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Rabi oscillations and adiabatic rapid passage measured in the 2s-3p transition of atomic hydrogen
Volume 64, number 1
OPTICS COMMUNICATIONS
1 October 1987
RABI OSCILLATIONS AND ADIABATIC RAPID PASSAGE MEASURED I N T H E 2s-3p T R A N S I T I O N OF A T O M I C H Y D R O G E N V. LORENT, W. CLAEYS, A. CORNET and X. URBAIN Dkpartment de Physique. Universit~Catolique de Louvain, Chemin du Cyclotron 2, B 1348 Louvain-la-Neuve, Belgium Received 8 April 1987
The transitions to the 3p~n and 3p3/2 levels of H have been observed by transverse cw laser excitation of the metastable component of an hydrogen atomic beam. The velocityof the atomic beam ( v~ 4 X l 04 cm/s ) was chosen to avoid spontaneous decay during the interaction time. The energy separation between the 1/2 and 3/4 levels is larger than the Doppler and transit time broadening and allows measurements of the two excitation processes separately, according to their transitions dipole moments. We report here measurementsof Rabi oscillationsand adiabatic followingcreated by spatial modulation of the laser field.
Rabi oscillations and adiabatic rapid passage have been observed in the infrared by Avrillier et al. [ 1 ] and Adam et al. [2] using bolometric detection in a transversaly crossed laser and supersonic molecular beam experiment. These processes have first been explored in the visible range by Kroon et al. [ 3 ] in neon (three levels system). Our experiment deals with atomic hydrogen. A high velocity ( v ~ 4 × 10 7 cm/s ) metastable 2s hydrogen beam is produced by neutralization of monoenergetic protons passing through a caesium vapour cell. This charge exchange is quasi resonant with the n-- 2 hydrogen level and gives about 20% of the neutral beam in the metastable 2s state [4]. The atomic beam is crossed at right angle by a cw ring dye laser working in the 656 nm visible range (fig. l ). By sweeping the laser frequency, we observe the transitions to the 3p~/2 and 3P3/2 levels separated by 3.25 GHz. The transition fwhm, for these beam velocities and for a 0.04 cm laser waist, is 0.8 GHz. This gives the possibility, in the case of a linearly polarised laser field, to excite each level with a well defined and unique dipole moment. Indeed, by expanding the in, j, m j ) in the In, l, m ) uncoupled spin-orbit basis, one clearly sees that only one Clebsch-Gordan coefficient is involved in each transition: I <2Sl/2. 1/2 Id'/~l 3p,/2. ,/2>1
=,,~1
(2sli'/*l 3 p > l ,
and 12s,/2 )/2 l i'/*l 3p3/2. i/2 ) [
= 2v/~l <2sl i.~l 3p> I. (The same coefficients apply for the mj= - 1/2,--,mj= - 1/2 transitions). The excitation efficiency is determined by measuring the metastable population as a function of the laser frequency (fig. 2). This measure is obtained by detecting a fixed fraction of the Ly,~ emission coming from 2s atoms electrically quenched 30 cm beyond the laser-atom interaction point. The travel time of the atomic beam over this distance corresponds to hyndred times the spontaneous decay of the 3p level ( 5 . 4 x 1 0 -9 s) and ensures counting arising solely from the 2s Stark induced Ly~ emission. The Bloch equation formulation may be used to describe the laser-atom interaction in an isolated two atomic level system. It requires, if applied to a real system, a low decay rate of the excited states during the transit time of the atoms through the laser field. In our experiment, the 3p lifetime of 5.4× 10 - 9 S is not very large c o m p a r e d to the 1.8 X 10 -9 s transit time corresponding to twice the 0.04 cm laser waist and 4.36 X 107 cm/s velodty of the atomic beam. We have to take this into account in the formulation. We
0 030-4018/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
41
Volume 64, number I
OPTICS C O M M U N I C A T I O N S
E-H I
=
1 O c t o b e r 1987
F =--c
2Wo
Fig. 1. Schematic view of the aparatus. Protons extracted from a Colutron source (1) are accelerated (2) up to 1 keV and mass analysed in a magnetic sector (3). The beam is partially neutralized in a caesium cell (4). (H + + C s ~ H ( 2 s ) + Cs +, AE= - 0 . 4 7 eV). Ions are removed from the neutral beam (5). Two diaphragms (6) 0.05 cm diameter limit the atomic beam divergence to 10 - 3 rad. Bias voltage on the plates (7) quench the 2s atoms in front of the detector. A tubular electron multiplier (8), shielded by a LiF window (9), counts the Ly, photons. The cw ring dye laser (A) is a 699 Coherent laser pumped by the 514 nm line of a Coherent argon ion laser (B). DCM is used as dye. Maximum stabilized single frequency laser power is 600 mW in the 656 nm range. Laser intensity is changed by rotating a half-wave plate (C) in front o f a Glan-prism (D). The waist and divergence of the laser beam are varied by moving a lens (E) of focal length 50 cm and by changing the distance from the lens to the interaction point with a set of two mirrors (F). A grating monochromator of 70 GHz accuracy ( G ) and a H2Ne discharge lamp (H) are used for tuning the laser frequency close to the resonances.
(a) Z=O 4--
..
,
(b) Z=2.75D
3Pvz
.....
:
,.:"
.Z5
:.
t-
3P312 =:*':::":'""
"E
":. .. ¢o
:'.
-:.: O
"
..
'...
..
5"
o L.J
":~.".
~t !
I
3.25 fihz E
3.25Ghz Laser frequency
)
':4
Laser frequency
>
Fig. 2. The depopulation of the mctastable level versus laser frequency. The two peaks are the transitions to the 3pz/z and 3P3/2 levels. (a) Situatuation of Rabi oscillation• One clearly sees that the depth of the 3p~/2 peak is much more pronounced than the 3p3/2 peak though a factor of x/~ between the dipolar moments is on behalf of the 3p3/2 transition. This clearly ensures that more than a ~ pulse has been accomplished for the transitions 2s~/2-3p3/~. The fwhm, mainly duc to transit time broadening, corresponds to 0.36 mm waist. (b) Signal obtained by adiabatic rapid passage. The 0.19 mm waist is offset from the atomic beam by a distance 2.75 times the confocal parameter b. The 3P3/2 peak will be always deeper than the 3pc/2 peak with increasing laser power. The spectral broadening due to the angular divergence of the laser is equivalent to the transit time broadening with a 0.19 mm waist. 42
Volume 64, number 1
OPTICS COMMUNICATIONS
assume no relaxation o f the 3p level to the 2s metastable level (a small fraction (11.8%) o f the 3p decay refeeds the 2s state, while most o f the decay (88.2%) leads to the ground state) a n d assume no a t o m - a t o m interaction. The generalized Bloch equation for the decaying system can then be written,
1 October 1987
where Rl = l u * + u l * ,
R2 = i ( l u * - u l * )
11"- u u * ,
R 4 = 11"+ u u * ,
R 3=
,
OR3/at=
-
(712 )( R3 - R 4 ) -121Rz ,
(1 stands for 2sl/2. 1/2 state, u stands for 3pl/2, 1/2 o r 3p3/2, 1/2 state), a n d y is the 1.86 × l 0 s s-1 transition rate for spontaneous decay o f the 3p state. t21 (t) = 2£2,1 U(vxt) with g2,1 the R a b i frequency a n d U(vxt) the laser b e a m profile along the a t o m i c b e a m path.
OR4/at=
-
(712)(R4 - R 3 ) ,
t23 (t) = a o ( o / a t ,
OR~/Ot= - (?/2)R~ + I 2 3 R 2 , OR2/Ot= - (X/2 ) R 2 - Q3RI -FQI R 2 ,
(a)
R~ / ,-"
/ /'
'X, \
!Ji
/ll ........
(9(0 = (o9 - o g . , ) t + O , (vxt) o 9 - ogul is the detuning o f the laser frequency. 01 (Vxt) is the relative phase between the a t o m i c coherence a n d the light as seen b y the atom. This accounts for the fact that the a t o m sees a wave front
"
I/j/
........
'\
i- . . . . . . .
', N \ \ 1R / ~ i i /
I R2
,/
.// /
iS i
.4--
o.
r'l
g_
Z I
-5 (b)
!~_~
/
/
/
/
/ I ~\
',
\
Ii/\ .......
5
"-.. \\
llI.~
/
/
/ I ,k / I i\
0 time ( 10-9S)
-v-i-¢¢'--¥-_~
co "re~ Q O_
",\ , ",
ii/ IIi
I /
I //
¢,q
I
-S
I
0
I
S
time (10-9 s) Fig. 3. Time evolution of the Bloch vector in our experiment conditions. (a) z=0.6b and the Rabi frequency in units of reduced transit time ~2u~Wo/V,= 1.45; (b) z= 2.75b and g2.~Wo/Vx=2.3. Vertical axis R3 is the difference of the populations uu ° - 11"and horizontal axis R~, R2 are the coherences lu* + ul*, i (lu*-ul*).
Fig. 4. Calculations of the time evolution of the 2s~/2 and 3p3/2 populations are shown, z/b=2.75 and the Rabi frequency in units of reduced transit time fJ.two/v~= 1.15. The calculation including 3p decay is represented by solid lines. Dotted line is without relaxation. 43
Volume 64, number 1
OPTICS COMMUNICATIONS
(a) 1-
g g_
0
1
2
~12
ILaser powerl (arbitrary units)
(b) 1-
o
c2 °o 0
1
2
(Laser powerV2larbitrary units)
Fig. 5. (a) Rabi oscillation of the metastab]e population. ~ transition to 3P3/2, ["] transition to 3p,2. Among (550 mW) ~/2 is
needed for 2sl~2~3p3/_, 2n pulse. Experimental data are adjusted to theoretical calculation at one point. (b) Transitions to the 3P3~z level; ~ adiabatic rapid passage situation, z = 2.75b; [] intermediate situation, z=0.6b. All solid curves represent calculations including relaxation.
curvature when the laser b e a m waist is placed at a distance z from the atomic beam. At the resonance with the transition and for a gaussian TEMo~ mode, one o b t a i n s 12, (t) = 212u~( W o / w ) e x p ( - v~t2 / w 2) , ~,,3( t ) = - ( v2 / w Z ) ( 4 z / b )t ,
where v~ is the velocity o f the atom, b = 2zt ~o/2 is the conformal parameter, and w2(z) = wE (1 + 4 z 2 / b 2 ) .
44
1 October 1987
A general description o f the coherent l a s e r - a t o m interaction can be found in ref. [ 5 ]. Fig. 3 shows the evolution o f the Bloch vector ( n o relaxation). The solution o f the generalized Bloch equation c o m p a r e d to the one without relaxation shows low discrepancy in the calculation o f the metastable level p o p u l a t i o n as it can be seen in fig. 4. O u r experimental results for R a b i oscillations a n d a d i a b a t i c r a p i d passage are shown in fig. 5. We find a relatively good agreement between theory a n d e x p e r i m e n t except for the high laser fields where the Rabi oscillations tends to disa p p e a r owing to the transverse (to the a t o m i c b e a m axis) field distribution. This signal averaging could be reduced by narrowing the gaussian field distribution in the direction parallel to the atomic b e a m by means o f cylindrical focusing [3]. The small discrepancy between the m e a s u r e d a n d calculated adiabatic r a p i d passage m a y be a t t r i b u t e d to the little inaccuracies in the measurement o f the waist size and position. We can conclude that the coherent excitation o f the 3p states o f hydrogen is well observed an understood. The possibility o f observing R a m s e y fringes in the visible range [ 6 ] on transitions between H levels can therefore be also anticipated. References
[ 1] S. Avrillier, J.M. Raimond, Ch.J. Bord~, D. Bassi and G. Scoles, Optics Comm. 39 (1981) 311. Ch.J. Bord6, S. Avrillier,A. van Lerberghe, Ch. Salomon, Ch. BrYant, D. Bassi and G. Scoles, Appl. Phys. B 28 number 2/3, (1982) [2] A.G. Adam, T.E. Gough, N.R. Isenor and G. Stoles, Phys. Rev. A32 (1985) 1451. [3] J.P.C. Kroon, H.A.J. Senhorst, H.C.W. Beijerinck, B.J. Verhaar and N.F. Verster, Phys. Rev. A 31 (1985) 3724. [4] F. Brouillard, W. Claeys and G. Van Wassenhove, J. Phys. B: At. Mol. Phys. 10 (1977) 687. [5] Ch.J. Bord6, Rev. Cethedec 20 (NS83-1) 1. [6] Ch.J. Bord6, Ch. Salomon, S. Avrillier, A. Van Lerberghe, Ch. Br6ant, D. Bassi and G. Scoles, Phys. Rev. A 30 (1984) 1836.
OPTICS COMMUNICATIONS
1 October 1987
RABI OSCILLATIONS AND ADIABATIC RAPID PASSAGE MEASURED I N T H E 2s-3p T R A N S I T I O N OF A T O M I C H Y D R O G E N V. LORENT, W. CLAEYS, A. CORNET and X. URBAIN Dkpartment de Physique. Universit~Catolique de Louvain, Chemin du Cyclotron 2, B 1348 Louvain-la-Neuve, Belgium Received 8 April 1987
The transitions to the 3p~n and 3p3/2 levels of H have been observed by transverse cw laser excitation of the metastable component of an hydrogen atomic beam. The velocityof the atomic beam ( v~ 4 X l 04 cm/s ) was chosen to avoid spontaneous decay during the interaction time. The energy separation between the 1/2 and 3/4 levels is larger than the Doppler and transit time broadening and allows measurements of the two excitation processes separately, according to their transitions dipole moments. We report here measurementsof Rabi oscillationsand adiabatic followingcreated by spatial modulation of the laser field.
Rabi oscillations and adiabatic rapid passage have been observed in the infrared by Avrillier et al. [ 1 ] and Adam et al. [2] using bolometric detection in a transversaly crossed laser and supersonic molecular beam experiment. These processes have first been explored in the visible range by Kroon et al. [ 3 ] in neon (three levels system). Our experiment deals with atomic hydrogen. A high velocity ( v ~ 4 × 10 7 cm/s ) metastable 2s hydrogen beam is produced by neutralization of monoenergetic protons passing through a caesium vapour cell. This charge exchange is quasi resonant with the n-- 2 hydrogen level and gives about 20% of the neutral beam in the metastable 2s state [4]. The atomic beam is crossed at right angle by a cw ring dye laser working in the 656 nm visible range (fig. l ). By sweeping the laser frequency, we observe the transitions to the 3p~/2 and 3P3/2 levels separated by 3.25 GHz. The transition fwhm, for these beam velocities and for a 0.04 cm laser waist, is 0.8 GHz. This gives the possibility, in the case of a linearly polarised laser field, to excite each level with a well defined and unique dipole moment. Indeed, by expanding the in, j, m j ) in the In, l, m ) uncoupled spin-orbit basis, one clearly sees that only one Clebsch-Gordan coefficient is involved in each transition: I <2Sl/2. 1/2 Id'/~l 3p,/2. ,/2>1
=,,~1
(2sli'/*l 3 p > l ,
and 12s,/2 )/2 l i'/*l 3p3/2. i/2 ) [
= 2v/~l <2sl i.~l 3p> I. (The same coefficients apply for the mj= - 1/2,--,mj= - 1/2 transitions). The excitation efficiency is determined by measuring the metastable population as a function of the laser frequency (fig. 2). This measure is obtained by detecting a fixed fraction of the Ly,~ emission coming from 2s atoms electrically quenched 30 cm beyond the laser-atom interaction point. The travel time of the atomic beam over this distance corresponds to hyndred times the spontaneous decay of the 3p level ( 5 . 4 x 1 0 -9 s) and ensures counting arising solely from the 2s Stark induced Ly~ emission. The Bloch equation formulation may be used to describe the laser-atom interaction in an isolated two atomic level system. It requires, if applied to a real system, a low decay rate of the excited states during the transit time of the atoms through the laser field. In our experiment, the 3p lifetime of 5.4× 10 - 9 S is not very large c o m p a r e d to the 1.8 X 10 -9 s transit time corresponding to twice the 0.04 cm laser waist and 4.36 X 107 cm/s velodty of the atomic beam. We have to take this into account in the formulation. We
0 030-4018/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
41
Volume 64, number I
OPTICS C O M M U N I C A T I O N S
E-H I
=
1 O c t o b e r 1987
F =--c
2Wo
Fig. 1. Schematic view of the aparatus. Protons extracted from a Colutron source (1) are accelerated (2) up to 1 keV and mass analysed in a magnetic sector (3). The beam is partially neutralized in a caesium cell (4). (H + + C s ~ H ( 2 s ) + Cs +, AE= - 0 . 4 7 eV). Ions are removed from the neutral beam (5). Two diaphragms (6) 0.05 cm diameter limit the atomic beam divergence to 10 - 3 rad. Bias voltage on the plates (7) quench the 2s atoms in front of the detector. A tubular electron multiplier (8), shielded by a LiF window (9), counts the Ly, photons. The cw ring dye laser (A) is a 699 Coherent laser pumped by the 514 nm line of a Coherent argon ion laser (B). DCM is used as dye. Maximum stabilized single frequency laser power is 600 mW in the 656 nm range. Laser intensity is changed by rotating a half-wave plate (C) in front o f a Glan-prism (D). The waist and divergence of the laser beam are varied by moving a lens (E) of focal length 50 cm and by changing the distance from the lens to the interaction point with a set of two mirrors (F). A grating monochromator of 70 GHz accuracy ( G ) and a H2Ne discharge lamp (H) are used for tuning the laser frequency close to the resonances.
(a) Z=O 4--
..
,
(b) Z=2.75D
3Pvz
.....
:
,.:"
.Z5
:.
t-
3P312 =:*':::":'""
"E
":. .. ¢o
:'.
-:.: O
"
..
'...
..
5"
o L.J
":~.".
~t !
I
3.25 fihz E
3.25Ghz Laser frequency
)
':4
Laser frequency
>
Fig. 2. The depopulation of the mctastable level versus laser frequency. The two peaks are the transitions to the 3pz/z and 3P3/2 levels. (a) Situatuation of Rabi oscillation• One clearly sees that the depth of the 3p~/2 peak is much more pronounced than the 3p3/2 peak though a factor of x/~ between the dipolar moments is on behalf of the 3p3/2 transition. This clearly ensures that more than a ~ pulse has been accomplished for the transitions 2s~/2-3p3/~. The fwhm, mainly duc to transit time broadening, corresponds to 0.36 mm waist. (b) Signal obtained by adiabatic rapid passage. The 0.19 mm waist is offset from the atomic beam by a distance 2.75 times the confocal parameter b. The 3P3/2 peak will be always deeper than the 3pc/2 peak with increasing laser power. The spectral broadening due to the angular divergence of the laser is equivalent to the transit time broadening with a 0.19 mm waist. 42
Volume 64, number 1
OPTICS COMMUNICATIONS
assume no relaxation o f the 3p level to the 2s metastable level (a small fraction (11.8%) o f the 3p decay refeeds the 2s state, while most o f the decay (88.2%) leads to the ground state) a n d assume no a t o m - a t o m interaction. The generalized Bloch equation for the decaying system can then be written,
1 October 1987
where Rl = l u * + u l * ,
R2 = i ( l u * - u l * )
11"- u u * ,
R 4 = 11"+ u u * ,
R 3=
,
OR3/at=
-
(712 )( R3 - R 4 ) -121Rz ,
(1 stands for 2sl/2. 1/2 state, u stands for 3pl/2, 1/2 o r 3p3/2, 1/2 state), a n d y is the 1.86 × l 0 s s-1 transition rate for spontaneous decay o f the 3p state. t21 (t) = 2£2,1 U(vxt) with g2,1 the R a b i frequency a n d U(vxt) the laser b e a m profile along the a t o m i c b e a m path.
OR4/at=
-
(712)(R4 - R 3 ) ,
t23 (t) = a o ( o / a t ,
OR~/Ot= - (?/2)R~ + I 2 3 R 2 , OR2/Ot= - (X/2 ) R 2 - Q3RI -FQI R 2 ,
(a)
R~ / ,-"
/ /'
'X, \
!Ji
/ll ........
(9(0 = (o9 - o g . , ) t + O , (vxt) o 9 - ogul is the detuning o f the laser frequency. 01 (Vxt) is the relative phase between the a t o m i c coherence a n d the light as seen b y the atom. This accounts for the fact that the a t o m sees a wave front
"
I/j/
........
'\
i- . . . . . . .
', N \ \ 1R / ~ i i /
I R2
,/
.// /
iS i
.4--
o.
r'l
g_
Z I
-5 (b)
!~_~
/
/
/
/
/ I ~\
',
\
Ii/\ .......
5
"-.. \\
llI.~
/
/
/ I ,k / I i\
0 time ( 10-9S)
-v-i-¢¢'--¥-_~
co "re~ Q O_
",\ , ",
ii/ IIi
I /
I //
¢,q
I
-S
I
0
I
S
time (10-9 s) Fig. 3. Time evolution of the Bloch vector in our experiment conditions. (a) z=0.6b and the Rabi frequency in units of reduced transit time ~2u~Wo/V,= 1.45; (b) z= 2.75b and g2.~Wo/Vx=2.3. Vertical axis R3 is the difference of the populations uu ° - 11"and horizontal axis R~, R2 are the coherences lu* + ul*, i (lu*-ul*).
Fig. 4. Calculations of the time evolution of the 2s~/2 and 3p3/2 populations are shown, z/b=2.75 and the Rabi frequency in units of reduced transit time fJ.two/v~= 1.15. The calculation including 3p decay is represented by solid lines. Dotted line is without relaxation. 43
Volume 64, number 1
OPTICS COMMUNICATIONS
(a) 1-
g g_
0
1
2
~12
ILaser powerl (arbitrary units)
(b) 1-
o
c2 °o 0
1
2
(Laser powerV2larbitrary units)
Fig. 5. (a) Rabi oscillation of the metastab]e population. ~ transition to 3P3/2, ["] transition to 3p,2. Among (550 mW) ~/2 is
needed for 2sl~2~3p3/_, 2n pulse. Experimental data are adjusted to theoretical calculation at one point. (b) Transitions to the 3P3~z level; ~ adiabatic rapid passage situation, z = 2.75b; [] intermediate situation, z=0.6b. All solid curves represent calculations including relaxation.
curvature when the laser b e a m waist is placed at a distance z from the atomic beam. At the resonance with the transition and for a gaussian TEMo~ mode, one o b t a i n s 12, (t) = 212u~( W o / w ) e x p ( - v~t2 / w 2) , ~,,3( t ) = - ( v2 / w Z ) ( 4 z / b )t ,
where v~ is the velocity o f the atom, b = 2zt ~o/2 is the conformal parameter, and w2(z) = wE (1 + 4 z 2 / b 2 ) .
44
1 October 1987
A general description o f the coherent l a s e r - a t o m interaction can be found in ref. [ 5 ]. Fig. 3 shows the evolution o f the Bloch vector ( n o relaxation). The solution o f the generalized Bloch equation c o m p a r e d to the one without relaxation shows low discrepancy in the calculation o f the metastable level p o p u l a t i o n as it can be seen in fig. 4. O u r experimental results for R a b i oscillations a n d a d i a b a t i c r a p i d passage are shown in fig. 5. We find a relatively good agreement between theory a n d e x p e r i m e n t except for the high laser fields where the Rabi oscillations tends to disa p p e a r owing to the transverse (to the a t o m i c b e a m axis) field distribution. This signal averaging could be reduced by narrowing the gaussian field distribution in the direction parallel to the atomic b e a m by means o f cylindrical focusing [3]. The small discrepancy between the m e a s u r e d a n d calculated adiabatic r a p i d passage m a y be a t t r i b u t e d to the little inaccuracies in the measurement o f the waist size and position. We can conclude that the coherent excitation o f the 3p states o f hydrogen is well observed an understood. The possibility o f observing R a m s e y fringes in the visible range [ 6 ] on transitions between H levels can therefore be also anticipated. References
[ 1] S. Avrillier, J.M. Raimond, Ch.J. Bord~, D. Bassi and G. Scoles, Optics Comm. 39 (1981) 311. Ch.J. Bord6, S. Avrillier,A. van Lerberghe, Ch. Salomon, Ch. BrYant, D. Bassi and G. Scoles, Appl. Phys. B 28 number 2/3, (1982) [2] A.G. Adam, T.E. Gough, N.R. Isenor and G. Stoles, Phys. Rev. A32 (1985) 1451. [3] J.P.C. Kroon, H.A.J. Senhorst, H.C.W. Beijerinck, B.J. Verhaar and N.F. Verster, Phys. Rev. A 31 (1985) 3724. [4] F. Brouillard, W. Claeys and G. Van Wassenhove, J. Phys. B: At. Mol. Phys. 10 (1977) 687. [5] Ch.J. Bord6, Rev. Cethedec 20 (NS83-1) 1. [6] Ch.J. Bord6, Ch. Salomon, S. Avrillier, A. Van Lerberghe, Ch. Br6ant, D. Bassi and G. Scoles, Phys. Rev. A 30 (1984) 1836.