the interference pattern. ^The diagram, at ji~ _w, ^? _w and 3_w ,how the _zero _contra~t

point;,

the

diagram,

at

~~

~

_and

~

_,how

w pha,e ^,hilt.

1 of the

incoming

atomic beam. The necessary average over the k-vector distribution

f(k)

of the

incoming

^beam affects both the observed

phase

shift

_f'll)

sin (~g(I)) dl

~ ~~~~~~

j (I

_cos _~g II dl

~

~~~

and the contrast of the ob;erved interference pattern

=

co I( _fir

_sin _(~Dii)i

^ij~

+

j(i)

co;

i~D(i)) kj~

=

cop

(w). (4)

~ ~

Cn

^is the contrast at zero

phase

shift and P (# is the relative loss of contrast as a function of the

applied phase.

This reduction of the interference _contrast _can al~o be viewed _as

being

caused

by

the finite coherence

length

_i~,,~~ of the beam. in _our _case of linear

dispersion,

if _we _assume

p~,,/12

that

f(k)

is _a Gaussian distribution with width _«,, _one finds _P p ) ₌ e The coherence

length

is then

given by

_i~,,~~ ₌

«,

(7)

2034 JOURNAL DE PHYSIQUE II N° 11

Averaging

_over the

incoming

k-vector distribution the contrast of the interference pattern as a function of the

velocity averaged magnetic pha~e

shift of the ~tretched state

#~

is

give by

:

C,~ #~ p~

C

(pj)

₌ ^P

(p~)

cos

(p~)

+ ? P / cos

j

+ (5)

4 2

For small

phases

shifts the

velocity averaging

will diminish and broaden the

rephasing

resonance~. For

large applied phase

shifts~

@~w

~

the average over the

velocity

ml

distribution of the atom beam reduce~ the contrast in the interference pattern to zero for all

atoms except ^those ⁱⁿ ^the two n>~ =

0 _states~ which

experience

no Zeeman

phase

^shift. The

contrast

damps

out to a level that is _one quarter ^of ^the

original

contrast.

Even with

averaging

over the

incoming

^(.-vector distribution the total

pha~e

shift is _zero except ⁱⁿ ^the

regions

between the

peaks

where _we find

again

a

phase

shift of _w. But the

regions

with _w

phase

shift get ^smaller ^with

increasing applied phase

shift, and

disappear

when _no _zero

contrast

points

_are reached in between the

rephasing peaks.

4.

Experiment.

4. EXPERIMENTAL SET-uP.

Experimentally,

rather than

applying

a

magnetic

field to

only

one arm of _our interferometer, it _was easier to

apply

a different field to each of the two beams.

A

magnetic

field B,~~j~~ ^of

equal

field

strength,

but

oppo~ite

direction, can be created

by sending

_a current

I,

down the septum,

perpendicular

to the

plane

of the interferometer

(Fig.

3).

We _now superpose a

nominally

uniform external

magnetic

field B~,~, in the ₌ direction

(along

the

beam)

_over the entire interaction region. The septum ^field ^will ^add to the bias field _on _one

side and substract from it _on the other side

(Fig.

3b). ^The ^Zeeman energy, ~ B, is _now

different for the two beams and different

phases

are accumulated for the _two

paths

around the septum.

The

paths

^of ^the two beams form _an

Amperian loop

around the current distribution. We _can

use

Ampere's

law _to find the differential

phase

shift for _a

given magnetic

moment

projection

(in Gaussian units) _:

~~

~°"'F ~

G

^~~ ^~~ ^~~

/

^~~

~~°'

_~ ~~~

j(0,

_y is _a

geometric

correction factor _w coming from

misalignments

between the atomic beam, B~,~, ^and B,~~j~~ ⁶ ^is ^the

angle

^between the bias field and the septum

magnetic

field, and y is the

angle

between the beam direction and the septum

field).

If B~,~, w B,~~~~~ and the septum ^is ^much

longer

then the beam

separation,

the

angular dependence

in

equation (6) simplifies

_to

f(6,

_y ₌ cos

(o

cos

(y

).

In our

experiment

the _current is fed to the septum at two

points, opposing

each other, located

longitudinally

in the _center of the foil

(Fig.

3a). B~,~, ^is

produced by

a set of _« Helmholtz-like _» field coils 20 _cm in diameter and 20 _cm apart, ^wound ^around ^the vacuum tube _at the second

grating flange

that houses the interaction

region.

A

typical

field

strength

_was 3-4 G. This field is much

larger

than the fields

generated by

the current in the septum (B,~~~~~ ^0.13 ^G ^at ^the

center for A

current)

and

provided

the

quantization

axis. We

employed

two other _sets of

orthogonal

coils to cancel field~ not

along

the beam axis (z direction).

Assuming

_an

infinitely long

septum ^of

height

^2d contacted at two

points opposing

each other and with the

interfering

beams at the center between the _two connection

points

of the septum, the

magnetic

^field ^of our setup ^takes a

simple

form

j 0

10

Bioiji = B,opium ⁺ Bb>j, +

j

0 ₊ 0 (7)

~~ ~b>J,

d~ ₊ _=~

(8)

(a)

coils for _l~~~~

~sep~m

~interferingbeams

z y

~

^~sepium

+ - - +

~ ₊ _- _-

- + ~

~

~~

(as

Fig. ~. Septum geometry, currents and

magnetic

field, _as u;ed _in the magnetic repha,ing experiment.

a) Schematic of the metal septum ~eparating ^the ^two interfering beam~, the current connection,, and the

current flowing

through

the septum i~ shown. For clarity the ~ide plate, of the interaction region where

omitted. b) The detail of the interaction region and magnetic ^fields ^and ^the ^bias ^fields ^for ^the repha~ing experiment. The dark _arrows _are magnetic ^fields, ^and ^the light arrows represent ^the atomic beams.

The +/- stands for the

left/right

_arm of the interferometer. We _we that the B,~~j~~ ^varies ^like

z~ ^~ near the end of the foil. In _our geometry, ^d ₌ 2 cm and ₌

=

5 _cm, the field _at the

edge

^is

only

about 10 % of B,~~,~~ ^at ^the ^center. ^Also ^the ^beam

separation

(55 ~m) is _much smaller than

the septum

length

therefore the above discussion of the

Amperian loop integral gives

a very

good approximation

of the observed

phase

shift.

4.2 MAGNETIC REPHASING. The data from _a

typical experiment

with the setup as described

above, _are shown in

figure

4

together

^with a fit from

equation

(5) which accounts for the

8

magnetic

states and the effects of the finite

velocity

distribution of the beam. Not

only

the relative

positions

of the contrast maxima but also their width and the

degradation

of _contrast of

~ubsequent rephasings

due to the finite coherence

length

are well accounted for

by

the model.

As

expected,

when _a small current was sent down the ~eptum (which

applies

_a small

phase shifty,

the _contrast

degrades.

The first

rephasing

_occur~ _at _a _current of

i

II 0 mA. Here the

(9)

2036 ^JOURNAL ^DE PHYSIQUE II N° I1

~

-

i s

~ Z

C~

© Q

~

rrent [mA]

Fig.

4.

the ontra,t from on,tructive rephJ,ing oi the independent interference pattern, oi the ~

different magnetic ,ub,-;tate, oi the ,odium _atom. The

lower

graph ,how, ^the pha,e _(in

ob,erved interference ^pattern.

phme

shift

applied

to the stretched _state I, 4w and the

phase

shift of the four states with

m~ =

is then ~

w. At thi,

point,

all of the interference pattern, ^have ^the ~ame

pha;e

^modulo

? qr, and

high

contra,t

fringe,

_reappear becau~e all of the interference pattern; ^line up with each

other. Similar

rephasing peak;,

wider, but of

succe~~ively

~maller

height

can be _;een around

235mA, 36~lmA~ 4~~mA and 610mA. With

higher

and

higher

current the contra;t

approaches

a

;teddy

value of 6 fl, one quarter ^the

original

value of about ?4 %.

In the first

region,

between the

repha;ing peak,

_one finds ,maller _contrast. There is _zero contra~t at a shift of _qr, ? qr and 3 _w for the n>~ =

? and _two distinct

peaks

appear at a

phase

shift of ~"

and ~"

At

higher

order

rephasings

these small

peaks

_are washed _out

by

^the

2 2

velocity

distribution of the beam and are

completely

_gone

by

the 4-th,minimum. In addition to

the contrast revivals _we also ob,erved that the

phase

shift is _zero except ⁱⁿ the

regions

between

the

peaks

^where ^the

phase

~hift is _qr, There _are sudden transitions between the,e

region~, exactly

at the

points

with _zero contrast.

A fit of the above model to our data

gives

a

period

for the

repha~ing peak~

of 123.7(4) mA.

Thi~

corresponds

to rod

phase

;hift for 9.84(~) mA. Thi; ha; to be

compared

with the

expected

^value ^of

9.69(3)

mA for Na atoms with 071(3) m/s

velocity,

as measured from _a diffraction pattern. ^Thi; ^I, ^not

surprising,

since due to some ancient

hi;tory

of our atomic beam machine, the

magnetic

fields in _our _vacuum chamber _are not uniform and the

applied

bia~ field I; not

homogeneous.

A _more reali,tic _treatment _mu~t therefore average tht;

misalignment

over

the

path arriving

at an

expre~sion

with _a _«

fudge

factor _»

( f'(

o, _y

)). Comparing

our

rephasing experiment

^with ^the ^velocities ^measured ^from diffraction patterns we find

(f(o, y))10.985(5).

From the

smearing

of the

rephasing peaks

_we find _a rms

velocity

d;,tribution of ^~' 5.6 %

compared

to 3,6 §& as evaluated from the diffraction pattern. ^Thi~

I

d;,crepancy

could be caused

by magnetic

field

gradients

near the

edge

^of ^the ~eptum, ^induced

by increasing

_septum currents.

(10)

In addition _we find that the whole

repha~ing

pattern ^I; ^offset

by

14,3(5) mA,

corre,pond- ing

to ~ome re;idu;tl

phase

~hift of _~g~

=

1.45 rad

coming

from _a

magnetic

field

gradient

in _our interferometer.

4.~ E-B _BALANCE. A; seen in

equation

(5) ;tnd from the data in

figure

4 the average over

the

velocity

di;tribution of the atom beam reduce~ the _contra~t in the interference pattern for

large

current~

through

the foil for all atoms except ^those ⁱⁿ ^the two n>~ = ,tates, which

experience

no Zeeman

phase

~hift. At

high enough

current the observed interference is

only

from the n>~ = ~l ~tates, any ^small

pha~e

,hift ob;erved when

applying

_an additional interaction will be

purely

from the m~ =

0 _,tate;.

One _can _now think of

compensating

a

large pha~e

shift

~g w in _one beam

by

a similar

f'i

one, with the _~ame

di~per~ion properties

but of

opposite

_,ign. If _we take for _one of the

pha~e

shift, the

magnetic

interaction, the

compensation

will be

only

for _one

~pectftc magnetic

substate (or two if

they

~hare the _~ame

magnetic

moment

projection).

The interference

contra;t will be

regained

for

only

one ma~netic sub;tate,

allowing experiments

with

polarized

atoms in an

unpolarized

beam.

To demon;trate this _we

performed

an

experiment

in which _we

compen~ated

the

magnetic

phase

~hift with _an electric

phme

~hift friJm the DC-Stark effect. If _one

applie,

_an electric field

on one ,ide of the interaction

region

one induce~ _a

potential

^due to the DC-Stark effect

U~,,,~

₌

aE~,

where _a I, the electric

polarizability.

If the field is

homogeneou;

the 2

re~ulting phase

^~hift i~

~~g~,,,~

₌

aE~

_where I I;

length

^oi the interaction

region

and fi _r ?

r is the

velocity

of the atom~. The measured

phase

shift is

quadratic

_in the

applied

electric field and allows _a

precise

determination of the electric

polarizability

of the

ground

state

(18(. By applying

a

large

Stark

phase

^shift of 70 radians to _one beam, well out~ide the coherence

length

of the beam, _we

destroyed

^the contra~t for all of the _m~ ,tates. We can now

regain

the

interference contrast of

specific polarization

state~

by applying

_a differential Zeeman

phase

shift with the

rephasing magnetic

field

(Fig.

5). For certain ranges of septum currents, and

therefore differential Zeeman

phase

shift~, first the

n>~=2

_components and then the

n>~ = l components ^had a total

(Zeeman plus Stark) phase

shift that _wa~ within the coherence

length

of the beam.

Figure

6 show~ the

phase

^of ^the

re~ulting

interference pattern as a function of ~eptum current. The

slope

^of the line reflects the

magnitude

of the

magnetic

moment

projection

and _one _can

clearly di~tinguish

between the

region

where the _m~

=

2 _<tate and the

n>~ ^state contribute to the interference pattern. ^Thi~ ^allows

experiments

to be

performed

with

polarized

atoms in _a

unpolarized

beam without the

difficulty

^of

optical pumping

(but

without the

gain

in

inten~ity

which such

optical pumping

could

bring).

A similar balance

between electric and

magnetic

interaction _was used

by

Beder;on _et al.

[?3(

to measure the

E-field

+ - - - +

» Bwp~m

Bbias

Fig. ^5. ^Schematic ^for [he field configuration, of the E-B balance experiment.

(11)

im~i=2

~~

~

l

(m (=1

# ^~

~

~ o

6 0 0 8 0 0 0 0 0

septum current

[mA]

Fig. 6. Magnetic phwe shift for _fi>~ 2 and _[11>~ state~ as ob~erved with _an unpolarized beam

in a « nia,qneii< iepfiaiiii,q _» experiment. An additional DC ~tark phase shift of 65 rad _is applied in _one

arm of the interferometer. For 400-7U0 mA septum current one of the ni~ 2 ~tate~. and between 700- 050 mA _two of the [n>F( ,tates are shifted back _in coherence with itself. The _two

~lopes

of the pha;e corre~pond to different

magnetic

moment

projections

being ^within their coherence

length.

electric

polarizabilittes

of neutral atoms. In his

experiment

he balanced the forces _on the atom in

inhomogeneous

^fields i in _our

experiment

_we balance the

phases

in _a

homogeneous

field.

5.

Applications.

5, MEASUREMENT _oF _THE _vELociTY OF INTERFERING ATOMS. Since the value of the Bohr

magneton ^is ^known ^with

high precision,

and if the geometry ^of ^the setup ^and ^the ^local ^field

direction is known with sufficient accuracy, ^the

only

free parameters ^that determine the

repha~ing

_« _resonances _» _are the

velocity

of the beam and its

velocity

distribution. Thi~, in

principle

can be

exploited

to make _a direct measurement of the average

velocity

and

velocity

di;tribution of the atoms

contributing

to the interference pattern. ^In

precision

experiments, ^{it is} very

important

to eliminate

systematic

effects

arising

from processes like the

clipping

of part of the beam

by

the septum, ^which cause the

velocity

of

interfering

atoms to differ from the average

velocity

of the atoms in the beam up~tream ^of the interferometer _as determined

by

a

measurement of the diffraction pattern. ^This ^effect ^I; ^small

(10~~)

but it will be

important

to

know the

velocity

distribution of the

interfering

atoms in future

high precision separated

beam measurements, like a

precision

mea~urement of the electric

polarizability

of neutral atoms. But a~ seen in the di~cussion of _our

repha~ing experiment

care has _to be taken _to have _a well-

defined geometry ^and ^well-known

magnetic

fields.

5.2 POLARIMETER FOR ATOMS. Another

application

could be _a _very sensitive

polarimeter

for atom interferometer

experiments.

From

experiments

with _an

unpolarized

beam (the

rephasing experiment)

_we _can evaluate the

phase

shift for the stretched state

including

the

average

geometric

^correction factor

(f

lo,

y))

with

good preci~ion

^(0.5 % in the above

experiment). Performing

the

experiment

with _a beam of unknown

polarization

will

give

an

average

phase

shift per

applied

current.

Comparing

that

phase

shift _to the

phase

~hift of the

stretched state as determined

by

^the

rephasing experiment,

we can get information _on the

average

polarization

of the beam,

especially

in the clo~e _to 100 % range, where other methods like Stern Gerlach magnets or fluorescence are not so sensitive.

(12)

5.3 MEASUREMENT _OF _THE _GEOMETRIC _PHASE. As another

example

^of a

pos~ible

extension of the

magnetic rephasing experiment

in _our

separated

beam interferometer _we will describe

here _a _new method to

explicitly

measure the

Berry phase

of the _wave function. When _a

quantum system ^evolves (even

adiabatically)

around _a

cyclic path

in

phase

_space it

gains

an

additional

phase,

as first described

by Berry [?4-26].

Various demonstrations of this effect have been

performed

with

photons [27],

_spin rotation

experiments

^with neutrons

[28],

and

atomic

hydrogen [29].

In a variation of the

magnetic rephasing experiment

_one _can

demonstrate the

geometric phase

of the _wave function

directly by

transporting the atom _on the

two arms of the interferometer _on different

paths

in

phase

_space. Such _an

explicit

determination of the

geometric phase acquired by

the _wave function of _a massive

particle

_w»

prohibited

in

previous experiments,

because of _an unknown and much

larger dynamic phase

shift between the two

paths.

For _a

geometric phase experiment

one

ideally

wants _a

magnetic

field

configuration

that

gives

the _same

dynamical phase (Eq.

)) but transports ^the

angular

momentum vector of the

particle

on different

paths

in the two

interfering

beams. The

dynamical phase depends

_on the

magnitude

of the

magnetic

field. The

geometric phase

^i~ determined

by

the evolution of the

state vector. If the motion is adiabatic this is identical to the evolution of the field direction.

The

magnetic

field of the _current sheet,

being

of the _same

magnitude

but

opposite

direction is _a

good starting point

for such _an

experiment.

We

plan

the

following experiment

based _on _our magnetic

rephasing

setup. ^This ^time ^the

quantization

axis is defined

by

a

guide

^field ⁱⁿ

j,-direction,

normal to the atomic beam which travels in z-direction.

Adding

a screw coil around the interaction

region, (Fig.

7) _we get ^the

following magnetic

field

~~

o

B,~<e" ^Sin

~i~ ~~

^°

~~~~~~ ~~~~~~~ ^~ ~'~~~~ ~ ~~~~~ ^~

~

~d

^~

~'~"

~°~

~i~

^~ ^~~~~' ^~~~

d~

₊ z~ ° 0

screw

coil

~septum

interfering beams

Fig. 7. Schematic for the configuration of the Berry phase experiment. ^The septum current and the

screw coil creating the rotating magnetic field _are ;hown.

(13)

o Bx

0

1

o.5

By _o

left _arm

right

arm

-o.5

-1

-2 -1 0 1 2

Bz

Fig. ~. Mo~ement oi the magnetic field vector, _on both ,itle, oi the ,eptum ^with di,tance through the

interaction region. The _maximum ol'the ,eptum ^field Î; ^twice ^the ^,ize ôf ^the B,~,~,,, ând B~,,,, ^0. ^The

field, _are given in unit; oi B,~,~,,. Entering ^the interaction region the atom find, the magnetic ^field vector at thc marked location (.). While the _atom _move; through the interaction region the magnetic ^field ~ector

move, along the two trajectorie, (-) _a,

given

by the _arrow,, and return, to the ,tarting point when the

atom leave, the interaction region. The ,olid angle hew.cei> the two _curve, _i, _a _mea,ure ot the

geometric

pha,e obtained by the adiabatic evolution oi the ;tate.

One _can show that when the septum ^field is normal to the other field,,

[Bj~,j_,~ i, the ~ame for both _arms of the interferometer and therefore the

dynamic phase

~g,j~,,j,,,~, ⁼ ~ _B~~,j,~ di h _c

acquired

in either _arm of the interferometer will be the _same. Hence there will be _no

dynamic phase

shift between the _two _arms of the interferometer, _a~ _we have ~hown

experimentally

using a combination of bid; field in the

_v-direction

and _our ~eptum ^field.

The

~pinor

wave function itself will evolve _on different

trijectorie,

in

pha;e

_space on

different sides of the septum ^and

acquire

a

phase

~hift ~~g~~,,,,,~,,,~ = n>~ ~fl~,~,j~,,,~,,, ^that ^I,

purely geometric

and I,

given by

the difference of the ,olid

anile;

_which _the _state _vector _~ubtends

IAD~,~,~~,j,,,,, = f1~~~, D,~~~,~). ^lf the motion is adiabatic D~,,,j~,j,~,,, ^is

given by

the ~olid

angle

subtended

by

the

magnetic

field _vector _B,~,,,,j _a; ;hown in

figure

8. We _can vary D~,~,~~,,,~,,,

by changing

the relative field

;trength~

for the variou; component, ^of our field. We _can al,o

change

the

adiabaticity

in _our

experiments by varying

the total field

,trength;. By optical pumping

to _a [n>~[ ₌ or [n>F( _~ ^? ,tate we can demon,trate the

,pin dependence

of the

geometric pha;e.

The

geometric pha,e

I,

only

_;een in ;1combination oi _;tt lea,( B,~~,~,,,, ⁺

B,~,~,,. Any

,mall

additional

dynamic pha,e

,hift of each component ^alone can be mea;ured

independently.

Therefore we will be able to ;eparate the

dynamic pha;e

from the

geometric pha,e,

and _can

make _a direct interferometric mea;urement of the

geometric phase

of the _wave function for

ma,sive

particles.

Ii is

intere,ting

to

point

out that an

experiment

like thi, _can al;o be _seen _as _a

(14)

demon;tration of

geometric

forces

(30(.

The

pha,e

;hift ha; the _same relation to the

geometric

force _a, the Aharonov Bohm

pha,e

shift has to the Lorentz force.

6. Conclusion.

We have ob,erved the

periodic rephasing

of the 8 different

magnetic

~ub,tate; of the Na

ground

state. We have ,hown that [he

magnetic

^~ub;tates form

independent

interferometers.

Thi;

experimental technique

allows various

applications

e;,ential for

precision experiment~

u;ing

atom interferometers of our type. It will allow _a direct measurement of the

velocity

of tho,e atom~ that contribute to the interference pattern, ^and ^allow u~ to

study

the

polarization

of the atomic beam inside the interferometer. Further exten,ion~ of this

experiment

will allow _a direct iiea~urement of the

geometric pha;e, completely ;eparated

^from the

dynamical pha;e.

Acknowledgments.

Our _recent work on atom interferometers and atom

optic;

_i;

,upported by

the

Army

Re;earch

Office _contracts DAALO~-89-K-008?, and ASSERT 2997~l-PH-AAS, the Office of Naval

Research _contract N00014-89-J-1?07, and the Joint Services Electronic~

Program

contact

DAAL03-~9-C-0001. T-D-H- _was

~upported by

a National Science Foundation

graduatre

re~earch

fellow,hip.

J. S. _was

partly ;upported by

an Eiu.ii Sc.fiiiidin _qei fel

low;hip

^of the Foi>d

=ii.

Fi)ideimi,q

dei

Wi.<.<en.<c.fiaftfic.hail Ffii.~c.rim>g

and

by

an APART

fellowship

of the Aii.itiian At ademi' _(>f' Siieiic.e.<.

References

ii Fizeau H., Awl Pfiv.~ Cfieni 3 (1~53) 457 Awl. Cfii,Jii. Flit, B 66 (1~62).

(2( Michel,on A. A., Alit .I .~i1. 22 II ~~ Ii 20 Morley E. W., Am..I S(1 3t (1~861377.

(3) Lord Rayleigh. Pin( Rot .jot. 59 (1896) l~~.

[4( Marton L.,

Simpwn

J. A. and Suddeth J. A., Pfii'.v Rei. 90 (1954) 490 Rev. St-i- /n.vt. 25 (1954) 1099.

[5] M611enstedt G. and Duker H., Nat~iiu.i.v.yen.vthajteii 42 _: Z. Phj'i 145 (1956) 377.

(6( Rauch H. Treimer W. and Bon,e U., Ph_i'~ Leit A 47 (1974) 369.

(7( Matter Wave

Interierometry.

a,pecial review _i,,ue of Pfiv~i<a B & C (S( (19~~i.

(~( Keith D. W.. EL,trom C. R., Turchette Q. A. and Pritchard D. E., Pfivi Ret- Lett 66 (1991) 2693, [9] Carnal O. and Mlynek J,, Pfij'.v, Rei Left, 66 (1991) 2689.

(10( Riehle F,, Ki,ter, Th., Witte A,, HelmcLe J. and Borde Ch, J., Pfiv,~ Ret- Li,tt, 67 (1991) 177, II II Ka,evich M,, Chu S,, Phi-i, Rei Leit 67 (1991) l~l.

ii2) Robert J., al al. Em Pfiv.~ Len 16 (199 ii 29.

ii 3( Shimizu F., Shimizu K. and TaLuma H., Phi-i Rei A 46 (1992) 46.

(14( Sterr U.,

Seng,tock

K., Muller J. H., Bettermann D. and Ertmer W., Aj>pl Phi'~. B 54 (1992) 341.

(15( For _an _overview of recent atom interferometer experiment, _,ee _« Optic, and Interferometry with

Atoms

»

Special

issue Appfi~d Phi'.w<..I B 54 (1992) J. Mlynek, V.

Balykin,

P. Meystre Eds.

(Plenum New York, 1986).

[16] Ekstrom C. R., Experiments ^with _a

Separated

Beam Atom interferometer, PhD Thesis (MIT,

1993).

[17] Schmiedmayer ^J. et al, _« Atom interferometry _», ^Lecture Notes in Physic~ 420, F,

Ehlotzky

Ed.

(Springer

1993).

[18] Ek~trom C, R. _et al., submitted _to Plij,.I. ^Rev. A (1994).

[19] Schmiedmayer J. et al., submitted to Pfii.I. R~i'. Len. (1994).

[20] Wei~; D.,

Young

B. and Chu S,, Phvi. R~i. Lat. 70 (1991j 2706,

(15)

j2II ^Keith ^D. Q., Soave R. J. and Rooks M, J,, I. Vac. St-i- Teihnol. 9 (1991) 2486 _; j22] Chang B. J., Alverness R. and Leith E. N., Appl Opt. 14 (1975) 1597.

j23] Molof R., Schwartz H., Miller T. and Bederson B., Phi'.I. Ret> A10 (1974) it ~l.

j24] Berry M, V., Pi<>c. R. Sac. (Londfinj A 392 (1984j 45.

j25] Berry M. V.,

Phy.I.

Toda_v (Dec. lQ90) 34.

[26] Geometric Phases in Phy~ics, Advanced Serie~ in Mathematical Physic~ Vol. 5, F. Wilczek and A,

Shapere

Eds. (World Scientific,

Singapore,

1989).

[27] Chiao R. Y, and Wu Y. S., Phys. Ret,. Lett. 57 (1986) 993 Kwiat P. G. and Chiao R. Y., Ph_i,s. ^Rev. ^Lent. ⁶⁶ (1991) 588.

[28] Bitters T. and Dubbers D., Phys. ^Ret'. Lent. 59 (1987) 251 _; Weinfurter H. and Badurek G., Phys Rev. Lent. 64 (1990j 1318.

[29] Miniatura Ch. _d al., Phy.~ Ret-. L~tt. 69 (1992) 261.

[30] Stern A., Ph_1,~. Ret. Left. 68 (1992j 1022 _;

Aharonov Y. and Stern A., Phj'.~ Ret'. Lent 69 (1992) 3593.