ION DETECTION TECHNIQUES APPLIED TO RYDBERG ATOMS IN EXTERNAL FIELDS

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ION DETECTION TECHNIQUES APPLIED TO RYDBERG ATOMS IN EXTERNAL FIELDS

F. Penent, C. Chardonnet, D. Delande, F. Biraben, J. Gay

To cite this version:

F. Penent, C. Chardonnet, D. Delande, F. Biraben, J. Gay. ION DETECTION TECHNIQUES APPLIED TO RYDBERG ATOMS IN EXTERNAL FIELDS. Journal de Physique Colloques, 1983, 44 (C7), pp.C7-193-C7-208. �10.1051/jphyscol:1983716�. �jpa-00223273�

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JOURNAL DE PHYSIQUE

Colloque C7, suppl6rnent au noll, Tome 44, novernbre 1983 page C7-193

I O N DETECTION TECHNIQUES A P P L I E D TO RYDBERG ATOMS I N EXTERNAL F I E L D S

F. Penent, C. Chardonnet, D. Delande, F. Biraben and J.C. Gay

Laboratoire de Spectroscopie Hertzienne de l 'Ecole Normale Supe'rieurer, Tour 12 E O I , 4 , Place Jussieu, 75230 Paris Cedex 05, France

Rdsumd - Au cours des cinq dernihres ann6es, le ddveloppement simultans de l'excitation par laser continu accordable et monofrdquence et des mdthodes de dd- tection thermoionique a permis de disposer d'une puissante technique expdrimen- tale pour les 6tudes en phase vapeur. L'une des applications les plus intdres- santes dans le domaine de la physique fondamentale concerne les problsmes ou- verts pour lesquels l'atome de Rydberg fournit une base d'dtude bien adaptbe, par exemple pour l'dtude des propridtbs de l'atome dans des champs extsrieurs.

Plusieurs situations choisies sont analys6es 1 la lumihre de modsles vectoriels gdndralisds du diamagndtisme, de l'effet Stark linsaire et du rdgime en champs dlectrique et magnstique crois6s. Bien que les rdsultats expsrimentaux soient en tous points rdvllateurs du comportement hydrog6no?de en champs extdrieurs, aucun de ces rdsultats n'a 6t6 obtenu sur l'atome d'hydrogsne lui-msme.

Abstract - The simultaneous development of single mode C.W. dye laser excitation and thermoionic detection over the last five years has supplied with one of the most powerful experimental technique for atomic physics studies under vapour phase conditions. One of the most interesting application concerns problems of fundamental physics which can be tackled on Rydberg atoms especially the ones dealing with the atomic properties in external fields. Several selected exam- ples are discussed to the light of generalized vectorial models of diamagnetism,

linear Stark effect and crossed electric and magnetic fields effects. Although the experimental results are highly illustrative of the pure hydrogenic behaviour in external fields, none of them have been obtained on the hydrogen atom itself.

Applying an external field on the atom leads to fundamental alterations of the atomic properties /1,2/. For example the atomic spectra may drastically differ from

the usual Rydberg type in zero field. In addition, a complete modification of the symmetry properties of the wavefunctions takes place.

Although these questions are somewhat basic ones in physics, they are to some extent unsolved. They have recently received renewed consideration. This is largely in consequence of the development of tunable dye lasers and of our ability of producing and detecting atoms in high Rydberg states.

Regarding those kinds of fundamental problems, the selection of situations in which the atoms behave as hydrogen atoms is worthwhile. This insures that the external field perturbation of the spectra is not hindered by the spurious effects of close range corrections to the Coulomb potential. Hence, the,alteration of the atomic spectra due to the external field action can be observed in nearly pure conditions allowing meaningful comparisons with theory in its simplest and most basic form.

A way of tending to such an achievement is producing atoms with high n and 1 values.

The higher the (n,l) values, the smaller the departures from the perfect hydrogenic behaviour as the system will tend to behave as an electron tightly bound to an ionic core. Providing a convenient choice of the excitation process, this is a way of producing atoms in Rydberg states having an almost hydrogenic behaviour, but without the complications of dealing with hydrogen atoms.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983716

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C7-l94 JOURNAIL D€ PHYSIQUE

After some elementary recalls on the symmetry properties of the hydrogen atom, we will present some experimental aspects of its spectrum under various conditions of external fields. The emphasis will be on the low field behaviour of the spectrum with discussions of the linear Stark effect, of the rovibrational structure of a diamagnetic manifold and of the atomic spectrum in crossed electric and magnetic field conditions. All these results have been obtained using the technique of thermoionic detection with C.W. single mode dye laser excitation in either Doppler-free or non-Doppler free excitation processes. Although those experimental results are illustrative of the hydrogenic behaviour in external fields, none of them have been obtained using hydrogen atoms.

1 - THE HYDROGEN ATOM IN EXTERNAL FIELDS

1.1 - The zero-field Coulomb problem

The most part of the basic aspects of the interaction of an atom with external fields can be understood from an elementary analysis based on the symmetry properties of the Coulomb spectrum in zero field.

The important feature in the Coulomb spectrum is the n2 degeneracy of each energy level with principal quantum number n, with respect to 1 and M. On more general grounds, this is associated with the existence of a supersymmetry for Coulomb-like potentials and with the peculiar structure of the symmetry group /3,4/. Indeed a quite convenient description for our present concerns can be derived from arguments of classical physics.

The classical trajectory (the Kepler ellipse) is a closed cyrve. This means that besides the conservative character of the angular momentum L (as in all situations involving centrally symmetric potentials), there must be an additional conservative quantity with vectorial character and directzd along the major axis of the Kepler ellipse. This is the well-known Lenz vector A, the modulus of which is proportional to the eccentricity of the e_llipse /4/. The quantum expression of 2 ^{is just}^:

with obvious notations. In a given energy shell with fixed grincipal quantum number n, it is worthwhile introducing a scaled definition of the A vector :

where H is the hamiltonian. One then gets : a.L =+O,

The (a,L) vectors allow a complete description of the Coulomb spectrum and of the degeneracies in a given energy shell n. Indeed, it is much more convenient to deal with tke nsw set of operators :

f 41 = (5 + $12

l z = (L - ^a)/2 ^{( 3 )}

These operators do commute with the hamiltonian in a given energy shell and possess all the properties of two independant angular momenta. They fulfil1 :

(2j + = - meq/2E

Hence n = 2j + 1 meaning that j is either an integer or an half integer for a given n value. It is then straightforward to derive the eigenfunctions for fixed n value.

A natural choice is that of the uncoupled representation of the two angular momentum

{j: j z j lZ jZz} from which it is clear that the degeneracy of the n subshell is n2.

An equivalent choice is { j : j$ L, A ~ ) in which the z components of the angular momentum and of the Runge Lenz vector,ase defined. Such a choice is associated with

the separability of Schrgdinger's equation in parabolic coo~dinates. A third possible description is through the coupling scheme {j: jg (jl+ -? J2)2(jl+jz)z) or

{j: j$ L' L ~ } which is associated with the familiar description through spherical

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harmonics. Of course, the passing from one basis fo the other one only involves Clebsch-Gordan coefficients 1 5 1 .

From such a description follows a vectorial model of the hydrogen atom recalled on Figure ( 1 ) .

Fig. ( 1 ) - Vectorial model of the Hydrogen atom.

1.2 - Hydrogen atoms in external fields. Generalized vectorial models a) External electric field - Linear Stark effect

Under such conditions, the system is well-known 1 6 1 to be separable in parabolic coordinates which in turn is associated with the existence of an exact dynamical symmetry / 3 , 6 / . We restrict the analysis to a given energy shell. Thg adiaba~ig invariant is then the energy of interaction with the external field E :

W = q E.; + ^-f (5)

From classical perturqation theory, <P> is proportional to A. Then, one gets (in atomic units) (the E field is directed along z axis) :

Fig. (2) - Stark vectorial model at low fields.

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C7-196 JOURNAL DE PHYSIQUE

Another constant of the motion is the projection of the angular momentum on the field axis :

tz = (jlz + jzz) 2 ( 7 )

Then, from (6) and (7), the n degeneracy of the shell is partially removed by the introduction of the electric field. The manifold is splittkd into in - 1 components, with a spacing 3 n E. Such a quantity w~

- +

is the linear Stark frequency associated with the precession of

T1

^and

T2

^{around E}

(see Figure (2)). Then, Stark effect is a linear phenomenon for the hydrogen atom at low fields. The eigenfunctions are obviously of the {jf jf, j lz-j2z=q, j lz+jh=~}

types. Writing

n = nl + nz + I M I + 1

A, = (nl - ^nz)/ n ⁽⁹⁾

one recovers the usual description in terms of the parabolic quantum numbers (nl, nz) /6/. From (6) each component q = nl - nz in the manifold presents a n - I q l

degeneracy in the M value,

1 3

This perturbative description is valid provided 7 >> wE = n E that is for field n

values smaller than the critical ionisation field /l/.

b) Symmetries in an external magnetic field. Paramagnetism and diamagnetism

The interaction hamiltonian with the magnetic field is composed of two parts /2,5/

2m 8m (10)

The first term (linear in B field) is resp~nsible for the usual o~bital Zeeman effect and associated with paramagnetism (L precesses around the B field at Larmor frequency ; see Figure (3)).

Fig. (3) - Vectorial model of paramagnetism

The second term (quadratic in B field) is called diamagnetic interaction. It is often negligible for low-lying states, but it can be important for high Rydberg states.

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In contrast to the previous situation, there is+no exact dynamical symmetry. The projection L, of the angular momentum onto the B field axis is conserved

L, = jl, + jzZ = M (11)

A correct description of the interaction (including the diamagnetic one) was performed for the first time in 1981 /7,8/ in the low field regime.

As iq the electric field case, an adiabatic invariant A can be derived in terms of the j 1 and j2 operators. From the invariance of the diamagnetic interaction under varioqs spatial transformations, its expression can be shown to involve only terms as jljz and jlZjzZ 191, while the derivation of the coefficients can be done choosing simple situations of quantum or classical physics 191. One gets exactly

2 + *

A = 3n2 + 1 - ^4M⁺^4(5jlZj2,- ^2~132) ⁽¹²⁾

while the complete expression of the energy in such conditions becomes (in atomic

units) Q

E =--!-+F+ Y L n 2 1 \

2112 2 16 (1 3)

withy= hw /2Ryd the reduced magnetic field (U, = qB/m is the cyclotron frequency) From (12),~it is clear that the diamagnetic interaction involves an unusual coupling scheme of the two angular momenta jl and j+ ; /5,8/. Then, it can be shown that the symmetry of the diamagnetic manifold is of rovibrational type.

For each M value, the manifold is splitted into about n/2 components due to the diamagnetic interaction. When yn3 11, the previous analysis breaks down, and the atomic Rydberg spectrum evolves to a Landau type spectrum /2,10/, that is the spectrum of a free electron in a magnetic field.

c) Symmetries in crossed electric and magnetic fields

Usually, there is no remaining constant of the motion. Nevertheless, if the diamagnetic interaction is negligible, the atomic spectrum is still organized.

This manifests the existence of a new class of motion, intermediate between Zeeman and linear Stark effects 1111.

This is readily seen from the expression of the energies : + + + +

W = 2 B.L + qr.E

2m _-f _-& (14)

-f +

introducing the (jl,j2) operators and the % (Larmor) and (+ (Stark) frequencies, one ggts (3tomjc unip) :+

W = (WL ⁺%)+jl ++(W - ^UE)'J~t ⁽¹⁵⁾

Writing $1 = UL w~ (16)

Q = +2u2)1/2 +

t

n2 ~ 2 ) 112 L E

where Q is the modulus of and $2 , ^{one gets}^:

W = Q(jiQ, + jzQ2) (17)

The manifold is then splitted into (2n - ¹⁾ components which do present a remaining degeneracy. The spacing between two adjacent components is proportional to Q. Then, in this intermediate regime, the spacing is neither proportional to B nor to E.

The wavefunctions,are of+(j : ^jz , ^j^l jn ⁾t pes associated with the independant quantizations of j~ and j2 along th8131 $&d 8 2 axis. The associated vectorial model is shown on Figure (4).

When the 3 and 3 fields are not crossed to each other, the degeneracies are comple- tely removed and the manifold is splitted into n2 components.

Other types of regimes, where the diamagnetic interaction is not negligible, in high field conditions, have been predicted /12,13,14/. These will not be discussed here.

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JOURNAIL D€ PHYSIQUE

Fig. (4) - V e c t o r i a l model of t h e spectrum i n c r o s s e d e l e c t r i c and magnetic f i e l d s

1 . 3 - V a l i d i t y of t h e a n a l y s i s f o r many e l e c t r o n atoms

The p r e v i o u s r e s u l t s a r e i n t i m a t e l y a s s o c i a t e d w i t h t h e supersymmetry of t h e Coulomb problem. For non hydrogenic atoms, t h i s c o n s t i t u t e s t h e b a s i s of t h e analy- s i s , provided t h e r o l e of quantum d e f e c t s i s n o t too important. This means t h a t one w i l l a s y m p t o t i c a l l y r e c o v e r t h e i d e a l hydrogenic s i t u a t i o n when :

. ⁶^<<^W

n

where W i s t h e e x t e r n a l f i e l d p e r t u r b a t i o n and 6 t h e quantum d e f e c t . P r a c t i c a l l y , t h i s means t h a t one must choose experimental s i t u a t i o n s i n which t h e atomic beha- v i o u r i s looking l i k e t h a t of hydrogen, o r f i n d i n g some t r i c k s f o r checking t h e p a r t of t h e spectrum behaving a s i n hydrogen.

2 - EXPERIMENTAL EVIDENCE OF NEW QUANTIZATION LAWS I N THE SPECTRUM OF MANY ELECTRON ATOMS I N EXTERNAL FIELDS

2 . 1 - Main c h a r a c t e r s of t h e experimental s t u d i e s

The e x c i t a t i o n of high l y i n g Rydberg s e r i e s of a l k a l i atoms i s performed through t h e use of a C.W. s i n g l e mode r i n g dye l a s e r . It d e l i v e r s about 800 mW power i n t h e 5 900 d region. This i s p r e s e n t l y t h e only c l a s s of experiments c e n t e r e d on e x t e r - n a l f i e l d s t u d i e s of t h e Rydberg spectrum using such kinds of C.W. dye l a s e r e x c i t a -

t i o n . The l a s e r l i n e w i d t h i s i n t h e range of 1 MHz. The l a s e r l i n e i s locked on an e x t e r n a l c a v i t y . This allows continuous s c a n s of t h e frequency on a 150 GHz range withouc any mode jumps while t h e l i n e s h a p e i s always p e r f e c t l y c o n t r o l l e d .

Experiments a r e performed under vapour phase c o n d i t i o n s u s i n g e i t h e r Doppler l i m i t e d e x a i t a t i o n schemes e q u i v a l e n t t o s t e p w i s e e x c i t a t i o n o r two-photon Doppler f r e e techniques. The d e t e c t i o n of Rydberg atoms i s performed through t h e use of a t h e r - moionic d e t e c t o r which i s b a s i c a l l y a space charge l i m i t e d diode allowing an e f f i - c i e n t a m p l i f i c a t i o n of t h e i o n s produced from t h e Rydberg atoms. A t y p i c a l arrange- ment i s shown on F i g u r e s 5 and 6. The f a c t t h a t t h e method i s e x t r e n e l y e f f i c i e n t i s well-known and r e c a l l e d i n t h e f o l l o w i n g . ~ u ' t th e d e t e c t i o n i s a l s o extremely r e l i a b l e i n t h e s e n s e t h a t t h e s t r a y e l e c t r i c f i e l d s a r e w e l l c o n t r o l l e d and s m a l l e r t h a n 20 mV/cm (depending on t h e geometry of t h e d e t e c t o r ) . C o l l i s i o n s e f f e c t s inhe-

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-

Fig. (5) - Experimental apparatus

Laser

+ +

Fig. (6) - Thermoionic detector for crossed (E,B) fields experiments.

rent to any vapour phase studies are always negligible.

An important advantage of experiments under vapour phase conditions lies in the possibilities of choosing "exotic" excitation schemes leading.@@ Bhe selection of states with almost hydrogenic behaviours. For example, the use of the so-called hybrid resonance process /15,16/ in caesium vapour allows, through abso.ption of two non-resonant photons, to populate nF Rydberg series. These series with small

(6 = 0.0335) quantum defects are almost hydrogenic ones. They are particularly convenient for external field studies with Doppler resolution. The selection of Doppler free two-photon processes on Rb atoms allows on the other hand studies in extremely good conditions of resolution although the nS or nD series are strongly non hydrogenic. As shown below, finding some tricks allows anyway to study the hydrogenic part of the spectrum, although under indirect conditions.

2.2 - The field free Rydberg spectrum

The efficiency of the excitation - detection process is illustrated in zero- field conditions in ~i~ures(7) and (8) /l 7,18/.

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JOURNAL DE PHYSIQUE

Fig. (7) - Doppler limited Rydberg spectrum (Caesium F series).

Figure (7) displays the Rydberg spectrum of Caesium got using the two-photon hybrid resonance process. Such plots have been obtained for n up to 162 1191 which is the ultimate value one can expect using Doppler limited techniques. Indeed the Doppler width being of the order of 1 GHz, it is of the order of the spacing 2 ~ / n ~ between adjacent Rydberg lines for n 2 170. For n Q 140, the collisional shift of the lines is of about 200 MHz (collisional red shift).

In Figure (8) is a plot of the Rydberg spectrum of Rubidium in two-photon Doppler free conditions /20/. This has been obtained using an original technique, ,modulating the laser frequency. The signal is then the derivative of the Lo- rentzian two-photon lineshape. This provides with great improvements of the signal to noise ratio compared to conventional AM modulation techniques.

The technique is presently limited to n values up to 150, in part due to Stray electric fields.

Direct measurements of the linewidth suggest that the stray electric field in the equipotential volume of the diode is smaller than 20 mV/cm. Im- provement of the shielding of the diode together with the use of an optiual cavity will allow to extend the technique to n values up to 300, as a gain of (50)' is likely to be obtain on the two- photon signals.

-. Fig. (8) - Doppler free spectrum (Rb atoms n=37S two-photon line) 2.3 - The quasi-hydrogenic diamagnetic manifold of Caesium

As discussed in 51.2the action of a magnetic field on the spectrum is two-fold.

Firstly the paramagnetic interaction is responsible for the well-known Zeeman effect, linear in B field. Secondly, the diamagnetic interaction proportional to B'P' cau- ses a deeper alteration of the spectrum which evolves from Coulomb to Landau type.

Such transition has been experimentally studied in great details /l71 especially demonstrating the existence of a new quantization law n3y = 1.56 obeyed at the zero- field threshold (y is the reduced magnetic field ; see § 1.2.b) and the evolution of the spectrum from a bound state Rydberg to a Landau resonances spectrum /21,22,23/.

One of the most interesting regime at low fields is the diamagnetic regime in which, due to the break-down of L' as a good quantum number, a "forbidden line structure1' appears in the excitation spectrum. Such a spectrum is shown in Figures(9) .and (10) as a function respectively of the laser frequency and of the B field /17,19/. The situation is that of M = 3, odd parity, quasi-hydrogenic states of Caesium. Compared to the pure hydrogenic situation, there are still some small perturbations in the

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B IkGI

Fig. (9) - Diamagnetic manifold in energy Fig. (10) - Diamagnetic manifold in B (Caesium M=3 odd parity states) field (CS M=3 odd parity states) at fixed

(B = 6.16kG) laser frequency

oscillator strengths due to the very small quantum defect of the F state. At fixed magnetic field values, the spacing between successive components is just proportional to .'B The manifold has about n/2 components. To each component is associated a value of the quantum number K which in turn is quantizing the spectrum of the invariant A in formula (12). The states with the smaller K values are located at the top ofthediamagnetic band. They are analogous to the states of the oblate spherical top with some rotational symmetry ($ee 5 1.2.b). States at the bottom of the band do have vibrational symmetry along the B field axis. The structure of the manifold is then of rovibrational type. A complete description of its properties can be found in references /5,24,25/.

In the previous high field experiment, the field was produced by the meansof a super- conductive solenord and the manifold was checked for low n 2 40 values. The same kind of phenomena will occur at low fields in the range of 500 Gauss provided one is able toproducestates with n up to 150, in a Doppler-free experiment. This is under current investigation, and quite clear to understand as the diamagnetic interaction is sca- ling as n $ ~ ~ .

2.4 - Substructures in the quasi-Landau spectra close to the zero field threshold

In the previous situation, the diamagnetic hamiltonian HD was a small pertur- bation of the Coulomb binding energy, that is 2 ~ / n ~ >> HD. Its main role was breaking the supersymmetry in the Coulomb manifold in a well controlled way described in 5 1.2.b. But other situations may exist for higher n or field values in which HD becomes comparable or even greater than the Coulomb binding energies. These regimes are called quasi-Landau and Landau regimes. The first one takes place near the zero field threshold /2,17,22/. It is characterized with the existence of equally spaced resonances (with a spacing 3/2hwc - wc the cyclotron frequency) while the quantization law in field, at constant energy of the electron is n 3 . ~ = 1 .56.Bc (Bc=2.35. 10' Gauss). Although the dominant structures in the spectrum are now well understood /i7,23/, it is quite clear that they present substructures as shown on Figure (11).

This is an experimental spectrum obtained at threshold on M = 3, odd parity quasi- hydrogenic Caesium states. Compared to the situation at low fields discussed in 5 1.2.b, the theory is here far from being achieved. But the existence of these,substructures+can be thought as due to an excited vibrational motion of the electron along the B field axis while dominant structures are associated with the radial quantization of the motion in the direction perpendicular to the field. The quasi-Landau spectrum is then band structured. A simple model allows to show that at threshold, the vibrational spacing is about one fourth of the radial one (1.5hwc) which is suffi- cient for rendering chaotic the general appearance of the spectrum. A numerical si-

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Fig. (11) - Quasi-Landau spectrum at threshold (zero energy of the electron) as a function of the B field strength.

mulation shown on Figure (12) reproduces quite well (at least qualitatively) the fea- tures of the experimental spectrum at threshold 119,261.

Fig. 12 - Numerical simulations of the quasi-Landau spectrum at threshold (based on a fully 3 dimensional semi-classical analysis)

More generally, such a regime shares numerous common characters with phenomena involving the dynamics of coupled anharmonic oscillators as in intramolecular energy transfer (classical resonance regions) and photochemistry 12,271.

2.5 - The quasi-hydrogenic Stark manifold

A convenient choice of the electrode arrangements in the detector allows to apply a small electric field in the interaction volume where the atoms are excited 128,291. In order to avoid any excitation of discharges, the strengths of the field are necessarily small ones. This is not a limitation as a proper choice of the n va-

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l u e w i l l make t h e i n t e r a c t i o n with t h e e x t e r n a l f i e l d comparable o r even g r e a t e r than t h e Coulomb i n t e r a c t i o n . Furthermore a proper choice of t h e s t a t e s under considera- t i o n and of t h e n v a l u e allows t o g e t a s i t u a t i o n a s hydrogenic a s p o s s i b l e . That i s t h e r o l e of quantum d e f e c t s becomes e n e r g e t i c a l l y n e g l i g i b l e compared t o t h e e x t e r n a l f i e l d s e f f e c t s . Then S t a r k e f f e c t w i l l become a l i n e a r phenomena a t low f i e l d s a s i n t h e p u r e l y hydrogenic s i t u a t i o n . This is e x p e r i m e n t a l l y r e p o r t e d below i n two d i f f e - r e n t s i t u a t i o n s , on Caesium and Rubidium atoms.

2.5.a- L i n e a r S t a r k e f f e c t of quasi-hydrogenic Caesium s e r i e s The e x c i t a t i o n o f Caesium M = 3 Rydberg s t a t e s through t h e s o - c a l l e d h y b r i d resonance process has allowed t h e experimental o b s e r v a t i o n of t h e l i n e a r S t a r k e f f e c t f o r n v a l u e s between 30 and 80, a t extremely low e l e c t r i c f i e l d s t r e n g t h s between 0 and 12 V/cm /29/. The c o n d i t i o n s a r e n e a r l y hydrogenic ones f o r a l l t h e s t a t e s involved i n t h e S t a r k mixing process. Indeed t h e upper bound of t h e quantum d e f e c t s corresponds t o t h a t of F s t a t e s ; t h e n i t i s f a i r l y small and t h e c o n d i t i o n f o r such an o b s e r v a t i o n :

6 3

- n3 << - 2 n ( n l - n2) E

i s e a s i l y f u l f i l l e d a t low e l e c t r i c f i e l d v a l u e s .

These experimental r e s u l t s 1291 a r e p r e s e n t l y t h e only d i r e c t evidence of t h e l i n e a r behaviour got i n u l t r a low e l e c t r i c f i e l d c o n d i t i o n s (about one thousand times smal-

l e r t h a n i n r e f e r e n c e /30/) u s i n g C.W. dye l a s e r e x c i t a t i o n i n vapour phase condi- t i o n s . The s t r u c t u r e of t h e hydrogenic n = 36, M = 3 S t a r k manifold i s d i s p l a y e d i n Figure (13) f o r E = 11.3 V/cm. The manifold i s s p l i t t e d i n t o 33 components a s s o c i a - t e d t o t h e v a l u e s n l = 0 t o n l = 32 of t h e p a r a b o l i c quantum number ( o r e q u i v a l e n t l y t o t h e 33 p o s s i b l e quantized v a l u e s of t h e z component of t h e Runge-Lenz v e c t o r ) . The spacing between a d j a c e n t components i s 3nE ( i n atomic u n i t s ) and l i n e a r l y depends onthe f i e l d s t r e n g t h , a s expected . The average e x t e n s i o n of t h e manifold i s then of about 3 n 2 ~ .

Fig. (13) - S t r u c t u r e of t h e (n=36, M=3) l i n e a r S t a r k manifold of Caesium (E=II.3V/cm) + The (n, M, n l , n2) p a r a b o l i c components of t h e manifold a r e m a n i f e s t i n g t h e breaking of t h e n2 degeneracy so a s t o comply w i t h t h e symmetry of t h e e x t e r n a l e l e c t r i c f i e l d p e r t u r b a t i o n . The p r e v i o u s c o n d i t i o n s a r e such t h a t l / n 3 >> 3/2n(nl - n2)E so t h a t t h e s u c c e s s i v e S t a r k manifolds a r e w e l l i s o l a t e d from each o t h e r .

Another a s p e c t of t h e spectrum i s shown on Figure (14) i n a regime where s e v e r a l ad- j a c e n t manifolds a r e i n t e r a c t i n g t h a t i s t h e c o n d i t i o n l / n 3 % 3 / 2 n ( n l - n2)E i s f u l - f i l l e d . This i s f o r n around 60 i n a range of e l e c t r i c f i e l d s t r e n g t h s between 0 and 12 Pcm. Although t h e manifolds a r e l i k e l y t o i n t e r a c t , t h e e n e r g y - f i e l d p l o t of Fi- gure (15) c l e a r l y shows t h a t t h e behaviour of each energy l e v e l i s s t i l l l i n e a r i n t h e e l e c t r i c f i e l d . Indeed t h i s i s t h e m a n i f e s t a t i o n i n a n e a r l y p e r f e c t approxima- t i o n of t h e e x i s t e n c e of a dynamical symmetry i n t h e S t a r k problem f o r t h e hydrogen atom 1291.

There a r e two k i n d s of independant c o n t r i b u t i o n s i n such a p l o t , a s s o c i a t e d r e s p e c t i -

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( 1 4 ) - L i n e a r S t a r k spectrum o f quasi-hydrogenic M=3 Caesium s t a t e s around Fig.

n - 2, 60 (E= 9.8V/cm)

F i g . (15) - L i n e a r S t a r k p l o t s around n = 60 i n t h e i n t e r n mixing regime.

v e l y w i t h M = 3 (quasi-hydrogenic) and M = 1 ( s t r o n g l y non-hydrogenic) s t a t e s , due t o some p e c u l a r i t i e s o f t h e o p t i c a l e x c i t a t i o n p r o c e s s . The S t a r k diagram a s s o c i a t e d w i t h M = 3 s t a t e s o n l y i n v o l v e s s t a t e s w i t h h i g h l y h y d r o g e n i c b e h a v i o u r s . T h i s i s q u i t e c l e a r l y s e e n from t h e b e h a v i o u r o f t h e l i n e s around n = 59, 60, 61. T h e i r po- s i t i o n s l i n e a r l y depend on t h e E f i e l d s t r e n g t h o n t h e whole range under i n v e s t i g a - t i o n . Furthermore, no s i g n i f i c a n t d e p a r t u r e from t h i s b e h a v i o u r o c c u r s i n t h e regime

where a d j a c e n t m a n i f o l d s a r e merging. I n d i v i d u a l l i n e s b e l o n g i n g t o t h e v a r i o u s ma- n i f o l d s can s t i l l be t r a c k e d w i t h o u t any a m b i g u i t y . T h i s p l o t e x h i b i t s t h e l i n e a r S t a r k b e h a v i o u r a s w i l l b e s e e n on t h e Hydrogen atom, i n c o n d i t i o n s where t h e second o r d e r t e r m i s a l m o s t n e g l i g i b l e . I t i s i n d e e d r e s p o n s i b l e f o r a s y s t e m a t i c q u a d r a t i c r e d s h i f t o f t h e p o s i t i o n o f t h e l i n e s a t maximum of 1 GHz a s can be s e e n on F i g u r e ( 1 5 ) .

O f c o u r s e , t h e b e h a v i o u r f o r non-hydrogenic s t a t e s , e . g . M = 1 s t a t e s , i s c o m p l e t e l y d i f f e r e n t 128,291. But p a r t o f t h e M = 1 s t a t e s , w i t h s m a l l quantum d e f e c t s , w i l l behave a s a n i n c o m p l e t e l i n e a r S t a r k m a n i f o l d ( s e e n e x t s e c t i o n ) . I n a d d i t i o n , v a r i o u s c l a s s e s o f quasi-Fano i n t e r f e r e n c e p r o f i l e s m a n i f e s t i n g t h e b r e a k i n g of t h e dynamical symmetry due t o non-hydrogenic c o r r e c t i o n s t o t h e p o t e n t i a l have been s e e n on t h e i n t e n s i t y s p e c t r u m 1281.

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2.5.b - Doppler-free anticrossing spectroscopy of the linear Stark manifold

The previous experiment allowed a direct investigation of the structure of the linear Stark manifold on quasi-hydrogenic series of Caesium. But we will show here that this is as well possible when dealing with strongly non-hydrogenic states, providing a proper adaptation of the experimental conditions.

The nS, nP and nD series of Rb have important quantum defects (respectively 3.13, 2.65 and 1.35). Consequently, they have a strong non-hydrogenic behaviour when applying an electric field. At low fields, they first exhibit a quadratic Stark effect.

In contrast,states with 1 > * 3 (F, G, H... series) have small quantum defects (smaller than 6(F)= 0.017) and are nearly degenerated in zero-field. Consequently, at low fields, they will behave as in the Hydrogen atom situation, exhibiting a linear Stark effect. Such states cannot be excited in a two-photon process from the ground state

for M > / 3 states while the states with M = 2 and M = 0 can be. But they will behave

as an incomplete hydrogenic manifold at low fields as the 1 = 2, 1 = 1 and l = 0 components are out of the manifold (due to their large quantum defects).

The M = 0, n = 42 S state exhibits a quadratic Stark effect as shown in Figure (16).

Fig. (16) - Diabatic energy curve of the (M=O, 425) state of Rubidium (Doppler-free two-photon experiment). The perturbing levels responsible for the anticrossings are those of the incomplete M=O quasi-hydrogenic Stark manifold which linearly behave with the E field.

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C7-206 JOURNAL DE PHYSIQUE

The energy dependance on the + E field is a smooth One while the oscillator strength remains concentrated on this line. The coupling between the (nS) and states of the manifold is at third order. The transfer of oscillator strengths is weak. The components of the manifold are not likely to appear on the excitation spectrum except close totheanticrossings points, with the (nS) perturbed state. This is shown in Figure (16). The anticrossings sizes are small, of the order of 100 MHz, as the coupling is weak. A strong but local transfer of oscillator strengths occurs near the crossing.

The positions and sizes of the anticrossings are in agreement with theory. T$e dis- tances between successive crossings are consistent with both a+quadratic in E behaviour of the energy of the perturbed nS state and a linear in E behaviour of the energies of the perturbing levels, issued f ~ o m the incomplete quasi-hydrogenic Stark manifold. The very weakly depending on the E field energy of the nS state is used as a baseline for checking the linear in field quasi-hydrogenic quantization in the manifold /20/.

2.6 - Experimental evidence qf+a new quantization law in the quasi-hydrogenic manifold, in crossed (E,B) fields.

The experimental situation on Caesium atoms is suitable f o ~ 5 direct test of the quantization law in the quasi-hydrogenic manifold, in crossed (E,B) fields. It is under progress. But we are here providing with this evidence on Rubidium atoms using the techniquedescribed in 5 2.5.b.

The idea is to use the diabatic energy curve of the (M = 0, nS) state and its anticrossings with the components of the quasi-hydrogenic manifold so as to check th~; new quantization which will settle in the manifold, in the presence of a crossed to E magnetic field.

In thepresenceof a B field alone, the two photon (M + = 0, nS) line is splitted into several components associated with the hyperfine structure of the ground stale. For fields smaller than 400 Gauss, there are no dependance of the energy on the B value as the diamagnetic+c~ntribution is still negligible. The Znergy of the (M = 0, nS) state in crossed (E,B) fields only weakly depends on the E field value (quadratically).

But as a consequence of the drastic change of She quantization in the manifold, the positions of the anticrossings will vary with B field strength, and copared to the situation in 5 2.5.b, new kinds of anticrossings will appear. At high B field value the number of anticrossings is twice the number in the absence of B field in agreement with the model of 5 1.2.c. Such an anticrossing is shown in Figure (17).

+- +

Fig. (171 - Two photons lineshape and anticrossing behaviour in crossed (E,B) fields (n = 375 transition) - (a) before the anticrossing (E=6.85 V/cm ; B=350 Gauss) - (b) at the anticrossing (E=7.08 V/cm ; B=350 Gauss).

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+ +

Fig. (18) - Experimental evidence of a new quantization law in crossed (E,B) fields.

As s$own in Figure (18), the evolution of the positions of the anticrossings with g

and B just follows a law showing that the spacing is proportional to

[$

^.+ ⁹₄^nz^EZ]^'12

in agreement with the model of § 1.2.c (y = BIBc is the reduced magnetic field). This is the first experimental proof of the existence of a new quantization law in the manifold in the regime intermediate between linear Stark effect and Zeeman effect. It is e§peciallyclear, as expected, that the distance between successive members of the manifold is neither proportional to B, nor to E, indeed a surprising consequence of the supersymmetry of the Coulomb problem.

3 - CONCLUSION

We have illustrated the usefulness of the thermoionic detection method for dealing with various aspects of the atomic Rydberg spectrum. Contrasting with intuition, it allows accurate determinations of a lot of atomic parameters. Major progresses in atomic physics have been obtained this way, especially as concerns the spectrum in external fields. In this area, the experimental results are presently among the best ever got in extremely varied field strengths and n values conditions. The extension ofplrt of the work towards the production of atoms in states with n around 300 is under progress.

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JOURNAIL D€ PHYSIQUE

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