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Physical Review A

Long-wavelength approximation in on- and off-resonance transitions

Yannis Komninos, Theodoros Mercouris, Cleanthes A. Nicolaides

Published Feb 24, 2005• 6 citations
Publisher

Abstract

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 $kr⪡1$, where $r$ is ``of the order of atomic dimensions'' and $k$ is the magnitude of the wave vector. On- and off-resonance transitions are considered, with $kr$ 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 $n=50$ and 80 hydrogenic functions as initial states and values of $k$ up to $27\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$, which is sufficient to produce $kr>1$, 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 $k$ and large $r$, 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 $n=50$ shell, calculation shows that the $50p$ state is coupled to angular momentum states up to $\ensuremath{\ell}=21$, compared to the electric dipole coupling of $\ensuremath{\Delta}\ensuremath{\ell}=\ifmmode\pm\else\textpm\fi{}1$.

DOI
10.1103/physreva.71.023410

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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 \ensuremathℓ=21\ensuremath{\ell}=21\ensuremathℓ=21, compared to the electric dipole coupling of \ensuremathΔ\ensuremathℓ=\ifmmode±\else\textpm\fi1\ensuremath{\Delta}\ensuremath{\ell}=\ifmmode\pm\else\textpm\fi{}1\ensuremathΔ\ensuremathℓ=\ifmmode±\else\textpm\fi1.
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.
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 nnn = 20 to that with nnn = 21, for which $\ensuremath{\langle}r\ensuremath{\rangle}\ensuremath{\approx}400$ a.u., the dipole approximation starts to fail already at XUV wavelengths ($\ensuremath{\lambda}<125\phantom{\rule{0.28em}{0ex}}\mathrm{nm}$).
The subject of this chapter is the quantum mechanical analysis of the interaction of electromagnetic radiation with atomic transitions. The analysis is based on the Schrödinger wave equation, and in the first section, the gauge-invariant form of the external electromagnetic field is introduced. The electric dipole interaction and the long-wavelength approximation for the analysis of this interaction are discussed. The perturbative analysis of both single-photon and two-photon electric dipole interactions is presented, and density matrix analysis is introduced. The interaction of radiation with the resonances of atomic hydrogen is then discussed. The analysis is performed for both coupled and uncoupled representations. In the last section of the chapter, the radiative interactions for multielectron atoms are discussed. The Wigner–Eckart theorem and selection rules for transitions between levels characterized by coupling are developed. The effect of hyperfine splitting on radiative transitions is also briefly discussed.
Using a method based on the Magnus expansion, the survival of Rydberg atoms in intense laser fields is studied. Usually, for atoms in intense laser fields, effects beyond the electric dipole approximation need to be considered. In the past few years, researchers have primarily paid attention to the nondipole effects induced by the electric quadrupole term of the optical field. The quadrupole term plays a more important role than the electric dipole term in the calculation of the survival probability of Rydberg atoms in intense laser fields even though the magnitude of the former is smaller than that of the latter in terms of the multipole expansion of the optical vector potential. Meanwhile, the effects of higher-order corrections beyond the quadrupole term remain ambiguous. In this paper, we confirm that corrections containing higher-order terms, except for the quadrupole term, can be neglected in the calculation of the transitions of Rydberg atoms in external fields. We then use the second-order time evolution operator containing the quadrupole term to study the scaling law of the transition probability of Rydberg He in laser fields with different parameters.
Extensive calculations of the electric-dipole matrix elements in alkali-metal atoms are conducted using the relativistic all-order method. This approach is a linearized version of the coupled-cluster method, which sums infinite sets of many-body perturbation theory terms. All allowed transitions between the lowest nsnsns, np1∕2n{p}_{1∕2}np1∕2​, np3∕2n{p}_{3∕2}np3∕2​ states and a large number of excited states are considered in these calculations and their accuracy is evaluated. The resulting electric-dipole matrix elements are used for the high-precision calculation of frequency-dependent polarizabilities of the excited states of alkali-metal atoms. We find ``magic'' wavelengths in alkali-metal atoms for which the nsnsns and np1∕2n{p}_{1∕2}np1∕2​ and np3∕2n{p}_{3∕2}np3∕2​ atomic levels have the same ac Stark shifts, which facilitates state-insensitive optical cooling and trapping.
Numerical calculations are presented for the interference spectra of a laser-excited three-level atom, where the strong and the weak atomic transitions are driven by resonant and nonresonant laser fields, respectively. The spectral functions describing the interference spectra for the electric dipole allowed excited state have been considered in the low- and high-intensity limit of the laser field operating in the strong transition. The interference spectra arise from the competition between short-lifetime spontaneous processes and short- and long-lifetime excitations induced by the strong and the weak laser fields, respectively. Both laser fields have been treated as quantized and as classical entities. The computed spectra have been presented graphically for different values of the Rabi frequencies and detunings of the weak laser field. It is shown that the decrease in the intensity of the short-lifetime excitation may provide a measure of the spectral width of the long-lifetime excitation.