The optical autler-townes effect in doppler-broadened three-level systems

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The optical autler-townes effect in doppler-broadened three-level systems

C. Delsart, J.-C. Keller

To cite this version:

C. Delsart, J.-C. Keller. The optical autler-townes effect in doppler-broadened three-level systems.

Journal de Physique, 1978, 39 (4), pp.350-360. �10.1051/jphys:01978003904035000�. �jpa-00208768�

THE OPTICAL AUTLER-TOWNES EFFECT

IN DOPPLER-BROADENED THREE-LEVEL SYSTEMS

C. DELSART and J.-C. KELLER

Laboratoire

Aimé-Cotton,

^C.N.R.S.

II,

^Bât.

505,

⁹¹⁴⁰⁵

Orsay,

^France

(Reçu

^{le 17 octobre}

1977, accepté

^{le 14 décembre}

1977)

Résumé. ²⁰¹⁴ On discute

quelques

propriétés de l’effet Autler-Townes

optique

^tel qu’il ^{est observé}

dans les systèmes à trois niveaux avec élargissement

Doppler,

^et

particulièrement

^la

signification physique

du dédoublement de raies observé, les conditions d’observation et l’écart de ce doublet, ainsi que les effets dus à ^un désaccord de fréquence ^{du laser saturant.} On propose ^une

représentation graphique

^du phénomène, ^{fondée sur} la recherche de classes de vitesse en résonance, et on en déduit

une discussion

qualitative

complète ^de l’effet; ^{on montre en}

particulier

que le doublet observé n’est pas vraiment un dédoublement de la résonance comme pour l’atome immobile, ^mais correspond ^à l’existence d’un trou de

fréquence

^{où le laser sonde ne voit aucune classe de vitesse en resonance.}

On

applique

^{ensuite cette} méthode à des cas typiques

qui

correspondent ^{à des} systèmes réels à trois niveaux du néon. Les courbes

théoriques

obtenues à partir ^{du calcul} semi-classique utilisant la matrice densité permettent de confirmer les résultats qualitatifs ^{de la} discussion

graphique.

^On

montre, de plus, que

l’approximation

^du

premier

ordre n’est _pas suffisante _pour de nombreux cas

expérimentaux.

L’ensemble de ces

prédictions théoriques

^est ensuite confronté à une nouvelle expé-

rience utilisant deux lasers à colorant, ^ainsi qu’à ^une expérience ^récemment publiée. ^{L’accord entre}

théorie et expérience ^{est tout} à fait satisfaisant.

Abstract. ²⁰¹⁴ Some features of the optical Autler-Townes effect as observed in Doppler-broadened

three-level systems,

including

^the

physical

significance of the observed line splitting, the conditions for the observation of the doublet, the doublet separation ^{and the} detuning effects, ^{are discussed}

in this paper. A

graphical

illustration of the phenomenon, ^{based on a} search for on-resonance

velocity groups, is proposed which allows a complete

qualitative

discussion of the effect and shows that the observed doublet is not really ^a

splitting

^{of the} resonance as for the atom at rest but rather

corresponds ^{to a}

frequency

hole where the laser probe ^{does not see} any ^resonant velocity group.

The method is

applied

^to real three-level systems ^{of neon as} typical ^{cases. A} semi-classical density-

matrix calculation is

performed

numerically ^and

definitely

confirms the results of the

qualitative graphical

discussion ; furthermore it shows that, for many experimental situations, ^{the first order}

approximation

^is

inadequate.

The observation of the Autler-Townes doublet in a new

experiment

using ^two dye ^{lasers is} reported. ^The

experimental

^results and that of one of our

previous experi-

ments are compared with the theoretical

predictions ;

^the agreement ^is

quite

satisfactory.

Classification

Physics ^Abstracts

32.90 ^- 32.80K

1. Introduction. ^- The extension of the

pioneering

work of Autler and Townes

[1]

^{to the}

optical

range has

recently

^{been the}

subject

^{of a} great deal of interest.

The

optical

Autler-Townes effect or

dynamic

^Stark

splitting

^{is most}

simply

observed in a three-level atomic system

interacting

^{with two} continuous-

wave monochromatic

optical

^fields

(optical analogue

of the

original

RF-microwave

experiment).

^The

energy levels of the first transition are

actually split by

^the

quasi-resonant powerful

pump field and this

splitting

^is

probed by

^{the weak}

probe

^field

quasi-

resonant with the second

coupled

transition.

The

experimental investigation

of this effect has been carried out either in an atomic beam

[2, 3]

^or

in a gas

cell,

^{inside a laser}

cavity [4, 5]

^{and outside}

[6, 7].

These three

experimental possibilities obviously

^cor-

respond

^to very different theoretical situations.

Many

papers have been devoted to the theoretical treatment of three-level systems

interacting

^{with two}

quasi-resonant

monochromatic fields

[see

^{e.g. refe-}

rences

[8-15]].

However few authors have discussed the

optical

Autler-Townes effect in

Doppler-broadened

three-level systems ^{in detail}

[11, ^13,

16,

17].

In

^our

previous

^work

[7],

^the

optical dynamic

Stark

splitting

^induced

by

^a

single

^{mode CW}

dye

laser

(Â.

^{= 5 945}

Á)

^was observed in an

extra-cavity

neon gas cell with a cascade level scheme and a

1.15

He-Ne

probe

^{beam. As}

expected,

^the

splitting

^was

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01978003904035000

found to be

proportional

^{to the} square ^root of the

saturating

laser power.

By changing

^the

polarization

direction of the

probe beam,

^{the existence of two} Rabi nutation

frequencies

^{due to level}

degeneracy

was

clearly

shown. Due to the fact that the

probe

^laser

was a gas

laser,

^{it was not}

possible

^to

study

syste-

matically

^the

detuning

^effects.

The present paper is concerned with the

experi-

mental as well as with the theoretical

investigation

of some

important

features of the

optical

^Autler-

Townes effect as observed in

Doppler-broadened

three-level systems :

i)

the effect of the

detuning

^of the pump

field ; ii)

^the effect of the ratio of the two transition

frequencies

^{on the} behaviour of the doublet.

To understand the

physical background

^{for these}

two

effects,

^some theoretical aspects ^{of the}

problem

are reexamined.

In section 2, ^we consider a three-level system ^at

rest

interacting

^{with two near resonant monochro-}

matic fields. The

equations

of motion of the

density

matrix of the system ^are written for a

quite general

case. In the weak

probe

^field

approximation,

^an

analytic expression

^{for the}

absorption

^{of the}

probe

beam is derived. Well known

properties

of the Autler- Townes

splitting

^{for the} atom at rest are deduced.

In section

3,

^the calculations are

generalized

^{to the}

case of

Doppler-broadened

systems. ^A

graphical

illustration of the

phenomena,

^{based on a search for}

on-resonance

velocity

groups, is

proposed.

^A

complete explanation

^{of the actual}

physical significance

^of

the observed doublet and of the conditions for the observation is obtained

using

this model. Our method is

compared

^{with a}

recently

introduced dressed-atom

picture including

^the

Doppler

^effect.

In section 4, ^two

typical

^cases

(cascade

three-level systems ^of

neon)

^{are studied. A detailed}

graphical

discussion is achieved to account for the

properties

of the Autler-Townes effect.

Furthermore,

^the

velocity integration

^{of the}

general

solutions of the

equations

of motion obtained in section 2 is

performed

^nume-

rically, taking

^{into account} the Maxwellian

velocity

distribution. A

comparison

is made between the exact solution and the first order

approximation.

In section 5, ^new

experiments performed

^{in three-}

level cascades of neon with two CW

single-mode dye

^{lasers are} described. The new

experimental

results as welf as those of a

previous experiment

of ours

[7]

^are

compared

with the theoretical dis- cussion of the above section.

2. Three-level

system

^{at rest. - In the whole} paper

we will

only

^consider

non-degenerate

three-level systems for theoretical

calculations,

^this

approxi-

mation

being

sufficient to

explain

^{most of our}

experi-

mental results. As a consequence the atom-field interaction can be

represented,

^{in the}

dipole approxi- mation, by

^a

single

Rabi nutation

frequency

^for

each

transition. ^However,

^{it must be}

emphasized

that a

complete

^{treatment with}

degenerate

^levels

would be necessary ^to

perform

^a

precise quantitative comparison

^between

experimental

^and theoretical

curves.

2.1 NOTATIONS AND EQUATIONS OF MOTION FOR THE ATOM-FIELD SYSTEM. ^- The

notations,

^which

are similar to those of reference

[11],

^{can be} established for all three-level systems ^{from the}

particular

^case

of the cascade level scheme illustrated on

figure

^1.

The

following sign

convention is used :

FIG. 1. ^- Energy-level diagram and characteristics of the two

driving ^fields.

Ei (i

⁼

0,

1,

2)

^is the energy of the

non-degenerate

level i. The level 1 is

always

^{the common} level of the .

two transitions of

angular frequencies

col, and _W2l.

Thus we have :

El - Eo

⁼

81 nWlO

^and

If we consider

only

^electric

dipole transitions,

these two transitions are

parity

allowed while the transition 2 ^H 0 is forbidden. For

generality,

^the

corresponding spontaneous

^emission

probabilities

are denoted yij

(i =1= j

⁼

0, 1, 2)

^{for the i}

- j

^transition with the

following

convention : yij ^{= 0 for}

Ei - Ej ^{0 ;}

720 ⁼ Yo2 = 0

(on figure

^{1 :} y 21 ⁼ 72 ; Ylo ⁼ Yi ; 712 = Y01 -

0).

ri (i

⁼

0,

1,

2)

^{is the total} relaxation rate of the

population

^{of the}

^{level i,} including

the effect of _spon-

taneous emission towards other levels and

may-be

some

quenching

^via

collisions ; r ij

^{is the}

corresponding

rate for the

optical

^coherence

_Pij;

^{we have}

Ai (i

⁼ 0, 1,

2)

^{is the} excitation rate for

level i,

the

populations being

created in a

discharge by

^various

mechanisms

(electronic,

^{ionic or atomic}

collisions,

spontaneous

decay

from other

levels, etc...).

The system ^is

coupled

^{with two}

driving

^fields

E1(t)

and

E2(t)

^{which are described in a classical} way :

ê1

^and

ê2

^{are the}

polarization

^{unit vectors of the two}

fields ; £51 = QI -

úJl0 and

b2

⁼

Q2 -

úJ21 ^are the

angular frequency detunings.

Using

^{for the} interaction Hamiltonian the usual

dipole approximation

^and

making

^{the usual resonant}

approximation (rotating

^wave

approximation),

^the

equations

of motion for the

density

matrix elements

can be written :

where :

, D is the

atomic-dipole operator.

The moduli of

h

^and

I2,

^i.e. W1 ⁼

JI1 Ii

^and

W2 ⁼

I212

^{measure the}

^coupling

between the atomic system and the fields. As well

known,

^the

quantities

2 W1 and _{2 W2} are

respectively

^{the Rabi}

nutation

frequencies

^{for the two} transitions.

2.2 STATIONARY SOLUTION IN THE WEAK PROBE- FIELD APPROXIMATION. - We are interested in the

stationary

^{state of the} three-level system ^{for which} the

quantities

poo, Pm P22, (XI0’ a21 and _a2o are

time-independent.

Although,

^{in this case,}

solving

^the system ^{of linear}

equations

^becomes very

simple,

^the

algebraic

expres- sions for the

general

^{case are rather}

complicated.

On the other

hand,

solutions in the weak

probe-

field

approximation give easily

^the

positions

^{and the}

nature of the

interesting

resonances. In our

experi-

ments, ^the observation of the Autler-Townes effect is achieved

by monitoring

^the

probe-field absorption,

and more

precisely

the modification of this

absorp-

tion induced

by

^the

applied

^{fields. But it must be}

emphasized

^{that the}

probe

^field is defined here as

the weak

field

^without any consideration to what the method of detection could be.

The

absorption

coefficient

[12]

^{for the}

probe

^field

is

given by :

where

P2

^is the power

density

^{of the}

probe ^laser, proportional

^to

w2.

In the weak

probe

^field

approximation,

^the

absorp-

tion coefficient

(X2(b1, Ô2)

^becomes

obviously

^inde-

pendent

^{of the}

probe

^beam

intensity.

^{Then the}

quantity

Im

((X21/I2)

^can be written :

where

and where _n10 and _n2j are the

population

differences for the two transitions without the two

fields.

Z . 3 AUTLER-ToWNES RESONANCES : STRONG

PUMP

,FIELD LIMIT. ^- The resonant terms in the expres- sion

(8)

^{are the Re}

[ ] quantities,

^{which are sim-}

plified

^{if we} consider the

strong

pump field

limit,

i.e. wi » Yij,

ri, rij.

Thus the classical Autler-Townes

resonances are

given by

^a

quadratic equation

^relative

tO Ô2 [11] :

or

where ⁹ _{= E1 E2 ;} ⁸ > 0 for a cascade level scheme and 8 0 for a folded

configuration.

A detailed discussion of the Autler-Townes

splitting

for an atom at rest can be found for instance in refe-

rence

[ 11 ].

^For

large detunings ôi

1 > 2 w

1 the

expres- sion

(9)

^{allows us to find}

classically

^the two-step

resonance and the

two-photon

resonance, shifted

by

^the

quantities gW2/ô

^{1 and -}

Ew 1 jb

1

respectively.

Finally

^{it must be}

emphasized

^{that these results}

as well as the

graphical

discussion of the next section

are valid whatever the method of detection could be and

depend only

^{on the}

approximations

^made

(strong pump-field

^{and weak}

probe-field limit).

3.

Doppler-broadened

three-level system. ^{- 3. 1} VELOCITY EFFECTS. ^- We

study

^{now a} collection of such three-level atoms in a cell at a

given

temperature ^T and we assume a thermal

equilibrium

^{due to} collisions.

Thus the number of atoms

having

^the

velocity

^v

along

^a

given

^{axis Oz follows the} Maxwellian

velocity

distribution :

where u is the most

probable velocity

^{modulus :}

1 u ⁼

(2 RT/M)2.

The two fields are

supposed

^{to be pure}

travelling

waves

(wave

^vectors

ki, k2), propagating along

^the

same axis Oz

(kl

^and

k2

^{are the}

algebraic

^{wave num-}

bers)

^{either in the same direction}

(kllk2

>

0)

^{or in}

two

opposite

directions

(kllk2 0).

The

probe

^field

absorption

^{must now be}

integrated

with

respect

^{to the}

longitudinal velocity

^{and the}

absorption

coefficient is therefore

given by :

where

a2(Ôl, b2, v)

^is deduced from

a2(bl, b2) by replacing bl by (bl - k, v)

^and

b2 by (b2 - k2 ^{v) ;}

v is the

projection

^{of the atom}

velocity along

^{the Oz}

axis. ^"

The

integral (10)

^can be calculated in the

large Doppler-width

^limit

using

^residue

techniques,

^and

analytical

calculations have been achieved in the weak

probe-field approximation [10, 16].

^{In the}

general

case, computer numerical calculations are the

only

convenient _way to obtain the

absorption

^{curves of}

the

probe

laser for

typical

^{values of the}

detuning ô,

^{of the} pump laser. Furthermore

they

^{allow the}

finite

Doppler-width

^{to be}

fully

^taken into account, and because of the

particular

^{values of the}

velocity

involved in some cases

(see

^section

4),

^this is very

important.

3. 2 GRAPHICAL DISCUSSION IN THE WEAK PROBE- FIELD APPROXIMATION. ^- When

velocity

^{effects are}

introduced in the resonance condition

(9) (valid

in the weak

probe-field approximation

^{and in the}

strong pump field

limit),

^the

velocity

^{class corres-}

ponding

^{to the} resonances can be determined.

Equa-

tion

(9)

becomes the

following quadratic equation

relative to v :

For

given detunings b1

^and

ô2,

^{two axial}

velocity

classes are found to be on resonance if the discri- minant of

(11)

^is

positive;

^{if it is}

negative

^{no atoms}

are

resonantly

^excited

by

^{the fields. A}

convincing analysis

^{of this}

phenomenon

^was

given by

^Chebo-

taev

[13] in

^{the case}

b 1

^{= 0}

(1).

We propose in the

following

^an alternative answer to the

problem, namely

^a

graphical

illustration which will allow a more

complete understanding

^{of the}

physical

^effects.

We introduce two functions

Zl(kl v)

^and

Z2(kl v)

^{for the} pump and the

probe

^fields

respectively :

The

velocity

classes of the

resonantly

^{excited atoms}

are

given graphically by

^the intersection of the two

corresponding

^curves

^(see Fig. 2).

^For

b

^{1 = 0 the}

hyperbola Zt(k1 v)

^is

asymptotic

^to the X-axis and to the Y ⁼ X

straight ^{line ;}

^{its two branches are} sepa- rated

by

^{2 w 1 on the Y-axis. A}

detuning £51

1 =1- 0

simply produces

^a whole translation

along

^{the X-axis}

of the

hyperbola

^{which is now centred at}

k, v

⁼

ô, (Fig. 2).

Z2(k1 v)

^{is a}

straight

^{line of}

slope S intercepting

the Y-axis at the

point Eb2.

When the laser

frequencies

are varied across the

Doppler-broadened

^atomic

lines the modulus of S is

nearly

^{constant and}

equal

to

úJ 21/ úJ 10.

^The

scanning

^{of the}

probe-laser frequency

(’) ^After this paper had been submitted, ^we learned of the work

by ^R. SALOMAA, Physica Scripta ¹⁵ (1977) ^{251 in which a similar} point ^{of view is} developped.

corresponds

^{.to a movement of} translation of the

straight

^line.

The intersection of the two curves

generally gives

two

points

^as

expected.

However it

clearly

appears that values of

Ô2

exist for which no

velocity

^classes

are

concerned ;

^{i.e. no}

absorption

^of the

probe

^beam

occurs as far as the resonant interaction is the _pre- dominant one. Therefore the Autler-Townes effect

(responsible

^{for the}

^splitting

^{of the level 1 and conse-}

quently

^{for the} existence of the twô branches of

Z 1 (k 1 v))

manifests itself in

Doppler-broadened

^three-

level systems ^{as a}

frequency

gap where a resonant interaction with _any of the atoms of the whole col- lection is not

possible.

^{This is}

quite

^{different from}

the line

splitting

observed for an atom at rest.

FIG. 2. ^- Representation of the functions Zl(k1 v) ^and Z2(k1 v) used for the graphical discussion of the Autler-Townes effect in

Doppler-broadened three-level system.

The existence of the

frequency

gap

implies

^a

straightforward

^{condition on the}

Z2(kt v) straight

line

slope :

^{0 S 1.}

Therefore,

the conditions for the observation of the Autler-Townes effect in

Dop- pler-broadened

three-level systems ^{are :}

i)

w21 WlO,* the

probe-beam frequency

^must

be smaller than the _pump field

frequency ;

ii)

^{for a} cascade level scheme

(e =

⁺

1),

^{the two}

fields must propagate ⁱⁿ

opposite

^directions

(k2l k

¹

0) ;

iii)

^{for a folded}

configuration (e

^{= -}

1),

^{the two}

fields must propagate ^{in the same direction}

(k2lkl > 0).

The

frequency

gap is bounded

by

^{the two} tangents

(of slope S)

^{to the}

hyperbola

and its width

ab2 obviously depends

upon S, i.e. upon the ratio

W21/W10 :

where

As mentioned

before,

^a

detuning b 1

^of the pump field

corresponds

^{to a} translation

ki v

⁼

ô,

^{of the}

hyperbola along

^the X-axis. This

geometrical

^trans-

formation has the

following

consequences ^on the above

physical

discussion :

.

i)

^{the resonance}

frequencies

^for atoms at rest

(kl

^{v =}

0)

^are

given by

^the intersection of the

hyper-

bola with the Y-axis

(Fig. 2).

^This

clearly

^{shows the}

different behaviour of the two resonances corres-

ponding

^to

(9)

^{for non zero}

detuning (see

^also

^Fig.

²

of [11]) ;

ii)

^{on the weak} pump field

limit,

^the

hyperbola

is reduced to its asymptotes ^{and the two} gap sides meet to

give

^a

single

^{resonance at a}

detuning b2 given by :

which is the classical result for the Laser-Induced

Absorption

^Line

Narrowing (A.L.N.)

^method

[8, 13, 17-19] ;

°

iii)

^the width of the

frequency

gap is

obviously independent

^{of the}

detuning 81

^{and the}

frequencies

of _{the gap} sides

(which correspond roughly,

^after

the

velocity integration,

^{to the}

absorption

^resonances

(see [13]

and section 4. 3 of the

present paper))

^are

given by :

iv)

^the gap sides are defined

by

^the tangents

(of slope S)

^{to the}

hyperbola

^{and the}

tangent points give

the two

corresponding velocity

^{classes :}

v+ and v- are

respectively

the axial velocities corres-

ponding

^{to the} gap sides

bi

^and

hï

^defined

by (15).

The relative intensities of the two

absorption

resonances will

clearly depend

upon the

positions of v’

in the

Doppler

distribution. The behaviour of the Autler-Townes effect for various _pump field

detunings

^{can be}

qualitatively

discussed in this _way

(see

^{sections 4 and 5 of the} present

paper) ;

v)

^the

frequency scanning

^{of the}

probe

^{field for}

a fixed _pump field

detuning 1

^{can be}

replaced by

a

frequency scanning

^{of the} pump field for ^a fixed

probe

^field

detuning b 2.

^{On the}

graphical

illustration of

figure 2,

^the translation movement of the

probe straight

^{line is}

simply replaced by

^a translation of the pump field

hyperbola along

the X-axis.

Therefore,

the Autler-Townes effect can also be

observed,

^but with a

frequency

^scale

multiplied by

^{a factor}

C010/C021 [20].

Under the same circumstances it is not

possible

^{to observe the}

splitting

^{in an atomic}

beam

(v

⁼

0).

3.3 ENERGY DIAGRAM WITH DOPPLER-EFFECT IN Té « DRESSED-ATOM » PICTURE. ^- Our

graphical interpretation

^{can be}

compared

^to the well-known dressed atom

picture,

^{which uses a} quantum ^treat-

ment both for the atom and for the field

(see

^for

instance

[21, 22]).

^{Such a}

picture, including

^the

Dop- pler

^{effect has been}

recently

^discussed

by

^{C. Cohen-}

Tannoudji [22].

^{We will}

briefly

introduce this dressed-

atom with

Doppler effect

model and show that our

graphical representation

^is very similar to it.

The _energy

diagram (versus

^the pump laser fre-

quency)

^{is shown on}

figure

^{3 for a} cascade level scheme but the transformation to the other

configu-

rations is

straightforward.

The dressed-atom state

| 0, n > (atomic

^{level 0 + n}

photons)

^{is taken as the}

energy

origin.

^Then

the 11, n > and 2, n >

^states

are

represented by straight

horizontal lines as in the atom energy

diagram. Thé 0, n

⁺

1 >

^{state contains}

one more

photon

^of

frequency Q,

^than

thé | 0, n

state and is therefore

depicted by

^a

straight

^{line of}

slope

^{1 which}

intercepts thé 11, n >

^{state at}

Qi

^{1 =} colo.

The atom-field

coupling implies

^a

perturbation

^of

the two

crossing level 0, n

⁺

1 > and ! |1, n >

^and

then an

anticrossing hyperbola represents

^the per- turbed levels

(the

asymptotes ^are

simply

^{the non-}

perturbed levels).

Let us assume that the atom has a

velocity

^{v in the}

opposite

^{direction to the} pump laser

(circular

^fre-

quency

S21) travelling

direction. In the atom

frame,

FIG. 3. ^- Energy diagram of the dressed-atom taking ^{into account}

the Doppler ^effect [22].

the laser

frequency

^is

land

^the

^probe

^laser

frequencies

^{in resonance are}

represented by

^{IJ and}

IK on

figure

^{3. If we tum back into the}

laboratory frame,

^{it is} necessary ^to

rectify

^{IJ and IK from the}

corresponding Doppler ^effect,

^{i.e. to add}

ro2l .£

(k1/k2

^is

negative

^{in our}

example

^and in any ^case IJ and IK are very close to

(02 1).

Therefore the

frequencies

are I’J’ and

l’K’,

^{where J’ and K’ are the} intersection

points

of the vertical line ro

= Q,

^{with the}

straight

lines of

slope

^{S =}

C021/CO10 coming

^{from J and K.}

The resonant

frequencies

^{for the}

probe

^transition

can be

determined,

^{for each}

velocity class,

^{with this}

geometrical

construction.

The

point

^{of view taken here is} somewhat the

inverse

of that

adopted

^{for our}

graphical approach

of section 3.2 where the

velocity

^{classes on resonance}

are obtained for fixed laser

detunings.

But the two

diagrams

^are

obviously equivalent (compare figure

^{2 and}

figure 3)

^{and our}

graphical

illustration

(section 3.2)

^is

nothing

^{but a dressed-}

atom

picture including

^the

Doppler

^effect.

4. Discussion of

typical

^{cases in neon. - 4. 1}

CHOICE OF THREE-LEVEL SYSTEMS IN NEON. ^- It iS

only possible

^to

study

the Autler-Townes effect with a few three-level systems of neon for

experimental

reasons such as :

i)

^the

necessity

^of

finding

coincidences of

single-

mode _gas lasers and

dye

^{lasers with neon}

transitions ; ii)

the existence of

Doppler backgrounds

^{due to}

collisions in the A.L.N.

spectra ;

^{the folded}

configu-

rations

involving 1 s-2p

transitions

especially

^suffer

from this

disadvantage [23, 24] ;

iii)

^{the A.L.N. resonance}

profiles

^{for weak fields}

are too

complicated

^{in the folded}

configurations;

on the contrary

they

^are

generally

Lorentzian for cascade

levél shemés (in

^{this case one}

^càn easily obtàin,n21 N 0) [9, 18] ;

B

iv)

^a

large transition probability

^is

required

^for

the _pump

transition,

which is the case for 1 s ^-+

1 p

transitions of

neon.

Taking account

^{of the}

^condition

^{0 S 1 for}

occurrence of the Autler-Townes

effect,

^three

typical

cases in the’cascade

schème

have been chosen :

These systems ^are

interesting

^{due to} the fact that the width of the Autler-Townes _gap

(2 pwl)

^is very different. The _p factors

(given by (14))

^are

respecti-

vely :

PA ⁼ 0.999 5, _pB ⁼ 0.235 _{and pc} ⁼

0.379,

corresponding

^to

frequency

^ratio

W21!W10 equal

^to

SA

^{= 0.516,}

SB

^{= 0.986 and}

Sc

^{= 0.963}

respectively.

From the system

(A)

^{to the extreme case} of the system

(B)

the gap width is reduced

by

^{a factor 4.2}

(for

^identical values of

wl).

^For convenience, ^the

following

discussion is restricted to these extreme

cases

(A)

^and

(B).

4. 2 GRAPHICAL DISCUSSION. ^-

Figure

^{4 A illus-}

trates the case

(A) ;

^the

frequency

^{ratio is}

nearly

^0.5

and the Autler-Townes _{gap has} almost the same

width _{2 wi 1} than for an atomic beam

experiment

^at

zero

detuning.

^Moreover the velocities v+ and v- concerned for _{the gap} sides have

nearly

^{the same value}

k 1 v t = b 1 (see ^equation (16)).

^{As a} consequence, the two

components

of the doublet should have the

same

intensity

^{even for non-zero}

detuning (ôi # 0).

The

corresponding

characteristics for the three- level system

(B)

^are

presented

ⁱⁿ

figures

^{4B and 5.}

The Autler-Townes

splitting

^{is reduced to 0.47} W1 due to the

frequency

^ratio very close to 1. The

positions

FIG. 4. ^- Graphical illustration of the three-level systems (A) and (B).

of the tangent

points

^{are shown on}

figure

^{5 for}

81

⁼

0 ;

the concerned velocities vl = _± 8.27

wl/kl

^{can be}

very

large.

^The

Doppler profile

^is also drawn on

figure 5 ;

^{in the}

particular

^case w 1 ⁼ 130 MHz

(experimental

^value of section

5),

^{we have}

to be

compared

^{to the most}

probable

^value

u ⁼

500 m/s.

FIG. 5. ^- Graphical illustration of the evolution of the resonance

intensities with detuning (three-level system (B)).

A

detuning c51 =F

⁰

produces important changes

in the Autler-Townes doublet

(see Fig. 5) :

i)

^{due to the different}

populations

^{for the two}

velocity

^classes

v t,

the doublet becomes

asymmetric

even for small

detunings ôi ;

ii) for c51

> 0 the

intensity

of the blue component of the doublet

(corresponding

^to

c5i

ⁱⁿ

(15)) ^rapidly

decreases with

b 1

and almost

disappears

^after

c51 6 wi ;

iii)

^for

c51

1 > 0, ^the

intensity

^{of the red} component of the doublet increases first with

c51,

^{has a maximum}

intensity for c51

1 ^-- 8 _wl and

slowly

decreases for

higher

^{values of}

c51 ;

iv) for b

i 0, the situations of the blue component and that of the red component ^{have to be}

interchanged.

This _very different behaviour of the two systems

(A)

and

(B)

^is

fully

corroborated

by

^{our numerical}

calculation of the resonance curve to be

presented

now.

4.3 CALCULATED CURVES. ^- The numerical cal- culation of _a2

(equation (10))

^{have been}

^performed

on the UNIVAC 1110 of the Centre

d’Orsay de

l’Université

Paris-Sud, along

^the

following

^{lines :}

i)

^the

population

difference _n21 has been

neglected ;

this

corresponds

^{to a situation where no}

absorption

of the

probe

^{occurs in the} absence of _{the pump} beam.

This

assumption

^is

quite

^{realistic in our}

experimental

conditions

(see

^section

5) ;

ii)

^{the numerical}

integration

^{over the}

velocity

distribution has been achieved

using

^the

Simpson

method with

prediction

^{of the}

integration step [25] ; iii)

the values of

Po, Ti, T 2,

^{Y 1}

and Y2

^{have been}

derived from recent papers

(see

^e.g.

[26-28]);

^the

relaxation rates

ru

^are deduced from the lifetimes :

rjj = ^(ri

⁺

rj)/2.

The resonance curves have been calculated

using

the exact solution of the

equations

^{of motion}

(1-6)

for the

stationnary

atom-field system ^{with two} strong fields

(Rabi frequencies

² W1 and 2

w2)

^{and also}

using

the weak

probe-field approximation (equation (8)).

The

prdtiles

^{so obtained are}

shown

^on

figure

⁶

for the system

(B).

^The

frequency

gap is

clearly

^seen

on the curve

corresponding

^{to the} first-order

approxi- mation ;

^{the same}

splitting

is obtained on the other

curve but an

important

reduction of the contrast

can be observed even for a laser _power ratio

P21Pl -

¹

% (nearly equivalent

^to

(W2/W 1)2

^{= 0.01}

in the case of system

(B)).

Furthermore the

amplitude

of the curve

(absorption coefficient) corresponding

to the exact solution is

considerably

^reduced

(by

^a

factor of

11).

FIG. 6. ^- Calculated resonance curves : comparison between the first-order approximation ^{and the exact solution} (three-level system

(B)). ^For convenience, the vertical scare (corresponding ^{to the} absorption coefficient) has been reduced by ^{a factor of 11 for the}

upper ^trace.

The

comparison

^{of the two curves}

clearly

^shows

that the first order

approximation

^{relative to the}

probe

^{can be}

inadequate

^under many

experimental

circumstances. At

higher probe

intensities the hole at the centre of the resonance curve can

disappear [20}.

Indeed,

^{in our}

experiments performed

^{with two}

dye

lasers

(section 5)

^{we had to} reduce the

probe-beam

power

density

^to less than 1

%

of that of the _pump beam.

All the characteristics of the resonance derived in the

graphical

discussion of the system

(B)

^are

confirmed in the calculated curves

(exact solution) of figure

^{7. For}

instance,

^the

splitting

is about 60 MHz with 2 _W1 ⁼ 260 MHz and the doublet behaves

as

predicted with () 1 .

FIG. 7. ^- Calculated resonance curves : evolution of the shape

and of the intensities with detuning c51 (three-level system (B)).

Our calculations also

give

evidence for the _pro-

gressive substitution, when 5i increases,

of the Autler- Townes intense

component

^{located at}

by

^the

two-photon

^{resonance located at}

h2

^{= -}

hl.

This resonance

corresponds

^to the direct transition 0 -+ 2 with two fields of

slightly

^different

frequencies propagating

ⁱⁿ

opposite

directions and

strongly

enhanced

by

^the presence of the intermediate level 1

[29].

^{The residual}

Doppler ^width,

^{which is the}

.

Doppler

^{width of the 0 --+ 1} transition

multiplied by

^{the factor}

(1 - S),

is very small for the system

(B).

Figure

^{8 shows that such a}

phenomenon

^{also occurs}

for the system

(A).

^{In this case the}

frequency

sepa-

Fixe. 8. ^- Calculated resonance curves : evolution of the shape

and of the intensity (three-level system (A)).