High resolution spectroscopy of the hydrogen atom. II. Study of line profiles

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Study of line profiles

J.C. Garreau, M. Allegrini, L. Julien, F. Biraben

To cite this version:

High

resolution

_spectroscopy

of

the

_hydrogen

atom.

II.

_Study

of line

_profiles

J. C. Garreau

_(*),

M.

_{Allegrini (**),}

L. Julien and F. Biraben

Laboratoire de

_{Spectroscopie}

Hertzienne de

_{l’ENS(***),}

Université Pierre et Marie Curie, 4 _place Jussieu, Tour _12, E01 75252 Paris Cedex _05, France

(Received

2

_February

1990, revised 15 June 1990,

accepted

27 June ₁₉₉₀₎

Résumé. 2014 Cet article est le deuxième d’une série de trois articles décrivant en détail notre récente

mesure de la constante de

_Rydberg.

_{Nous y}

_présentons

le calcul

_théorique

des formes de raies

atten-dues dans notre

_expérience

de

_{spectroscopie à}

deux

_photons

des transitions 2S-nS et 2S-nD de l’atome

d’hydrogène.

Les différents effets

_{d’élargissement}

et de

_déplacement

des raies ont été

_pris

en _compte,

et nous avons calculé la

_probabilité

de transition à deux

_photons

et le

_déplacement

lumineux sous

l’effet de l’interaction des atomes metastables 2S avec le laser. Les contributions au

_signal

de toutes

les

_{trajectoires atomiques possibles}

ont été sommées. La

_comparaison

de

_chaque

_{profil expérimental}

avec le

_profil

_théorique

_{correspondant}

donne la

_position

de la raie avec une

_précision

de

_quelques

10-11.

Abstract. 2014 As second

part of the detailed

_description

of our recent work on the

_Rydberg

constant

measurement, here

_{(paper II)}

we _present our calculations of the line

_profiles

of the

_hydrogen

2S-nS and 2S-nD

_{Doppler-free two-photon signals}

observed in our

_experiment.

Various

_broadening

and shift effects have been considered and detailed calculations of the

_two-photon

transition

_probability

and of the

_light-shift

have been done. The contributions of all

_{possible trajectories}

of the atoms in the metastable beam have been summed to _get the detected

_signal.

The fit of each observed line with the

corresponding

calculated

_profile

_provides

the line

_position

with a relative _accuracy of a few _parts in

1011.

Classification

Physics

Abstracts 35.10 - 06.2OJ - 32.80K 1. Introduction. ’

This _paper is the second of a series of three

_(referred

to as _papers

_I,

II and

_{III) giving}

a

com-plete description

of the method we have used to achieve a determination of the

_Rydberg

constant

with a relative _accuracy of 1.7 x

10-lO

_{[1] ,}

limited

_by

the current

_{optical frequency}

standard.

Such a

_high

_precision

measurement needs a _very careful

_study

of the observed

_signal

_profiles

and

understanding

of

_physical

effects which _may broaden or shift the lines.

Our method is to induce the 2S-nD

_{(n>8) Doppler-free}

_two-photon

transitions in an atomic

beam of

_hydrogen

or deuterium. In _paper 1

_{[2] ,}

we have described the

_experimental

_set-up

used to

_produce

the metastable atomic beam and to

_perform

the

_optical

excitation in order to observe

very narrow

_signals.

In this _paper, we discuss the

_possible

causes for the

_broadening

and shift of

the lines

_(sect.

_2),

we

_present

the detailed calculation of the line

_profiles

_{(sect. 3)}

and

_finally

we

compare the theoretical curves with the

_experimental

ones

_{(sect. 4).}

2.

_Broadening

and shift of the lines.

As detailed in reference

_{[2] ,}

the 2S nD

_Doppler-free

_two-photon

transitions take

_place

in an

optical

excitation chamber where a metastable atomic beam is collinear with the

_standing

wave obtained

_by

_{superposition}

of two

_{counterpropagating}

laser beams. After a

_two-photon

excitation

from the 2S state, about 95% of atoms in the nD states

_undergo

radiative cascades to the 1S

_ground

state. The

_two-photon

transition can then be detected

_by

_observing

the

_{corresponding}

decrease of the metastable beam

_intensity.

A

_{typical recording}

has been shown in

_figure

7 of _paper 1

_[2]

in the

case of the 2S-10D

_two-photon

transition in

_hydrogen.

The observed linewidth is about 1 MHz

(in

terms of the total

_two-photon

transition

_frequency),

to be

_compared

te the natural width of the 10 D level which is 296 kHz. Several effects can

_explain

this

_broadening,

the most

_important

being

light-shift

and saturation. We discuss the various effects in turn.

2.1 LASER LINEWIDTH. - The

_jitter

of the

_dye

laser has been reduced to about 50 kHz

_[2]

and

therefore the laser linewidth is

_responsible

for a line

_broadening

of 100 kHz.

2.2 FINITE TRANSIT TIME. - The transit time of the atoms

_through

the laser beams is _very

_long

because the laser beams and the metastable beam are

_collinear.

In the interaction

_region,

the metastable beam is defined

_by

two

_diaphragms

7 mm in diameter and 956 mm

_apart

as

_represented

in

_figure

1. For an atomic

_trajectory

_making

an

_{angle 8}

with the laser beam the line

_broadening

[3]

is 2v sin

_{fJVÏÏÏ2/1rWO}

_(total

width at the half

_maximum)

where v is the atomic

_{velocity (about}

3 _km/s for

_hydrogen)

and _wo the beam radius

_(about

0.6

_mm).

For the

_largest

_possible

_angle

of atomic

_trajectories

with

_respect

to the laser

_beam,

the line

_broadening

due to the finite transit time is 19 kHz.

Fig.

1. - Schematic view of atom

2.3 SECOND-ORDER DOPPLER EFFECT. - The second-order

Doppler

effect decreases the line

v2

frequency

by

the

_{quantity b}

=

;c2

_vif where _{vif is} the atomic

frequency.

In order to estimate

2C-2

Vi f

precisely

this

_effect,

we have measured the

_velocity

distribution

_{f (v)}

of the metastable atoms in the beam

_{by looking}

at the

_{Doppler profile}

of the Balmer-a line

_{[4] .}

In a metastable atomic

beam the

_velocity

distribution should _vary as

v4

exp( -v2 12(2)

where is related to the beam

temperature u =

(kTIM)112

[5].

We have measured 6 = 1525 f

lOm/s

for

hydrogen

atoms

6 = 983 f

lOm/s

for deuterium atoms.

Since the

_two-photon

transition

_probability

is

_proportional

to the transit time L/v of the atom

in the atomic beam

_(L

is the atomic beam

_length),

we can deduce a first estimate of the mean

second-order

_Doppler

shift b of our lines :

For the 2S-10D

transition,

the mean decrease of the line

_frequency

due to this effect is :

b = 40.8 kHz for

_hydrogen

atoms s _ 17.0 kHz for deuterium atoms.

Due to the

_spread

of the

_velocity

distribution,

there is also a

_broadening

of the same order of

magnitude.

2.4 LIGHT-SHIFT AND SATURATION. In the laser

_standing

wave, the

_light

_{power may} be 50 W in each

_propagation

direction. For an atom in the center of the laser beam the

_light-shift

is about

560

_kHz

_(case

of the 2S-10D

_transition).

The

_light-shift

induces also a

_broadening

due to the

variation of the

_{light intensity}

seen

_by

the atom

_along

its

_trajectory.

For the same

_light

_power there

is another

_broadening

effect due to the saturation of the

_two-photon

transition. For

instance,

for

an atom

_{travelling exactly along}

the laser beam

axis,

the excitation rate is 1.7 x

105

s-1 while the transit time is 3.3 x

10-4

s. For this

_trajectory,

the saturation is

_large,

while it can be small

for

_trajectories

which are

_parallel

to it but offset with

_respect

to the laser beam axis. These two

effects are _power

_{dependent. Figure}

2 shows the variation of the

_{expérimental}

linewidth of the

2Si/2 - 10Ds/2

transition in deuterium versus the

_{light intensity.}

It can be seen that the _power

dependent

effects are

_appreciable,

but cannot

_explain

the total linewidth.

Th account

_precisely

for the combined effect of natural

_lifetime,

_light-shift

and saturation we

have

_performed

calculations of the line

_profiles

_taking

into account all

_possible

_trajectories

of

metastable atoms

_through

the laser beams. Details of this treatment are

_given

below in section 3.

2.5 SlARK EFFECT 2013 The

_stray

electric _{fields may} shift and broaden the line. The shift is due to

the

_quadratic

Stark effect and varies as

n?

_(n

is the

_principal

_quantum

_number).

The

_broadening

is due to the linear Stark effect and varies as

n2.

Great care has been taken in the

_experimental

set-up

to reduce

_stray

electric fields. In order to evaluate their

effects,

we have excited transitions to

_{higher Rydberg}

levels which are more sensitive to the Stark effect.

_Figure

3 shows the

_signal

for the 2S-20D transition. The two fine structure

_components

have the same

_{expérimental}

linewidth

(950

kHz)

while the natural width is

_only

38 kHz. For the

_20D3/2

level

_{(respectively}

_20D5/2

Fig.

Z -

_Example

of the variation of the

_experimental

linewidth as a function of the

_light

_power inside the enhancement

_cavity.

Data are relative to the

_{2Si/2 " lODs/2}

transition in deuterium.

is 306 MHz cm/V

_{(respectively}

217 MHz

_{cm/V) :}

the Stark

_broadening

is about 50%

_larger

for

the

_D3/2

fine structure level than for the

_D.5/2.

In order to obtain an _upper limit to the residual

electric

field,

we assume that the linewidth is

_entirely

due to the

_{light-shift broadening (about}

700

_kHz)

and to the Stark

_broadening.

If we also assume a

_quadratic

sum for these

_broadenings,

we deduce that the Stark

_broadening

is 640 kHz and we obtain an _upper limit of about 2 mV/cm.

Compared

to our

_preliminary

_set-up

_{[6] ,}

where the

_stray

electric field was - 30

_mV/cm,

the ob-served

_improvement

is due to the new vacuum chamber

_[4]

evacuated

_by

two

_cryogenic

_pumps. As an

_example,

the effect of a 2 mV/cm field on the

_{2Si/2 " 1OD5/2}

transition is a

_broadening

of

- 100 kHz and _a shift of ~ 1 kHz.

Fig.

3. - Recorded

signal showing

the fine structure _components of the 2S-20D transition in deuterium.

2.6 OTHER MINOR EFFECTS. 2013 The contributions of other minor effects are not

_easily

isolated.

three

_pairs

of Helmholtz coils and the residual field is about 20 mG. This field induces a Zeeman

splitting

of about 170 kHz for the

_{2Si/2 2013 nD5/2}

transition.

We now tum to the theoretical calculation of the

_light-shift

and saturation

effects,

since

_they

are the most

_important

causes of

_broadening

and shift of the lines.

3.

_{Theoretical lineshapes.}

Atom

_trajectories

in the laser-atom interaction

_region

are sketched in

_figure

1. Since collisions are

negligible

in this

_region,

these

_trajectories

are

_straight

lines

_{passing through}

the two

_diaphragms.

For each

_trajectory,

we have to calculate the survival of metastable atoms

_through

the laser beams. Because of the Gaussian distribution of the

_light

_power, an atom

_{travelling along}

a

_given

trajec-tory

is affected

_by

a

_varying

excitation rate and a

_{varying light-shift.}

We will first calculate the

coefficients of these two

_effects,

and then sum the contributions to the

_signal

of all

_possible

trajec-tories in the metastable beam. In this calculation we take into account the natural life

_time,

the

light-shift

and the saturation effect. We

_neglect

the other effects discussed in section 2.

3.1 TWO-PHOTON EXCITATION RATE AND LIGHT-SHIFT. - Consider two atomic

_{statues i}

> and

1

f

> with

_{respective energies}

Ei

and

_Ef

_{(E f}

>

_E¡).

Due to

_two-photon

_excitation,

the transition

rate between these two states is

_{[7] :}

where n is the number of

_photons

_per unit volume in each

_{counterpropagating}

_{wave, w f= _}

_(E/

-Ei)/n, 1 r >

are all

_possible

intermediate states of _energy

_Er

_(including

the continuum states for

which the sum must be

_replaced

_by

an

_{integral), r f}

the natural width of the excited state

_(we

assume that the natural width of the initial state is _very

_small),

d the

_dipolar

momentum

_operator

and E the

_polarization

vector of the incident

_{light (the}

two

_{counterpropagating}

waves have the

same

_{polarization).}

The excitation rate can be rewritten as:

In

_equation

_(3),

_frequencies

are measured from the null _power resonance

_{w j ; ( 0) ,}

and S2 = 2w

-w fi (0).

The

_light-shift

effect _appears

_explicitly

_{in : wf i =}

_{w f= (0)}

-I-

ci, I,

where I is the _power

density

related to the

_{photon density}

n : I = n hw c

_(c

is the

_velocity

of

_light).

The coefficient yfi

i

is :

where ao is the Bohr

radius,

a the fine structure _constant, and _-y is

_expressed

in atomic units

(fi

= ac = m _

_1).

When the two waves are

_{linearly polarised,}

as in our

_{experiment, -y}

is

From second-order

_{perturbation theory,}

it turns out that the

_light-

shift coefficient cl. is

given

by :

where

and to

(with n

= i

f ).

In

_equations

₍₅₎

and

_(7),

the summation

_over

₁

r > states includes an

_integral

over the continuum states. We have chosen here the r.E _gauge rather than the

_p.A

one to de-scribe the electric

_dipole

interaction.

_Although

the final result must _{be the same, it is known that} convergence is better in the r.E gauge

[8].

We have

performed

a numerical summation over the

intermediate states

_{|r >.}

The formulas involved in the calculation of the

_dipole

matrix elements are listed in the

_appendix.

We

_give

the 2S-nD and 2S-nS transition

_amplitudes

in table la and Ib

respectively.

The contribution due to the _{intermediate 3P state, discrete} states and continuum are also

_given.

For the 2SmD transition

_amplitude,

the

_major

contribution to the sum

₍₅₎

is due to the

sum over the discrete states. In this case the

_{approximation}

of

_only

one intermediate state

_(3P)

fails

_by

about 30%

_(except

for the 2S-3D

_transition).

For the 2S-nS transitions

_(much

weaker than the 2S- nD

_ones),

we find that the continuum

_gives

a dominant contribution and that the

approx-imation of one intermediate state

_completely

fails. Table Ic

_gives

the

_light-shift

coefficients. We

find that the nS and nD shifts are _very similar. All these results are consistent with a calculation

by

D. Delande with the Sturmian function method

_[9].

Table Ib. - 2S-nS

two-photon

transition

_amplitude

y in atomic units : contribution

of

the various

intermediate states.

3.2

_{DESTRUCTION PROBABILITY OF THE METASTABLE ATOMS. From the light-shift coefficient}

and the transition rate

_given

above,

we are able to deduce the contribution to the

_signal

of a

_given

atom

_trajectory.

Ut us consider a

_{straight trajectory (see Fig. 1), starting}

at the

_point

_{(rl , 01)}

on the first

_diaphragm

and

_ending

at the

_point

_{(r2 , 02 )}

on the second

_diaphragm

_{(the cylindrical}

symmetry

of ihe

_problem

allows us to

_put

_01=0).

The laser

_{intensity along}

this

_trajectory

varies

with the

_longitudinal

_component

z of the atomic

_position

as :

where p

is the radial distance to the laser beam axis and w the radius of the laser beam

_(the

laser beam is

_supposed

to be

_perfectly

_aligned

with the two

_diaphragms).

For an atom

_travelling

with

velocity v

(vz

is the

_component

of this

_{velocity along}

the laser beam

_axis),

the

_probability

that a transition has taken

_place

_during

this

_trajectory

is :

In our

_experiment

the

_spatial

distribution of the laser beam

_intensity

is determined

by

the

Fabry-Perot enhancement

_cavity,

made of two mirrors 1.3 m

_apart

and with curvature radü of 4 m. The distance between the end mirror of this

_optical

_cavity

and the first

_{diaphragm defining}

the metastable beam is 134 mm.

lb

_get

_equation

_(9),

we have assumed that the laser

_intensity

seen

_by

the atom

_along

its

Thble Ic. -

Light-shift coefficients

(in

atomic

_units)

_{3i

_for

the 2S level

_(Bi

_depends

on n

_through

the

laser frequency w),

and,8f

for

nS and nD levels.

i)

Hyperfine

structure

_of

the 2S metâstable state. In our

_experiment

we resolve the

_hyperfine

structure of the 2S metastable state and we have studied the most intense

_component,

_namely

the transition

_281/2(F

=

1)

--->

nD.5/2

in

_hydrogen

and the transition

_2S1/2(F

=

3/2) -

nD5/2

in deuterium.

However,

our detection method -

_quenching

of all the metastable atoms down-stream the atomic beam - does not differentiate between the various

_hyperfine

sublevels. As the

metastable atoms are created

_by

electronic bombardment with a

_large

_energy

_spectrum

around

the 1S-2S transition

_{(10.2 eV),}

we make the

_assumption

that the

_hyperfine

sublevels of the

_281/2

are

_equally

excited i.e. with a

_weight

2F + 1

_{(F = 0}

and 1 in

_hydrogen,

F = 1/2 and 3/2 in

_deuterium).

Thus the transition

_probability

_{given by}

_{équation (9)}

for one

_hypertine

sublevel must be

_multiplied

by

a factor

Chas

to

_get

the transition

_probability

related to all metastable _atoms,

_{independently}

of the initial

_hyperfine

sublevel.

_Chs

is the ratio between the

_population

of the

_hyperfine

sublevel in

optical

resonance with the laser and the total metastable

_population:

Chs

=

3/4

in

hydrogen

and

_Chs

=

2/3

in deuterium.

ii)

Fine structure

_{of the}

excited level. The fine structure of the excited level involved in the

tran-sitions observed in our

_experiment

is also resolved and we have

_only

studied the most intense

component

corresponding

to _{J = 5/2.} As the

_two-photon

transition

_probability

is

_proportional

to the

_degeneracy

_{2J + 1 of} the fine structure

_sublevel,

we have to

_multiply

the _{coefficient IJi} in

equation (9)

by

the factor 0.6.

iii)

Repopulation

of

the 2S state. After

_optical

excitation,

the nD states

_undergo

radiative

Thble II. -

71vo-photon

cascade

_{probability from}

nD to 2S level.

Fig.

4. - various

pathways

for

_two-photon

radiative cascade from the laser excited nD level.

have verified that the effect of the

_four-photon

radiative cascades is

_{only -}

5% of the

_preceding

one

_{[10] .}

It is also _necessary to take into account the effect of the 2S state

_{hyperfine pumping :}

after the

two-photon

excitation and the radiative

_cascade,

the atom _may be in the F = 0 _{or F} = 1

hyperfine

sublevel of the 2S state with the

_{probabilities}

po and pi

(respectively

_Pl/2 and _P3/2 for

deuterium).

When the

_two-photon

excitation is made from the F =1 sublevel

_{(F = 3/2}

for

_deuterium)

we have

calculated

_{[10] :}

for

_hydrogen

for deuterium.

To take into account this

_pumping

effect,

we have solved a

_system

of rate

_{equations involving}

the two

_hyperfine

sublevels of the 2S state and the excited state

_[10].

The final result for the

The first factor of

_{equation (10)}

describes the

_optical

_pumping

from the F = ₁

_hyperfine

sublevel

to the F = 0 one. The factor 0.6

_{( 1- pal)}

takes into account the fine structure of the nD state and the

_inefficiency

of the

_two-photon

_process when the atom radiates to the initial F = 1 sublevel.

For deuterium po and pi have to be

_{replaced by}

_P1/2 and _P3/2

_{respectively.}

3.3 STATISTICS OF TRAJEGTORIES AND RESULTS. 2013 To calculate the

_signal

due to all the

metas-table _atoms, we would have to sum

_{equation (10)}

over all velocities and over all

_possible

trajecto-ries. In fact we have not

_performed

the summation over the actual

_velocity

distribution but have made the calculation with a mean

_velocity

_vm:

This

_{approximation}

is valid in the limit of low laser

_intensity

when it is

_possible

to

_expand

the

exponential

term of

_equation

_(10).

We have tested this

_{approximation by making}

the summation over the

_velocity

distribution in a

_particular

case. For a _power of 50

_W,

the difference between the

_lineshape

obtained with the exact

_velocity

distribution and the one calculated with the mean

velocity

is about 3%. We have also evaluated the effect of the second-order

_Doppler

effect. In

this case the term Q -

_cLsI(z)

has to be

_replaced

in

_equation

_{(10) by :}

In the case of the 2S-10D transition we obtain a

_global

second-order

_Doppler

shift of -40.6 kHz for

_hydrogen

and -16.5 kHz for deuterium

_{[10] .}

Thèse values are _{in very}

_good

_agreement

with the result of the

_{approximation}

made in section 2.3

_(-40.8

kHz and -17.0 kHz

_{respectively).}

For

these reasons, we have

_performed

the calculations with the mean

_velocity

_and,

in order to take into account the second-order

_Doppler

_effect,

we have corrected the line

_position

for the

_quantity

given

in

_{equation (1).}

To sum over all

_possible

_{trajectories,}

it is _necessary to know their

_spatial

distribution. The re-coil involved in electronic excitation causes the metastable

_trajectories

to have a

_{large dispersion}

around a mean direction

[4] .

For this reason we assume a uniform distribution for the

_impacts

of all

_{possible trajectories}

on the surface of the

_diaphragm

_ending

the

_optical

interaction

_region.

However,

at the

_beginning

of the metastable atomic

_beam,

the

_spatial

distribution of the

trajec-tories is

_probably

more

_complicated,

because it is determined

_by

the various characteristics of the electronic excitation

_(for

instance the

_spatial

distribution of the

_stray

_quenching

electric field or the

_spatial

distribution of the electronic

_density).

We have then

_approximated

this unknown spa-tial distribution

_by

an uniform one on a virtual

_diaphragm

centered on the real

_diaphragm

at the

beginning

of the atomic beam but with a smaller radius

_R1.

_Finally

the

_lineshape

is

_{given by:}

We have assumed that we have a

_cylindrical

_symmetry

and that the laser beam is well

_aligned

with

_respect

to the atomic beam. The radius

R2

is the radius of the real

_diaphragm

at the end of

the atomic beam

_{(R2 = 3.5 mm).}

As

_explained

below,

the value

Rl

= 1.5 mm was selected

_by

comparison

of the theoretical curves with the

_{expérimental}

ones.

A set of theoretical curves obtained for various

_light

_power is shown in

_figure

5 in the case of the

Fig.

5. - Theorcticat

lineshapes

for the

_{2Sl/2 -}

_IOD,5/2

transition in deuterium.

Fig.

6. - Theoretical line

intensity

_vs.

light

power for the same transition as in

_figure

5.

for

_high

_power. There is also a

_{significant broadening.}

The line

_intensity

varies from 6% at 10 W to 23% at 50 W. There is a

_strong

saturation which is illustrated in

_figure

6. In

_figure

7 we have

plotted

the line

_position

_(center

at the

_{half-maxùnum)}

versus the

_light

_power. The

_apparent

light-shift is

_slightly

_non-linear,

due to the saturation of the

_two-photon

excitation

_probability.

4.

_Study

of the

_{expérimental}

line

_profiles.

4.1 FIT PROCEDURE. - Our _purpose is to _{détermine very}

_precisely

the transition

_frequency

without

_light-shift

with

_respect

to our _very stable reference

_cavity.

With this aim it is _necessary to

determine the

_light

_power seen

_by

the atoms.

Fig.

7.

_{Theoretical light-shift}

vs.

_light

_power

_(same

transition as

_Fig.

5 and

_6).

The dotted line indicates

a linear

_light

shift.

Fig.

8. -

_a)

Fit of the

_experimental

line

_profile

_{(2Sl/2 - 1OD5/2 in}

_hydrogen)

with the calculated one.

_b)

Difference in an

_expanded

scale between the

_experimental

and theoretical curves shown in

_a).

The

_frequency

scale is in terms of the atomic transition

_frequency.

calibration in order to determine the

_light

_power, which is

_directly

connected to the line

_intensity.

The

_{following procedure}

is used. We have included all

_broadening

effects not taken into

ac-count in the theoretical calculation

_(i.e.

those considered in sections

_{2.1, 2.2, 2.3,}

2.5 and

_2.6)

_by

convolution of the theoretical line

_profile

with a Gaussian curve. For each

_{signal recording}

we can fit the theoretical curve

_by

_adjusting

four

_{parameters :}

the metastable

_yield

when the laser is off resonance, the

light

power

P,

the line

_position

without

_{light-shift (with}

_respect

to our _very stable

reference

_cavity)

and the Gaussian

_broadening.

As an

_example,

_figure

8a shows the fit of an

_experimental

_signal

relative to the transition

_2Si/2

profile

and the theoretical one is

_plotted

in

_figure

8b which shows that there is no

_systematic

error

in the fit.

In order to _compare

_precisely

the

_experimental

and theoretical

_profiles,

we have

_averaged

sev-eral

_recordings

obtained with the same _power so that the

_{signal-to-noise}

ratio is increased. The

best value of

_Ri

used in

_{equation (11)}

has been deduced from such a

_comparison.

This is illus-trated

_by

_figures

9a and 9b which show the result for

_Rl

= 3.5 _{mm and}

_R,

= 1.5 mm. In the first

case there is a

_systematic

effect because the

_asymmetry

of the theoretical curve is too

_large.

In the second case

_{(Ri =1.5 mm)}

this

_{discrepancy disappears}

and the relative difference between the two curves is much smaller

_{(signal-to-noise}

ratio -

_100).

The whole set of our

_experimental

results have been

_analyzed

with the value

Ri

= 1.5 mm. We will

_give

another reason for this choice

in 4.3.

Fig.

9. -

Comparison

between the

experimental

curve and the theoretical one, calculated for two R1

val-ues :

_a)

Ri = _3.5

mm;

b)R1

= 1.5 mm

_{(2S1/2 -}

_8D5/2

transition in

_hydrogen).

In order to determine the

_Rydberg

constant we have studied the three transitions

_{2S1 / 2}

-nD5/2

(n

=

8,10,12)

in

_hydrogen

and deuterium. For each

transition,

we have recorded 56

expérimental

line

_profiles

for various laser _powers. Thé fitted values of the

_light

_power P and of the

_light-shift

corrected line

_position

CLP

_give

two series of data which we will now

_analyze.

4.2 LIGHT POWER DETERMINATION. - In our

_{expérimental}

_set-up,

we can monitor the

_light

power inside the enhancement

cavity

with a

_{photodiode placed}

after the end mirror of this

_cavity.

Figure

10 shows the fitted _power

_(P)

versus the

_photodiode

_signal

_(IT)

in the case of the

_{2S1 2}

-lODS/2

transition. There is a

_good

_agreement

between these two sets of data. Nevertheless these two

_quantities

_(P

and

_IT)

are not

_exactly

in a linear ratio as can be deduced from the best fitted

straight

line which does not _pass

_through

the

_origin.

This

_stray

effect is

_systematic

and shows the limits of our fit

_procedure.

Table III

_gives

the

_slope

of this

_straight

line for the six transitions

studied. For a

_given

excited level

_(n

=

8,10,12),

_{there is a}

very

good

agreement

between the values

obtained for

_hydrogen

and deuterium

_although

the line intensities are different in these two cases

(for

deuterium the line

_intensity

is

_larger

than for

_hydrogen,

because of the smaller

_velocity

of deuterium

_atoms).

In table

_III,

the

_comparison

between different excited levels is not

_significant,

Fig.

10. - Fitted

_{power P venus}

the

_{photodiode signal}

IL Data are relative

_{to 48 recordings of}

the

_2S,/2

-lODs/2

transition in deuterium.

Table III. Ratio between

_{the fitted power}

P and

_{the photodioâe signal IT (arbitrary}

_units)

4.3 LINE POSITION. The

_light-shift

corrected line

_{position (CLP)}

_{given by}

each fit can be

compared

to the half-maximum center

_(HMC)

of the line. For each

_transition,

CLP has been

ex-trapolated

to null

_light

_power as shown in

_figure

11. In this

_figure,

the

_straight

line is

_{least-squares}

fitted : it is

_horizontal,

_showing

that the

_light-shift

is well corrected. The observed half maximum

center of the line is also

_plotted

versus the

_light

_{power :} a dotted line has been

_superimposed

_given

the

_{light-shifted}

line

_position

calculated from the fitted values of P and CLR In table

IV,

the mean

light-shift

per unit power is

compared

with the

_slope

of the

_straight

line

_fitting

_CLP,

for the six transitions studied. The

_light-shift

correction is better than 91%

except

for the 2S-12D transition in deuterium. In this _{case, the}

_disagreement

is

_probably

due to the _poor

_{signal-to-noise}

ratio. The

atomic

_frequency

is deduced from the CLP

_{extrapolation.}

Figure

12 shows the same

_{extrapolation}

as in

_figure

11 when the theoretical curves are calcu-lated with

Ai =

2 mm

_(cf.

_Eq.

_(11)).

The calculated

_light-shift

is too

_large

and we see that the

slope

of the CLP/P

straight

line is _very sensitive to the

R,

radius value. For

_R,

= 1.5 mm this

_slope

is _very close to zero. Th take into account the

_nonlinearity

between the fitted

_{power P}

and the

pho-todiode

_signal

_IT,

we have also made these

_{extrapolations}

versus the

_photodiode

_signal

IT Table

V

_gives

the

_{extrapolation}

results in these different cases. The first column

_gives

the _CLP/P

Fig.

11. -

Comparison

of the

_{light-shifted}

line

_position

for various

_light

_powers

_{(stars) with}

the

correspond-ing

light-shift

corrected line

_{position given by}

the fit

_(dots)

in the case of the

_{2Sl/2 - lODs/2}

transition in

hydrogen.

The

_straight

line is

_{least-squares}

fitted.

table IV -

_{Comparison of the}

mean

_{light-shift per unit power}

MLS/P with the

_{slope of the}

_light-shift

corrected

_{line position}

CLP versus the

_{light power.}

extrapolations.

There is a

_systematic

effect : the mean difference is -11 kHz. For the

_Rydberg

constant

_{determination,}

we have used the mean of the CLP/P and CLP/IT

_{extrapolations}

and

as-sumed an error bar

_including

the two

_values,

Le. an

_uncertainty

of db 8 kHz. The third column of table V

_gives

the difference between the CLP/P

_{extrapolation}

calculated with

_R1

= 2 mm and with

_{R1 =}

1.5 mm : there is a

_systematic

_difference,

5 kHz on _average, which shows the

depen-dence on

_R1

of the

_{extrapolation.}

We estimate _{in this way} that the

_uncertainty

due to the choice

of the value 1.5 mm for

_R1

is 5 kHz. Our estimate of the total

_uncertainty

due to the

extrapola-tion

_procedure

_(henceforth

called

_"uncertainty

of the theoretical line

_shape")

is the sum of these

two

_{contributions,}

i.e. 13 kHz. The

_{corresponding}

relative

_uncertainty

is 1.7 x

10-11,

rounded to

2 x 10-11

for the

_Rydberg

constant determination. This

_uncertainty

is différent from the statistical

Fig.

12. - Same

comparison

as that of

_figure

_11, but the theoretical curve is calculated with RI = 2 mm.

Table V. - Results

_of

the corrected line

_position

_(CLP)

_for

the six studied transitions. Column 1 lists the

_{uncertainty of}

CLP vs.

_light

_power

_{(P) extrapolation for}

_Ri=

1.5 mm. Column 2

_gives

the

differences

between the CLP vs.

_{photodiode signal}

_(IT)

_{extrapolation}

and the CLP vs. P

_{extrapolation.}

Column 3

_gives

the

_difference

obtained when CLP vs. P

_{extrapolation}

is calculated

_for

_{R1 =}

2 mm

and

_Ri=

1.5 mm.

5. Conclusion.

In

conclusion,

we have

_{carefully analyzed}

the

_lineshape

of the

_two-photon

transitions. We have

performed

theoretical calculations of the

_two-photon

transition

_probability

and of the

_light-shift.

The theoretical curves have been

_compared

to the

_experimental

ones and we have

_precisely

de-scribed the

_procedure

used to take into account the

_light-shift.

The final result is the

Appendix

Calculation of the

_dipole

matrix elements.

We

_present

here the formulae that are used to calculate the matrix elements

_{R( n,}

_{1; n, 1}

+

₁₎

=

n’, 11 r

1 n, l

>

_were

_n,

1>

_and

_{n ’,}

l + 1 > are _{discrete states,} and their

_{generalization}

to

discrete-continuum

_coupling

_{[11] :}

where :)

_{(Porchamer’s symbol)}

(hypergeometric fonction)

(if

one of the two levels is a discrete one, the

_{hypergeometric}

functions are

_{polynomials).}

(these

expressions

of zi and x2 are useful to calculate

_{R(n, l;}

_{n’, 1}

+

₁₎

for

_highly

excited

_levels).

Formula

_{(A.1) applies}

in the case

_{n +}

n’. For n =

n’,

we have:

(which

is often written with a _wrong

_sign,

that is with a

_phase

definition

_incompatible

with that of

(A.1)).

Let us now

_generalize

this formula to the case where one of the levels lies in the continuum. It

can be shown

_[11]

that

an

_analytical

continuation of

_equation

_(A.1)

can be made into the

contin-uum

_by

_replacing

n

_{by in}

where

(Ry

is the

_Rydberg

constant in _energy units and E the _energy of the continuum

_state).

This

_implies

a normalization for the continuum states such that the continuum wave functions are related to

the discrete ones

_by:

where k =

ao- 1 (E/Ry) 1/2

_(ao

is the Bohr

_radius).

This is in fact

_equivalent

to

_saying

that the bound-continuum matrix elements differ from

bound-bound _{ones, after}

_{replacing n by}

_i7J,

_by

a factor _exp

_{(-ir/4)[1 -}

_exp

_{(-2rq)] -1/2 ;}

the

which is an

_analytical

continuation of the discrete states relation:

with.

Acknowledgements.

The authors are indebted to Professor B.

_Cagnac

for _many

_stimulating

discussions

_during

this

ex-periment

and to Doctor D. Delande for much fruitful advice on the calculation of

_light-shift

and

two-photon

transition

_{probabilities.}

This work is

_{partially supported by}

the Bureau National de

Métrologie.

One of us

_(JCG)

is

_grateful

to the Brazilian Financial

_Agency

CAPES for a

scholar-ship.

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