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Physical Review A

On- and off-resonance radiation-atom-coupling matrix elements involving extended atomic wave functions

Yannis Komninos, Theodoros Mercouris, Cleanthes A. Nicolaides

Published Jan 24, 2014• 4 citations
Publisher

Abstract

In continuation of our earlier works, we present results concerning the computation of matrix elements of the multipolar Hamiltonian (MPH) between extended wave functions that are obtained numerically. The choice of the MPH is discussed in connection with the broader issue of the form of radiation-atom (or -molecule) interaction that is appropriate for the systematic solution of various problems of matter-radiation interaction. We derive analytic formulas, in terms of the sine-integral function and spherical Bessel functions of various orders, for the cumulative radial integrals that were obtained and calculated by Komninos, Mercouris, and Nicolaides [Phys. Rev. A 71, 023410 (2005)]. This development allows the much faster and more accurate computation of such matrix elements, a fact that enhances the efficiency with which the time-dependent Schr\"odinger equation is solved nonperturbatively, in the framework of the state-specific expansion approach. The formulas are applicable to the general case where a pair of orbitals with angular parts $|{\ensuremath{\ell}}_{1},\phantom{\rule{1.0pt}{0ex}}\phantom{\rule{1.0pt}{0ex}}{m}_{1}\ensuremath{\rangle}$ and $|{\ensuremath{\ell}}_{2},\phantom{\rule{1.0pt}{0ex}}\phantom{\rule{1.0pt}{0ex}}{m}_{2}\ensuremath{\rangle}$ are coupled radiatively. As a test case, we calculate the matrix elements of the electric field and of the paramagnetic operators for on- and off-resonance transitions, between hydrogenic circular states of high angular momentum, whose quantum numbers are chosen so as to satisfy electric dipole and electric quadrupole selection rules. Because of the nature of their wave function (they are nodeless and the large centrifugal barrier keeps their overwhelming part at large distances from the nucleus), the validity of the electric dipole approximation in various applications where the off-resonance couplings must be considered becomes precarious. For example, for the transition from the circular state with $n$ = 20 to that with $n$ = 21, for which $\ensuremath{\langle}r\ensuremath{\rangle}\ensuremath{\approx}400$ a.u., the dipole approximation starts to fail already at XUV wavelengths ($\ensuremath{\lambda}<125\phantom{\rule{0.28em}{0ex}}\mathrm{nm}$).

DOI
10.1103/physreva.89.013420

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In view of the possibilities of laser spectroscopy with very short pulses and relatively high frequencies, we examine aspects of the theory of atom-field interactions that are related to violation of the condition of the long-wavelength approximation (LWA) according to which kr⪡1kr⪡1kr⪡1, where rrr is ``of the order of atomic dimensions'' and kkk is the magnitude of the wave vector. On- and off-resonance transitions are considered, with krkrkr being larger than unity due to the large extent of the two wave functions involved in the coupling matrix element. The implementation of the analysis uses bound-free transition matrix elements with the n=50n=50n=50 and 80 hydrogenic functions as initial states and values of kkk up to 27eV27\phantom{\rule{0.3em}{0ex}}\mathrm{eV}27eV, which is sufficient to produce kr>1kr>1kr>1, thereby rendering the LWA inoperative. In spite of this, it is shown that, for on-resonance transitions, the results from the use of the well-known multipole operators resulting from the LWA [e.g., the electric dipole approximation (EDA)] agree with those from the exact expressions derived here from the application of the multipolar Hamiltonian. This numerical agreement is proven analytically. As a test of the kind of convergence of the multipole series expansion for small values of kkk and large rrr, it is shown that the lowest-order ratio of electric dipole to quadrupole matrix elements decreases rapidly within 1.0 atomic unit above threshold. Finally, it is shown that off-resonance couplings lead to differences between the full-interaction operator and the EDA which cannot be neglected. In the extreme case of intrashell couplings for the n=50n=50n=50 shell, calculation shows that the 50p50p50p state is coupled to angular momentum states up to $\ensuremath{\ell}=21$, compared to the electric dipole coupling of $\ensuremath{\Delta}\ensuremath{\ell}=\ifmmode\pm\else\textpm\fi{}1$.
We develop practical formulas for the calculation of the matrix elements of the interaction of the electromagnetic field with an atomic state, beyond the long-wavelength approximation. The atom-plus-field Hamiltonian is chosen to have the multipolar form, containing the electric, paramagnetic, and diamagnetic operators. The final workable expressions include the interactions to all orders and are derived by first expanding the fields in partial waves. The electric-field operator reaches a constant value as the radial variable becomes large, contrary to the result of the electric-dipole approximation (EDA) where the value of the corresponding operator increases indefinitely. Applications are given for Rydberg states of hydrogen up to n=50n=50n=50 and for free-free transitions in a Coulomb potential. Such matrix elements are relevant to a number of real and virtual processes occurring during laser-atom interactions. The computation is done numerically, using a combination of analytic with numerical techniques. By comparing the results of the EDA with those of the exact treatment, it is shown that the former is inadequate in such cases. This finding has repercussions on the theory and understanding of the physics of quantum systems in high-lying Rydberg levels and wave packets or in scattering states.
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We investigate a controlled beyond-electric-dipole extension of the quantum Rabi model in which the leading electric-dipole interaction is supplemented by an electric-quadrupole contribution. A central issue is the parity selection rule: for two exact atomic parity eigenstates of opposite parity, an electric-quadrupole transition matrix element vanishes. We therefore do not interpret the additional longitudinal coupling as an E2 transition between an opposite-parity pair. Instead, we consider an inversion-asymmetric (or parity-mixed) two-level system for which the two retained states may have different diagonal quadrupole moments. Projection of the multipolar interaction then produces a state-dependent term proportional to σz(a + a†), while the ordinary transition dipole produces the transverse term proportional to σx(a + a†). A state-independent quadrupolar contribution is removed by a bosonic displacement, leaving the effective Hamiltonian H =ωa†a+∆σz/2+gσx(a+a†)+gQσz(a+a†). This construction connects the beyond-dipole physical problem to the well-studied class of Rabi models with simultaneous transverse and longitudinal coupling, including asymmetric or biased Rabi models. We make this relation explicit by a constant spin rotation, which maps the model to a biased Rabi Hamiltonian with effective coupling κ = √g2 +g2Q and bias δ = ∆gQ/(2κ). We then derive the corresponding second-order Bargmann differential equation and solve its exceptional polynomial sector using a Zhang-type functional Bethe ansatz. The resulting Bethe equations, root-sum constraint, exceptional energy branches, and the first nontrivial parameter condition are obtained explicitly. In the limit gQ → 0, the construction continuously recovers the ordinary Rabi exceptional sector. From an experimental perspective, the connection with demonstrated quadrupolar light--matter coupling in engineered quantum systems suggests that the present framework may have relevance for spectroscopic studies of beyond-electric-dipole interactions, although a direct experimental realization of the specific effective model considered here remains to be established [20].
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By use of the linear-response-vectors method, the Kramers-Heisenberg matrix elements for arbitrary bound-bound transitions of a nonrelativistic hydrogenlike atom have been evaluated. To illustrate the discussion, Rayleigh and Raman scattering have been selected as relevant examples of second-order processes. We have derived, in a unitary manner, A- and \ensuremath{\varphi}-gauge compact analytic formulas, both for angular momentum and Stark states. Our results are expressed in terms of two new generalized hypergeometric functions with four variables, 1FE_{1}\mathrm{F}_{\mathrm{E}}1​FE​ and 2FE_{2}\mathrm{F}_{\mathrm{E}}2​FE​, which are finite sums of Gauss hypergeometric functions. Special attention is given to the two-photon transitions involving the atomic ground state. For angular momentum states, we mention the connection that we have established between our general results and earlier ones.