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HIGHER ORDER STIMULATED RADIATIVE CORRECTIONS IN HYDROGEN
A. Maquet
To cite this version:
A. Maquet. HIGHER ORDER STIMULATED RADIATIVE CORRECTIONS IN HYDROGEN.
Journal de Physique Colloques, 1982, 43 (C2), pp.C2-451-C2-453. �10.1051/jphyscol:1982239�. �jpa-
00221851�
JOURNAL DE PHYSIQUE
Colloque C2, supplément au n°ll, Tome 43, novembre 1982 page C2-451
HIGHER ORDER STIMULATED RADIATIVE CORRECTIONS IN HYDROGEN
A. Maquet
Laboratoire de Chimie 'Physique*, Université Pierre et Marie Curie, 11, rue Pierre et Marie Curie, 75231 Paris Cedex OS, France
Résumé : Nous présentons un calcul des déplacements et élargissements radiatifs des niveaux d'énergies d'un atome d'hydrogène irradié par le rayonnement non-résonnant d'un laser intense monomode. Le calcul est effectué dans le cadre de la théorie des perturbations, jusqu'au quatriè- me ordre inclus, en utilisant une représentation sturmienne de la fonction de Green coulombienne. Les ordres de gran- deurs deces corrections ainsi que plusieurs cas limites intéressants sont considérés et discutés.
Abstract:We present a calculation of the radiative shifts and broadenings of the energy levels ofa hydrogenic atom, irradiated by a strong, non-resonant and monomode laser field.
The calculation is performed within the framework of the time- dependent perturbation theory, up to fourth-order included, using the sturmian representation of the Coulomb Green's func- tion. The order of magnitude of these corrections as well as several interesting limiting cases are considered and d i s c u s - sed .
Though essentially asymptotic and even strongly divergent in many situations of physical interest,the perturba- tion expansion remains one of our best tools for investigating the behavior of atoms in intense fields. As a matter or fact, several powerful techniques such as Pade1 Approximants or P a d e - Borel summation methods are at one's disposal for extracting phy- sical significant quantities from a divergent perturbative series ( 1 ) . Within this framework it thus seemed interesting to perform accurate quantitative calculations which could p r o - vide some useful reference marks for future discussions and comparisons with other (non-perturbative) approaches ( 2 ) .
In this paper we present a time-dependent perturba- tive calculation of the radiative level-shift and broadening experienced by a non-relativistic hydrogen atom irradiated with a non-resonant laser beam.In such a situation the inten- sity-dependent stimulated corrections (ac-Stark s h i f t , or light- s h i f t , or ...) can become dominant with respect to the L a m b - shift .
In the limit of large t i m e s , t—-> oo , the level shift AE. of a state |i> , eigenstate of the unperturbed hamiltonien H with eigenenergy E. : H | i ^ = E . | i > , may be conveniently expanded in terms of powers of the perturbing potential V :
(1)
*Laboratoire associé au C.N.R.S.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982239
JOURNAL DE PHYSIQUE
where :
Note that for finite values of t
,
the level shift is actually time- dependbnt and one has for instance( %
= 1) :and similar expressions for AE: (4)(t) 1.
. . .
The level shift of an atomic state In> in the presence of the laser and in the limit of t-00
,
becomeswhere O and I are respectively the frequency and intensity of the incoming laser, 1 ~ = 7 , 0 1 6 x 1 0 ~ ~ ~ / c m ~ is a characteristic atomic field strength intensify and Tn are atomic matrix elements with structures similar to those entering the Eqs. (2). Those matrix elements were evaluated exactly by using a sturmian representation of the
Coulomb Green's function,(3),which enables us to reexpress them in terms of (possibly multiple) series. The expressions obtained in that way are valid whatever are the initial hydrogenic state In) and the laser fre- q u e n c y ~
.
We have checked that if a - o o n e recovers the usual dc- Stark shift value. If 2 0 is greater than the ionization energy of the state In),
the fourth-order shiftAE
( 4 ) ( 0 ) acquires an imaginary part proportional to the two-photon ioRization cross section, providing an extension of the optical theorem to the case of multiphoton forward scattering.In a H-atom in the ground state irradiated by a neodymium laser (O = 4.310-~a.u) our results show that the Lamb shift is compensa- ted by the ac-Stark shift at intensities of W I O ~ ~ W / C ~ ~ . At higher inten- sities n, 1O15~/crn2,already attainable experimentally, (&),the magniru- de of the fourth-order contribution becomes comparable to the second or- der one. We are then very close to the limit of validity of the pertur- bative approach
...
Table
I -
Is stateW (4.a.)
-d2)(W)
-T(~)(W)0 2.c-5)
I. (-3) CO, 4.30(-3)
I.(-2) Neod. 4.30(-2) Ruby 6.57(-2)
Values of the second- and fourth-order atomic matrix elements ~ ( 2 ) ( 0 ) and T(~)(w) for the ground state of hydro- gen at various laser frequencies W
.
The corresponding level- shift can be obtained after replacing into the formula Eq.(4).References
( 1 ) See, for instance : H.J.Silverstone, E.Harrel1 and C.Grot, Phys. Rev.
A
((1981) 1925; W.P.Reinhardt,Int. J.Q.Chem. (1982) to appear.
(2) E.Arnous, J.Bastian and A.Maquet, Phys. Rev. A (1982) to be published.
(3) L.Hostler, J.Math. Phys.
11
(1970) 2966; A.Maquet, Phys.Rev.= ,( 1977) 1088.(4) G.Mainfray, This volume.
