Adiabatic transfer in j ↦j and j ↦j - 1 transitions

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Adiabatic transfer in j �j and j �j - 1 transitions

Constance Valentin, Jin Yu, Pierre Pillet

To cite this version:

Constance Valentin, Jin Yu, Pierre Pillet. Adiabatic transfer in j �j and j �j - 1 transitions. Journal de

Physique II, EDP Sciences, 1994, 4 (11), pp.1925-1937. �10.1051/jp2:1994240�. �jpa-00248096�

J. Ph_vs. II France 4 (1994) 1925-1937 _NOVEMBER 1994, _PAGE 1925

Clas~ification PAj,,ri£s A hsfi.a< _I-r

42.50P 32.80P 03.65

Adiabatic transfer in j

-

j and j

-

j I transitions

Constance

Valentin,

Jin Yu and Pierre Pillet

Laboratoire Aim6 Cotton (*), Bit. 505, C-N-R-S- II,

Campus

d'orsay, 91405 Orsay Cedex.

France

(ReLeived /9 May 1994, ieceii,ed in final form 2 Augusi /994, accepted 8 Aug~ls/ /994)

Abstract. Coherent transfer _in multilevel ~ystems by adiabatic pa~sage through _a trapped state has

experimentally

~hown very different efficiencies for the two

hyperfine

_components

F 4 -F'= 4 and F 4 -F'= ~ of the cesium D~ ^{line. If} an

efficiency

_up to 55 % is

observed for ni~ = 4

- mj = 4 population ^{transfer with} the F 4

- F'

=

4 component, it is

never better than 25 % for the F 4

- F' 3 _one. We report ^here a theoretical interpretation of these

experimental

results. We ~how in particular that the F 4

~

F'

= 3 transition is much _more

~ensitive to any diffu~ed light, which _can destroy totally the

trapped

state.

1, Introduction.

Adiabatic tran~fer in _a multilevel system opens novel

perspectives

in coherent _atom

optics particularly

attractive for the realization of interferometers, where mirrors and beam

splitters

are

obviously

the

key

elements. The coherent adiabatic

population

transfer has been first

demonstrated in _a three-level Ii system ^{in the}

Na~

molecule with

partially overlapped

«~

-polarized

laser

fields,

which propagate ^{in the} same direction and induce stimulated Raman

scattering [I].

In _a recent

experiment [2],

we have

generalized

this method to the multi-A system ^{of the F} ₌ 4

~ F'

= 4

hyperfine

_component of the

D~

^{line of Cs} atom and have also extended it _to the _case of

counterpropagating

laser fields. This last

configuration

had been

pointed

out a~

permitting

coherent momentum transfers between

light

and atom

[13]

and the coherent _momentum tran~fer has

recently

been put ^{in evidence}

experimentally using

atoms in _a

Cs molasses

sample [4j

and in _a metastable helium atomic beam

[5].

In this article _we discuss the theoretical

interpretation

for _our

experimental

results _on the adiabatic transfer induced

by partially overlapping

«~-laser fields for the two F

= 4

~

F'

= 4 and F

= 4

~ F'

=

3

hyperfine

components of the D~ ^{line of cesium} atom. In

particular

we discuss the limitations for the measured tran~fer rates. In the

following paragraph,

we

generalize

first the three-level II system to the _j

~

j

and

j

~

j

atomic transitions. Then in

(~) The Iaboratoire Aimd cotton i~ msociated with the Univer~itd Paris-Sud.

1926 JOURNAL DE PHYSIQUE II N° 11

the next one, we compare the

experimental

results for the transfer from F

= 4, m~ =

+4 Zeeman sublevel to F

=

4,

_m~

=

-4 _one in both transitions, F =4~F'= 4 and

F

= 4-F'= 3. We show that the F =4~F'= 3 _case is much less suitable for the

adiabatic transfer.

Finally

in _a fourth section _we discuss the different mechanisms

leading

to limitations in the adiabatic transfer

efficiency.

In _a

previous

letter

[?],

_we had noticed that the presence of the

neighboring hyperfine

levels in the excited state makes the

trapped

stable not

totally

isolated, which _causes the limitation in transfer

efficiency.

The effect of diffused

light,

which makes the laser beams _not

perfectly

_« ⁺ or _«

polarized,

_can ^{also break} the

adiabaticity

condition. We show that the F

~ F transition with

large

F is much _more sensitive to this

spurious effect, making

the

adiabaticity

conditions, _more difficult _to fulfill.

2. Adiabatic transfer in _a multilevel

system.

For _a A-three level system ⁱⁿ interaction with

«~-polarized

^laser

^fields,

a

nonabsorbing

trapped

_or dark state results from _a linear coherent

superposition

of the two

ground

states,

(g~)

and

(g~) [6-10]

_:

(NA)

=

(fl~/fl) [g~) (fl~/fl

)

[g_ )

,

(II

with fl

=

(flj

₊

fl))"~

where

fl~ (resp.

J2_) represents ^{the Rabi}

frequency

of the

«+-(resp.

«~-)

polarized

laser field.

Using

a

configuration

where atoms interact with time-

delayed partially overlapped

«~-laser

pulses,

let's _assume that the «*-beam

precedes

the

«~-one, for instance. An atomic

population initially prepared

in the

[g~)

_{state can} be

regarded

as

trapped

in the

nonabsorbing

state

NA)

because _{at t}

= oJ this _state is identified

with

[g~ ^(J2_

₌

^0).

^{As the} «~ component increases, the

trapped

state evolves with the

change

of the

ellipticity

of the

resulting

laser field

polarization.

If adiabatic condition is fulfilled, TN

fly'

_where

T characterizes the rise time of the laser

pulses,

the atomic

population

^will stay ^{in the}

trapped

state without any real transition _to the excited state. At t ₌ + oJ, the laser field becomes

purely

«~

-polarized.

The

trapped

state is then switched _to the g_

)

state and the atomic

population

transfered into it. The transfer

efficiency

_can reach

nearly

loo fb _even for _a

high

spontaneous ^emission rate from the excited state

[e),

because

[e)

remains

practically unpopulated during

the process, The adiabatic condition

implies

that the characteristic time of evolution RI ^' should be smaller than the characteristic time of spontaneous ^{emission r~ For}

large detunings

of the laser field the characteristic time of

evolution becomes

(RI

₊

3~)~

^"~ and should _now be

compared

to _T.

Figure

shows _an

example

of the evolution of the

populations (b)

for different relative

positions

(D measured in units o1'A, _see

Fig. 3)

of the two laser

pulses (a).

The spontaneous emission from the excited level is here considered _{as a} leak for the system.

Typically

for D/A

= I, we obtain _a transfer

rate of 9511 for _a maximum saturation parameter ^{s=12.5 and} more than 99% if

s =

50

(s

=

2

(fl

~

/r)~ _).

_{This rate decreases when D/A have}

_{larger negative}

_{values. It is}

_nearly

zero for D/A

positive.

The _case where D/A

= 0 is

particular,

we do not obtain _a

trapped

state but _a final state which

depends

on the _area of the laser field

pulses,

_on the contrary ^{of the other}

cases where the final _state is

independent

of this parameter.

The three-level A system can be

generalized

to more

complex

systems

given by

j

_{~j or}

j

-

j

I atomic transitions

always

in _a

configuration

of

partially overlapping

«~-laser fields ~j or

j

I _are the kinetic _momenta associated to the different levels of the

transition).

Such transitions _as shown in

figure

2

present respectively

_one and _two multi-A systems

only coupled

to each other

by

spontaneous ^{emission. For} any ^state of the

N° it ADIABATIC TRANSFER IN j

~ j AND j

- j I TRANSITIONS 1927

is ~

(~i)

t1_ t1$ _t1~ 1)~

~

Of A .23 DIA 27

(

Cl

0

~' loo _{0 too}

~i

^~~ ₂

2

i~

~j ^{ii 11 O}

~ ⁺ $3

Cr DIA ~l 0 °S

t#l ~

~>~

Clw u

O

C 0

-~_

cL~

4J 0 la>

~~j ²⁵

l~ 11 t1~

llf II

p_ ⁺ +

DIA=00 DIA= 0 3

~j

~l~

0

0 too o too

Time (r-1)

~~~

DIA -2.3

DIA _{2 7}

o too too

DIA -1 0

rt

~

nc

11

-ii~ +nc+n

loo

DIA 0 0

DIA 0.3

o loo o loo

Time jr-i)

Fig.

I. -Calculated temporal ^evolution (b) of the

populations

_{ii~, n_} and _ii~ of the _~tates

g~, g_ and g~, for different overlapping (D/~ j of «~ and «~ laser beams ((he associated _{curve~ are} shown in

figure

a). The ~aturalion parameter iq 50, the shape of the laser

pulses

_is choqen _to be Gau~sian with _a width (FWHMj J ?0 r~ ^' In the calculations, the spontaneous ^{emission is treated} as a leak of the population.

1928 JOURNAL DE PHYSIQUE II N° II

electromagnetic

field

(f1~, f1~)

there exists _one

trapped

state for _a

j-j'

transition

~j'= j

or

j

I) that is

given by

NA

)

=

£

(-

f1~

_>II ^"'~/~

(f1_

16 ^{+ "'l'~} x

~v

n<=<.1-2. -<

~

~ ^l~~ _(ll~"+ ^~

~

m (I ~i _1« ~

m=>,-~ n<+~ '~ ~'~

fl

means that _we make the

product

of

3j symbols

^with m'

running

from

-j,

Q

2), _up to

(m

2), for the first _one, and from (m + 2) _up to

j

2 then

j,

for the second

one. If _m

=

j (resp.

+

j

), ^{the first} jresp. ^the

second) product fl

is

replaced by

I. There exists _a second

trapped

state for the

j

_-

j

I transition

given by

jt~~j

~

l

=

£ (j~il~)U~~~~Y~(Q )~~~~~~~~

~~n>

~~ ~' ~ '~~

~

~

^{J '}

^J' )j

x

n?' i -(nl'+i)

ni=-i>-il-I<-li m-~

x

fl~

~~

il÷ -~i j/- _i)lj ^~~'~l~~~

(a)

$

^{~~ '~} -l 0 +1 +2 +3 +4

,,

f t

- -

' ~~~ '' '~~~°

,

,''"

_Ill

,

,'~',

7/20 '~

' ',

,'j

_'

, ,

,'

, ^,

,, _, ',

' ' ,

,

' i ,' '

',

,

,'

_,

',

,

,''

' '

, '

, , , , , ^{, ,}

' , , _, ,

, ,

"~

,

~/(

,, ^, , , , , ^, , ^,

'

' , / ^' , _, ,

, , ^' ,' , _,

, , ' _, , _, j, ^{, , ,} , ,

,,, ^,,, ,, ^{, ,,}

-~ -

fi

_4-

~~ ~~ ~2 ~i 0 +I +2 +3 +4

~~~

-3 ~2 -I 0 +I +2 +3

~~~~

,

~~~ 5/18

'

~~~~

, 7/9

7/36

'

,

'

t

Ill

,

' ,

i

,

,

,

, ^,

'

'

,

,

'

'

-4 -3 -2 _-1 0 +1 +2 +3 +4

Fig. 2. Examples ^{of multi-,(} ~ystems for the (a) F 4

- F' 4 and (hi F

= 4

- F'

= 3 transitions.

The squares of the Clebsh-Gordan coefficients _are

given

in the

figure.

N° II ADIABATIC TRANSFER IN j

- j AND j _- j TRANSITIONS 1929

where .A' and _{~A" are} normalization constants. trivial

trapped

states _are obtained in presence of

«+ ^~~~'P field alone

(f2~

~,~,~ ~ ⁼

0) _:

(NA(«+ ))

=

[j,

₊

j)

and

[NA(«~))

=

[j, j).

The adiabatic transfer consists of

preparing

the atomic

population

in the

trapped

state

m = +

j (f1~

= 0), then

switching

off the _« ⁺ field _at the _same time _as the _{« one} is

switching

on. As in the three-level _case, if the adiabatic evolution conditions _are fulfilled, the _atomic

population

evolves while

statying

in the

trapped

state. At the end of the process, we get an

adiabatic

population

^transfer from the _m

= +

j

state to the _m

=

j

one in the

ground

_state without _ever

populating

_any excited _states. It is clear that spontaneous emission _can be

completely ignored during

the adiabatic transfer. _{We note at} this

point

that the _two transitions

j

_-

j

^and

j

_-

j

I have

quite

^{different Clebsh} Gordan

coefficients,

which, we will _see, leads

to very different behaviors in the adiabatic transfer.

It is also easy ^to realize that in the

counter-propagating

field

configuration

2

j

units of

photon

momentum

(hk)

have been transfered to atoms,

corresponding

to

j

absorbed ~r~

photons

and

j

^emitted _« ⁺

photons

in _a stimulated process. The result _can be

interpreted

either in _a dressed

state

picture including

the atomic external

degree

of freedom

[7]

_or in terms of the stimulated

Raman

scattering

II

j.

The momentum transfer allows the

possible

coherent

manipulation

of

atoms

[3-5],

which

justifies

the

large

interest of the researchs _on this

subject.

3.

Experimental

set-up and results.

In _a

previous

letter

[2

),

experimental

^{results have}

already

^been

partly

discussed. We recall here the

experimental

setup ^and

procedure,

and _we

give

_more details of the data

concerning

the

comparison

of the results for the

hyperfine

components F

= 4

- F'

= 4 and F

= 4

- F'

=

3 of the Cs

D~

line (insert ^of

Fig. 3).

A scheme of _our

experimental

set-up ^{is shown in}

figure

3 for laser beams

propagating

in the _same jai and

opposite

(bl directions. The atoms escape ^out

an oven (120 °C)

through

a vertical 100 _~m wide slit. Three laser beams, denoted

respectively Lp, L~

~

and

L~

cross the thermal atomic beam _at

right angle.

The «+

-polarized

laser beam

Lp, provided

from _a feedback-stabilized HITACHI diode

II1,

12], illuminates the atomic

beam _at 120 _mm away from the _oven. Its

frequency

is locked to the saturated

absorption

line

6s

~Sj,~

^F = 4

-

6p ~Pj~

F'

= 5 from _a cesium cell. After

interacting

with

Lp,

the atomic

population initially

distributed among the 9 Zeeman components ^{of the F} ₌ ⁴

ground

state

hyperfine

level is

optically pumped

^{into the} _{m~ =} + 4 Zeeman level. The

polarized

atomic

population

crosses then the adiabatic transfer _zone

(Zj

j, located at about 10 _mm downstream.

This _zone consists of

L~

~, « ⁺

-polarized

and

L~

~,

«

-polarized.

Their

frequencies

_are tuned either _on the F

= 4

- F'

= 4 transition _or the F 4

- F'

=

3 _one. For coherent transfer _to

occur,

L~~

must

precede L~~

with

partial overlapping

between them. In this

configuration,

the

prepared

_{m~ =} + 4 atomic

population

crosses first the _« ⁺

-polarized

beam which does not excite any real transition. The

temporal

as well _as the

spatial

coherence of the _two transfer beams _is very

important.

In fact

they

are

split

from _a

single

laser beam

through

a dual-beam

polarizer (Rochon polarizer).

The

single

entrance beam is

provided

^from a 50 mW STC diode stabilized

by optical injection

from _a feedback stabilized HITACHI diode. The resulted line

width is about 50kHz, which _ensures the coherence of the transfer beams

during

the

interaction. The _two

separated

^beams are

superimposed through

a

beamsplitter prism

cube,

which is mounted _{on a}

micro-displacement

unit

providing

_a relative

displacement

between

L~~

and

L~~

with _a resolution of

50~m.

A

cylindrical telescope

^{is used} to focus

L~~

and

L~~

to slit-formed spots on the atomic beam with _a width

(FWHMj

of

typically

~

=

500 _~m and _a

height (FWHM)

of about 2 _mm. The total powers for

L~~

and

1930 jOURNAL DE PHYSIQUE II N° II

(~)

^{~ ~}

~

~-+

z~

^x

m~ ^{= +4 = ~}

Cs

O~ O~ O" O' O~

LP LT ~A Lo

4

- 5' 4 _~ 4' 4 ~ 4' 4 _- 5'

4 _~ 3~

1,;fi~

Oven ~~~~~~ ~~~~

)

^~

(~)

^{~ ~ ~}

= A

~'~~3

~

Zi

mF"~~ mF=~

Cs

Lp

LT~

4 ~5~ 4~4'

Fig. 3. llnsert) Involved Cs level~. la, b)

Experimental

~etup~.

LT-

are

typically

1.20 mW and 0.65 mW

respectively.

Most of the

experiments

have been done with these values which

optimize

the transfer

efficiency.

The

non-perfectly-symetrical shapes

of the laser _cross sections make the

slopes

^of the

opposite

variations of the

«~ and «~ fields different and

explain

these _non identical values. Mean Rabi

frequencies

calculated for these

typical

_{parameters are}

f1~

=2.5r and f1~ =1.85r

(r=

2w _x 5.22

MHz)

for the

respective

F

= 4 mF ~ ± 4

- F'

= 4 _m~,

= ± 3 transitions and

fl~

₌ ^3.2 r and fl~

=

2.4 r for the F

= 4 m~ = ± 4

- F'

=

3 m~, = ± 3 _ones. For _our

thermal atomic beam

(v~280m/s)

the interaction duration is in the order of

T = A/v

=

1.8 _{~s »}

fly

_~. The adiabatic condition is therefore well satisfied.

To

analyze

the adiabatic

population

transfer, _{we use an}

analyzing

laser beam

L~ crossing

the atomic beam _at about 10 _mm away from the transfer _zone in the _case where both laser beams

LT±

Propagate in the _same direction. It is

«

-polarized

and is in fact

separated

^from the STC laser beam

by

_a non-coated

beamsplitter plate.

Its

frequency

is therefore the _{same as} the

transfer beams (tuned on the F

=

4

- F'

= 4 _or F

=

4

- F'

= 3 transition). The role of this beam is to

destroy

^the atomic

populations

in all of the Zeeman sublevels except ^the

population

in mF "

4 level which is

populated by

the adiabatic transfer. A

free-running

HITACHI diode

provides

the detection laser beam

L~

which is used _to measure the

population

in the

N° II ADIABATIC TRANSFER IN j - j AND j

- j TRANSITIONS 1931

mF " 4 level 50 _mm downstream

(Z~).

Its

frequency

is swept across the F

=

4

- F'

= 5

transition, and averages are

performed

_{to overcome} the

signal

fluctuation. If the presence of

the laser

L~

does _not

modify

the

population

in the level F

=

4 _as

analyzed by

the laser

L~

after the adiabatic transfer

(which

is the _case in _our

experiments),

then this _means that all

the atoms in the level F

= 4 _are in the Zeeman sublevel m~ =

4. In the interaction and the detection _zones, the earth

magnetic

field is

compensated by

three

pairs

of Helmholtz coils. A

small

homogeneous magnetic

field (, 100 mG is

applied along

the laser beam

propagation

direction _to dominate residual external fields and _to stabilize the

polarized

atomic

population.

The Zeeman shift induced

by

this field is 35 kHZ which is

negligible compared

to the line width of the Raman transition involved in the transfer process. This line width is, in _{our case,}

determinated

by

the atomic beam transit time

through

the transfer laser beams, ~v

l/r 600 kHz. In

figure

4, we show the measured

population

transfer

efficiency

for both transitions, F

=

4

- F'

=

4 and F

= 4

- F'

=

3, as a function of the

displacement,

D, between the _axes of the two transfer beams. D is measured in the unit of A and its

negative

values

correspond

to the _case where

L~~ precedes LT-

The

intensity

of the fluorescence

induced

by L~

^{is measured for each value of D and is normalized with}

regard

to the total. We have obtained _an adiabatic transfer

efficiency

_up to 55 fb in this _case of F

=

4 ~ F'

= 4 but

never better than 25 fb in the _case of F

= 4

~ F'

= 3.

Figure

5 shows _a

typical

evolution of the transfer rate 10ei"sus the laser

detuning

in the _case of the transition F

= 4

- F'

=

4. We _note that the adiabatic transfer is not very sensitive to the laser

detuning.

The transfer

efficiency

decreases _to half of that at the _resonance for _a

detuning

of 8 r, which

corresponds

to the

decreasing

of the saturation parameter. ^The same kind of behavior is observed for the F

= 4

- F'

= 3 transition.

Simultaneously

with the detection of the transferred

population,

we observe in the transfer _zone

jZj

^the fluorescence induced

by L~

~

and

L~ using

_a CCD

>~

#

~i

~ Sl

c

1J

1J

l~

1J

~

f

~ 52

1J

>

t

"

1J ^°

Ct$

7 6 5 4 3 2 0 2

D/A

Fig. 4. Adiabatic population ^{transfer for} both transitions F 4

- F'

=

4 and F

=

4

-

F'

=

3. The

error bar is 5 %.

1932 jOURNAL DE PHYSIQUE II N° Ii

50

~

~ ₃₀

~

fi

u u~

20 0 20 40 60 80 100

Detuning (MHz)

Fig. 5. Population ^transier rate iei,ni.q laser detuning for the F 4

-

F'= 4 transition.

/&

O~

~

~~

~

$*N ~

b

~

(

~

.~~

4~

~ Cd

tJ

tJ

~

tJ

~j

~-

O Q3

~Iq

x x

Fig.

6.-Fluorescence

quenching

for the F =4-F'= 4 transition (~ee te,t) la) _no overlapping

between <,+ and «~

polarizations,

(b)

partial

overlapping.

N° Ii ADIABATIC TRANSFER IN j

- j AND j

- j TRANSITIONS 1933

camera

(Figs.

6a,

b).

When there is _no

overlapping

between the

L~

±

lasers, _we observe first _a small

signal

due to the

L~

~

laser (a). This fluorescence

signal corresponds essentially

to the excitation of the F'

=

5 level because of the

high

saturation parameter.i~. ^{We observe then} a

second fluorescence

signal

due to the laser

L~

This

signal corresponds

to the emission of _a

very small number of

photons

_per atom before it is

optically pumped

on the F

=

3 level.

Figure

6b shows the evidence of the

quenching

of the fluorescence induced

by

the

L~~

laser when the

L~~

laser beam

overlaps partially

with the

L~~

beam

(L~~ precedes

L~

). The correlated

signals (quenching

of fluorescence and transfer of

population)

indicates the coherent character of the process

[2].

In the

counterpropagating configuration (Fig. 3b),

we

have observed the

quenching

of fluorescence

only

for _atoms with _{a near-zero} transverse

velocity,

because the _transverse

velocity Doppler

effect shifts atoms out of the Raman transition _resonance

[2].

4. Theoretical

interpretation.

To

interpret

these results _we have

developed

theoretical calculations

approximatively corresponding

to the

experimental

conditions for both transitions F =4-F'=4 and F

=

4

- F'

=

3. The calculations have been

performed

in the _wave function formalism. The

spontaneous ^{emission is taken into account}

by using

an

imaginary

_energy number

(- ir/? for the excited

hyperfine

sublevels. The evolution

equations

in the RWA

approxi-

mation and for _a

configuration

^{where both lasers} propagate ^{in the} same direction _are written

as

~°~'"'

= (r/2

+13~

_a~

,,~

-1fl~

^{(F, ni'-} : F',

m')

_a~

dt ~,

-1fl~

jF,

m'+ F',

m')

_a~,~~

~

~°~'"~

=jj[-in~jf,m;F',m+ I)a~,,~~j

-ifl~jf,m;F',m-

ja~,~~~j]

(4) dt

~

where F'

= 3, 4 and 5 (see insert of

Fig.

3), _a~

,,~

and _a~,

~~

are

respectively

the coefficient~

associated to the

ground

F, _m

)

and excited F',

m')

sub-states in the atomic _wave function,

3~.

=

jw~,

_w~ w is the laser

detuning

between the atomic

frequency

^of the tran~ition

F - F' and the laser

frequency

_w. The Rabi

frequencies

are

given by

_:

f1±jF,m;F',m')=(-1)~~"'( ^~', /~ ~)fl(F,F'). (5)

The

frequency f1(F,

F') is

given by

f1(F,F')=-~(-')~'~~~~~~~'~"'12F+')~~~'+~~~~/ ^~2 ~~~~

x (- _)~' ^{+ ~ ~'~ ~'~}

,li

°~

~

(6p

i d' _Iii

6s) 16j

3/- 1/2 II-

Ejj

^{is the field}

amplitude.

The reduced matrix element is

given by

j9

_Fjj hrA

(6p _((dl

⁽⁽

6s)

=

(7)

16gr

1934 JOURNAL DE

PHYSIQUE

II N° II

In such formalism the atom is lost _{as soon as} it emits _a spontaneous

photon,

which is well

adapted

to the kind of

problem

_we treat, where

only

the coherent processes are

interesting

and have _to be considered. The

problem

with the multilevel system ^{is that} as the number of excited

states increases, the number of

population

leaks increases and _we need _to increase the

intensity

of the lasers. The

shape

of the laser intensities is chosen _a Gaussian _curve and the maximum intensities _are here taken _to be

equal

for both

polarizations. Figure

7 shows the calculated

population

transfer _{rate i>eisus} the Rabi

frequency

for the transitions F ~F

(a)

and F

~ F

(b),

F

varying

from _to 4. We _see that _we need _a

higher

laser power for

large

F and that the F

~ F transitions lead to a more efficient transfer than the F

~ F _ones. We

obtain for instance for the transition F

= 4

~ F'

= 4 and for _s

=

2

f1~/r~

=

50 _a transfer of

population, superior

to 90 fb and

larger

than the

experimental

_one

[13]. Taking

into _account in the calculations the presence of other

hyperfine

levels F' of the

6p

~Pi/2 level, we show that the transfer rates go down _to 50 fb

(see Fig.

8). This result is in agreement ^{with the}

experimental

result and shows the

importance

of _a well isolated transition to realize an efficient adiabatic transfer

[141.

The _same calculations

performed

in the _case of the transition F

=

4

~ F'

=

3

give

a decrease of

efficiency

from 80 % to 55 fb, much

larger

than the

experimental

result. We remark nevertheless in

figure

9 that the Rabi

frequency

to reach _a

good efficiency

is

higher

for the F

=

4

~ F'

= 3

transition,

which _means that the adiabatic conditions _{are more} difficult _to fulfill. In

particular

_{we are}

going

_to show that it is

really

crucial _to switch off (or on) the

«~ laser

intensity

to

(from)

a

really

_zero value.

Figure

lo shows the effect of _a residual

spurious badly polarized light

outside the transfer _zone. We simulated here the

shape

of the laser field intensities _not

by

a Gaussian _curve but

by

^the

superposition

of _a Gaussian _curve with

a Lorentzian _one. The maximum of each «+ and «~ field

intensity corresponds

to a

normalized Rabi

frequency

of f1

=

(F'

₊ 1)~/~ n

(F, F')

= 4.35 r. For each

polarization

the

two curves used have their maximum _at the _same time and have the _same width

(FWHM).

We

call B the ratio between the maximum intensities of the Gaussian and Lorentzian _curves. The calculations have first been

performed

without

taking

into _account the presence of the other

(a) (b)

~__,__

~ j

'."

~

/

~

; +~ j /,

cy~ ⁱ /~j :~

/

ii i I /

~ / / /

~£ ~'i ~ /

C©

ijj

~ l

~ i ~ /

fi

z~ ; j

~

~$

'I

F=I-F=I

Ii

F=i-v=i F=i-F=I

'i F=i-F=1 ~' F=i-F=1

/ Fm4-F'=4

/

_F=4-F=i

o s io is o 5 io is

Normalized Rabi Frequency (DIP) ^{Normalized Rabi Frequency lQ/F)}

Fig.

7. Maximum

population

transfer _rate i<I.qii.q the normalized Rabi frequency 12 (see text) for different transitions F

- F (a) ^and F (b).

N° 11 ADIABATIC TRANSFER IN j

- j AND j - j I TRANSITIONS 1935

(a) (bl

Q/r=I Q/r=4 Q/r=j Q/r=4

0 o

flJ

e i

$

i

/~

^{Q/r=2 Q/r=5 Ct~ Q/r=2 Q'r=5}

~

~ flJ

1J

tj

i#

_c

c

~

~

~ ~

~ o

l I

Q/r=3 Q/r=6 Q/r=3 Q/r=6

0 _o

4 2 0 4 2 0 _~ ~ 0 4 2 °

D/A D/A

Fig. 8. Simulations of the

population

transfer _{vet iii.<} D/A for the transition~ F 4

- F'= 4 and

F 4

- F'= 3 of the Cs D~ line. All the hyperfine levels _are taken into account.

F=4-F'=4 F=4-F'=3

0,8

QJ~

Ct$ _{".._}

~ L~~

I

;'

'.,

d

~

o s io is

Normalized Rabi Frequency (Q/r)

Fig.

9. -Maximum of the tran;fer _{rate ieiqii.i} the normalized Rabi

irequency

fl for transitions

F 4

- F' 4 (full line) and F 4

- F' 3 (da,hed line) oi the C; Dj line for different value~ of tiff.

JOUR~AL DE PHYSIQUE II T 4 N' II ~O"ENfBER 19"4

1936 JOURNAL DE PHYSIQUE II N° 11

16) (a)

fi

(

~2 ^C4

# _u~

~- ~j

~ _l fi

fi _~

~ l ~

'

fl

3

$ ~ j

~ l _~

£ j ,fl )

~#

l

[

0 2 3 4 0 2 1 4

B Ratio B Ratio

Fig. 10.

Decre»ing

oi the population tran~fer rate ior tran~itions F

= 4

- F'= 4 (full line) and

F 4

- F' 3 (dashed line) _iei.qii.v the diffused light ratio & (see text), 12

= 4.35 F, la) without and (b) with the ~puriou~ hyperiine levels.

hyperfine

levels

(Fig.10a).

For Mm I, which _{means more} than 5 % of diffused

light

at 3 ~/2 from the maximum of the

profiles,

the

population

transfer rate reduces down _to about

45 % for the F

=

4

~ F'

=

4 tran~ition, but less than 2 % for the F 4

~ F' ₌ 3 _one. For

ti

= 0.25

(2

% of diffused

light)

_{we get a}

respective

rate of 70 % and 25 fb for the _two transitions.

Taking

into _account the presence of all the

hyperfine

levels

(Fig. 10b)

_{we get} for Mm _a transfer rate

superior

to 25 fb for the transition F

=

4

- F'

=

4 but le~s than % for the F

=

4 - F'

=

3 _one. For 8

=

0.25

respective

rates are 45 % and ?0 % for _the transitions F

=

4

- F'= 4 and F

=

4

~ F'= 3. The~e last values _are

quite

ⁱⁿ agreement ^{with the}

experimental

ones and the poor

efficiency

for the F

= 4

~ F'

= 3 transition _seem~ be due to the residual diffused

light.

We have put ^{in evidence}

experimentally

the effects of the diffused

light by

several ways. First it has been necessary to put ^the

optical

elements far away from the

coated windows of _{our vacuum} chamber to decrease the diffused

light

and increa~e the

efficiency

rate.

Second,

the _use of _a 10 _cm

length

Cs cell _to reduce the diffused

light by spatially

re~olved saturated

absorption

has been crucial for the observation of the adiabatic

transfer in the _case of the F

=

4

~ F'

=

3 transition.

We understand the

origin

of the different behaviors in the adiabatic transfer

by considering

the squares of the Clebsh Gordan coefficients of the _two transitions

(Fig.

?). We _see that for the F

=

4

~ F'

= 4 transition these coefficients _{are more} and less in the _same order of

magnitude.

On the contrary

they

are

quite

different for the F

= 4

- F'

=

3 transition. We have _a ratio 28

between them for the _two Zeeman transitions F 4 m~ = + 4 _~ F'

= 4 m~. = + 3 and

F

= 4 m~ = + ?

~ F'

=

4 m~. = + 3. We understand that _a

spurious

diffused

light

of 5 % is therefore not all

negligeable

and the

trapped

state cannot be formed.

5. Conclusion.

In conclusion, the

experiments involving trapped

states are

quite

sensitive to the

quality

of the

used lasers, _we have

put

^{in evidence here that} a transition F~F- with

high

N° II ADIABATIC TRANSFER IN j

- j AND j

- j TRANSITIONS 1937

F is much _more sensitive _to the diffused

light

than F

~ F _one. The condition of

adiabacity

is

more difficult to fulfill and _so the

experiments

_are much _more difficult to

perform.

The adiabatic transfer in the _case of the cesium atom is also limited

by

the presence of other

hyperfine

levels,

leading

to _a maximum transfer _rate of 55% for the transition F

=

4

~ F'= 4. This result has been confirmed in the reference

[4j.

This _rate limits the

possibility

^for

repeating

_many times the

operation

in order to make _an atomic mirror _as

previously

mentioned. The cesium atom is

perhaps

not such _a bad candidate. We have

theoretically

shown that the F 4

~ F'

=

4 transition of the

Dj

line, which is much _more isolated, should

give

a transfer rate of _more than 90 % with _our

experimental

conditions,

neglecting

_any ^diffused

light.

In the _same way the Na atom should not

give

_any adiabatic transfer due to the fact that it is _not

possible

to

sati~fy

in the _same time the condition of

adiabaticity

and that of _a weak

optical pumping

leak. If the cesium atom is

probably

the best alkaline atom, the metastable _rare gmes, which have _no

hyperfine

structure, are also

quite good

candidates.

The adiabatic transfer

provides

a

promising

mechanism for the realization of coherent

atomic mirrors _or

beam-splitters.

To repeat ^the

operation

or to conceive _an atomic

interferometer with several of such atom

optics

elements, it is suitable to find _a well isolated transition. We have also shown in this article that the use of _a

j

_~

j

transition is much more

difficult than a j -

j

one, for

large j.

References

Ii Gaubatz U., Rudecki P, Schiemann S and Bergmann K.,./ Chem Pfi_».1. 92 (1990) 5363.

[2] Pillet P., Valentin C., Yuan R.-L. and Yu J.. Phy.v. Rev A 48 (1993) 845.

[3] Marte P., Zoller P. and Hall J. L., Phys. ^Rev A 44 (199 Ii 4118.

[4] Goldner L. S., Gerz C., Spreeuw R. J. C., Rolston J. L.. WestbrooL C. I. and Phillips W. D., Pfii.v.

Ret. Left 72 (1994) 997.

[5 Lawall J. and Prentiss M., Ph_v.v. Ret> Lett 72 (1994) 993.

[6j Arimondo E. and Orriol; G., Lent Nur)i~o Cimento 17 (1976) 333.

[7]

A~pect

A.. Arimondo E.. Kai,er R.. Van;teenkiste N. and Cohen-Tannoudji C., Phi-v Rei> Lett 61 (1988) 826.

[8j Kulin~ki J. R,, Gaubatz U., Hioe F. T, and Bergmann K., _{Pfi_10v} Rei, A 4011989) 6741.

[9j Oreg J,, Hioe F, T, and Eberly J, H., Pfiv.I Rei' A 29 (1984) 690.

[10] Hioe F, T. and Carroll C, E., Phv.v Rei, A 37 (1988) 3000,

[1ii Valentin C.,

Gagnd

M.-C., Yu J. and Pillet P., Europhys. Lent. 17 (1992) 133.

II?] Yu J., Gagnd M.-C., Valentin C.. Yuan R.-L. and Pillet P., I. Ph;.v III Fian<.e 2 (1992) 16I5.

[13] Here fl

= IF' + )'~~ fl (F. F') 5 F I, called the normalized Rabi

frequency.

In _our experiment~

fl =5.6F and 4,1F ior respectively the <,+ and <,~ polarizations in the _case of

F 4

- F' 4.

[14]

Experimentally,

_we observe _an adiabatic tran;fer _{on a} larger _range of overlapping (D/A 3 _m theoretically (D/A 2 ). This _is due _to the fact the laser pulses do _not have exactly a Gaussian

curve. We _can obtain _a better re;ult by con;idenng each pulse _{as a sum} of two Gau~sian _curves,