Journal of Experimental and Theoretical Physics, Vol. 96, No. 6, 2003, pp. 1006–1018. Translated from Zhurnal Éksperimental’noÏ i TeoreticheskoÏ Fiziki, Vol. 123, No. 6, 2003, pp. 1145–1159. Original Russian Text Copyright © 2003 by Bolgova, Ovsyannikov, Pal’chikov, Magunov, von Oppen. ATOMS, SPECTRA, RADIATION Higher Orders of Perturbation Theory for the Stark Effect on an Atomic Multiplet I. L. Bolgovaa, V. D. Ovsyannikova, V. G. Pal’chikovb,*, A. I. Magunovc, and G. von Oppend aPhysics Department, Voronezh State University, Voronezh, 394006 Russia bNational Research Institute for Physical–Technical and Radiotechnical Measurements, Mendeleevo, Moscow oblast, 141570 Russia *e-mail: [email protected] cInstitute of General Physics, Russian Academy of Sciences, Moscow, 119991 Russia dTechnical University, Berlin, D-10623, Germany Received November 28, 2002 Abstract—The contribution of higher order corrections to the Stark energy is calculated in the anticrossing region of atomic multiplet sublevels. Perturbation theory for close-lying levels is presented that is based on the Schrödinger integral equation with a completely reduced Green’s function. Analytic formulas are obtained for the splitting of two interacting fine-structure sublevels as a function of the field strength. These formulas take into account fourth-order resonance and nonresonance corrections to both the diagonal and the off-diagonal matrix elements of the dipole moment operator. By the method of the Fues model potential, a numerical anal- ysis of radial matrix elements of the second, third, and fourth orders is carried out that determine a variation in 3 3 the transition energy between n P0 and n P2 sublevels of a helium atom for n = 2, 3, 4, 5 in a uniform electric field. It is shown that the contribution of the fourth-order corrections in the vicinity of anticrossing of levels for n = 2, 3, 4, 5 amounts to 0.1, 5, 10, and 15% of the total variation of energy, respectively. A comparative anal- ysis is carried out with the results of calculations obtained by the method of diagonalization of the energy matrix, which, together with resonance terms, takes into account other states of the discrete spectrum with n ≤ 6. © 2003 MAIK “Nauka/Interperiodica”. 1. INTRODUCTION A comparison of the numerical data obtained by these expressions with the results of measuring the field The Stark effect in a constant electric field was cal- dependence of the probability of radiative transitions culated for a hydrogen atom and described in detail in between highly excited Rydberg states [11] has shown textbooks and monographs almost immediately after that the first three terms in powers of the field strength the origination of quantum mechanics (see, for exam- are quite sufficient for calculating the transition proba- ple, [1, 2]). Nevertheless, this phenomenon has not yet bilities in virtually all situations of interest up to the been completely studied and still attracts attention in field value when the above-barrier ionization of the our time. The experimental studies of the Stark effect upper level becomes possible. have been stimulated by the development of precision By the end of 1970s, methods of laser spectroscopy methods of laser spectroscopy [3–6]. Theoretical calcu- of Rydberg levels had been developed that allowed one lations of the shift and broadening of the Stark states in to accumulate a large volume of experimental data on hydrogen atoms carried out by the beginning of the the Stark effect in highly excited multielectron atoms 1980s were based on the iterative solution of the [12–15]. These data stimulated the development of sim- Schrödinger equation involving the separation of vari- ple semiempirical calculation methods for the polariz- ables in a parabolic system of coordinates [2]. The abilities of atomic levels [16, 17] (including Rydberg development of computer programs for analytic com- levels [18]), as well as the development of exact ab ini- putations that allowed one to derive general expressions tio methods that enable one to consistently take into for the coefficients of power series in the field strength consideration relativistic and quantum-electrodynamic for both the shift and the broadening of atomic levels effects, which play a significant role in the spectra of [7, 8] has served as a powerful incentive for the calcu- ions of high degree of ionization [19–22]. In [13–15, lation of higher order corrections to energy. Analytic 23], it was observed that the Stark effect on multiplet programming has also enabled one to obtain general sublevels deviates from the quadratic law even in weak expressions for perturbation theory series for the wave fields. This phenomenon is accounted for by the Stark functions, matrix elements, and radiative transition prob- interaction between sublevels that leads to their repul- abilities between the Stark states of hydrogen [9, 10]. sion in a field and is determined by the hyperpolariz- 1063-7761/03/9606-1006 $24.00 © 2003 MAIK “Nauka/Interperiodica” HIGHER ORDERS OF PERTURBATION THEORY FOR THE STARK EFFECT 1007 ability of an atom. It was demonstrated that the main hyperpolarizabilities are rapidly increasing functions, (resonance) contribution to the hyperpolarizability of a α ~ ν7 and γ ~ ν17 [32], the anticrossing for higher levels multiplet sublevel can be expressed in terms of the ten- occurs in weaker fields, and the contribution of higher sor polarizability of this multiplet [24]. Therefore, pre- order corrections becomes more significant. The first cision calculation [25] and measurement [3, 5, 26] of calculations of the nonresonance hyperpolarizability irreducible components of the polarizability tensor of for the triplet state 3 3P in helium atoms [30] has shown multiplet states of atoms becomes topical. that the contribution of the fourth-order field correc- Precision information about the Stark effect in tions to the diagonal elements may be several times higher order terms of perturbation theory plays a key greater than experimental errors. Therefore, the deter- role in modern optical frequency standards based on the mination of corrections to the off-diagonal elements application of magnetooptical traps in combination also becomes topical. with the methods of laser cooling of atoms up to several The main goal of the present paper is to determine nanokelvins [27], as well as in the problems of fre- the contribution of the fourth-order corrections in the quency stabilization in atomic standards of a new gen- field strength to the energy of multiplet states of an eration—atomic fountains [28]. atom near the anticrossing of the fine-structure sublev- The interaction between sublevels of a multiplet in a els. In Section 2, we give a generalization of higher field may lead to an important phenomenon, the so- order perturbation theory for degenerate states [33] to called anticrossing [29]. This phenomenon manifests the case of close-lying levels that have nonzero splitting itself when the polarizability of a higher energy sub- in zero field. We obtain expansions of the matrix ele- level is greater than the polarizability of a lower energy ments of the interaction Hamiltonian of an atom and a state. Then, in a weak field, the sublevels move closer field in powers of the field strength F up to the fourth to each other; i.e., the fine-structure splitting decreases. order. In Section 3, the coefficients of these expansions In a stronger field, the sublevels are repulsed from each are expressed in terms of irreducible parts—scalar and other; this repulsion is determined by the resonance tensor polarizabilities and hyperpolarizabilities— part of the hyperpolarizability. The field strength at which, in turn, are represented by linear combinations which attraction turns into repulsion and the minimal of radial matrix elements of the second, third, and value of splitting are uniquely determined by the fourth orders. In Section 4, we derive general expres- atomic susceptibilities—the components of the polariz- sions for the field-dependent splitting of two interacting fine-structure sublevels. The fine-structure splitting of ability and hyperpolarizability—and can be calculated 3 theoretically. Usually, the field value corresponding to n P states of helium (n = 2, 3, 4, 5) is calculated numer- the anticrossing of levels is sufficiently high; therefore, ically as a function of F; in zero field, these states have an appreciable contribution to the energies of states total momenta of J = 0, 2 and a projection of M = 0. The may be made not only by the polarizability and the res- contribution of the fourth-order corrections in the field onance part of the hyperpolarizability but also by non- region of level anticrossing is determined. resonance additions to the hyperpolarizability. In [30], fourth-order nonresonance corrections have 2. HIGHER ORDER PERTURBATION THEORY been calculated to the energy of separate sublevels of a FOR CLOSE-LYING LEVELS helium atom multiplet. In this case, the resonance The calculation of higher order corrections to the γ(res) hyperpolarizability , which is determined by the Stark effect for atomic multiplet sublevels requires a off-diagonal matrix elements of the interaction operator consistent development of perturbation theory for between atom and field through the fine-structure states ˆ and is expressed in terms of the tensor polarizability αt, close-lying levels. Suppose that H0 (r) is a one-electron is calculated only in the second order of perturbation Hamiltonian of a valence electron with eigenfunctions φ theory. This is associated with the fact that, formally, nLSJ(r) and eigenvalues EnLSJ that correspond to the the fourth-order corrections to an off-diagonal element sublevels of an atomic multiplet with the principal make a correction of order F6/δ, where δ is a fine-struc- quantum number n and the quantum numbers of orbital ture splitting value at F = 0, to the expression for the L, spin S, and total J momenta (the Hamiltonian Hˆ 0 (r) fine-structure splitting δ(F) depending on the field takes into account the spin–orbit interaction).