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arXiv:2110.07363 · Physical Review A

Gauge effects in bound-bound Rydberg transition matrix elements

Alisher Duspayev, Georg A. Raithel

Published Jan 31, 2022• 2 citations
arXivHTMLPDF

Abstract

Accurate data on electric-dipole transition matrix elements (EDTMs) for bound-bound Rydberg-atom transitions have become increasingly important in science and technology. Here we compute radial EDTMs of rubidium in length, velocity, and acceleration forms for electric-dipole-allowed transitions between states with principal and angular-momentum quantum numbers $n$ and $\ensuremath{\ell}$ ranging from 15 to 100. Wave functions are computed based upon model potentials from [Phys. Rev. A 49, 982 (1994)]. Length-gauge EDTMs of strong low-$\ensuremath{\ell}$ Rydberg transitions, often employed in research and technology, are found to deviate from the fundamentally more accurate velocity-gauge form by approximately-$n$-independent and series-dependent shifts. The shift amount peaks at about $0.34\phantom{\rule{4pt}{0ex}}e{a}_{0}$ for the Rb $nD\ensuremath{\leftrightarrow}\phantom{\rule{4pt}{0ex}}(n+1)P$ series, a particularly strong Rydberg-transition series in Rb. The shift corresponds to relative EDTM corrections of up to $\ensuremath{\approx}{10}^{\ensuremath{-}3}$, which can be of concern in high-precision applications. We discuss the physical reasons for the observed gauge differences, explain the conditions for applicability of the velocity- and length-gauge forms for different transition series, and present a decision tree of how to choose EDTMs.

DOI
10.1103/physreva.105.012825

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Accurate data on electric-dipole transition matrix elements (EDTMs) for bound-bound Rydberg-atom transitions have become increasingly important in science and technology. Here we compute radial EDTMs of rubidium in length, velocity, and acceleration forms for electric-dipole-allowed transitions between states with principal and angular-momentum quantum numbers nnn and \ensuremathℓ\ensuremath{\ell}\ensuremathℓ ranging from 15 to 100. Wave functions are computed based upon model potentials from [Phys. Rev. A 49, 982 (1994)]. Length-gauge EDTMs of strong low-\ensuremathℓ\ensuremath{\ell}\ensuremathℓ Rydberg transitions, often employed in research and technology, are found to deviate from the fundamentally more accurate velocity-gauge form by approximately-nnn-independent and series-dependent shifts. The shift amount peaks at about 0.34ea00.34\phantom{\rule{4pt}{0ex}}e{a}_{0}0.34ea0​ for the Rb nD\ensuremath↔(n+1)PnD\ensuremath{\leftrightarrow}\phantom{\rule{4pt}{0ex}}(n+1)PnD\ensuremath↔(n+1)P series, a particularly strong Rydberg-transition series in Rb. The shift corresponds to relative EDTM corrections of up to \ensuremath≈10\ensuremath−3\ensuremath{\approx}{10}^{\ensuremath{-}3}\ensuremath≈10\ensuremath−3, which can be of concern in high-precision applications. We discuss the physical reasons for the observed gauge differences, explain the conditions for applicability of the velocity- and length-gauge forms for different transition series, and present a decision tree of how to choose EDTMs.
Microwave electric-field measurements with Rydberg atoms require accurate Rydberg-transition electric-dipole matrix elements. We calculate these matrix elements for Rb and Cs using a Dirac-Fock plus core-polarization method, which core-valence exchange is treated explicitly and core-valence correlation is represented by a cutoff core-polarization potential. A smooth cutoff-radius model fitted to quantum-defect energies reproduces the target energies with MHz-level residuals for states up to n=90n=90n=90. Comparisons with model-potential matrix elements and internal length-velocity consistency show that the sss-ppp transitions are generally the most robust, whereas the ppp-ddd and ddd-fff transitions show stronger atom- and branch-dependent sensitivity to short-range modeling. The largest relative differences arise from radial-integral cancellation, where short-range phase changes are amplified by the small final matrix element. These results provide a practical reliability assessment for Rydberg-transition electric-dipole matrix elements used in field-sensing applications.
The radial electric dipole moments of bound-bound, bound-free, and free-free transitions of high-lying (n>15)(n>15)(n>15) Rydberg states and low-lying continuum states $(\ensuremath{\epsilon}<400$ ${\mathrm{cm}}^{\ensuremath{-}1})$ are calculated for hydrogen and the alkali-metal atoms. All dipole matrix elements for the transitions between low-lll (s(s(s, ppp, ddd, and f)f)f) states of H, Li, and Rb are presented. Results of similar calculations for the other alkali-metal atoms (Na, K, and Cs) are summarized. These numerical results, using quantum defect theory, are compared with semiclassical approximations. The dipole matrix elements cross zero at certain combinations of initial and final states. The matrix elements at these so-called Cooper minima in Li, K, Rb, and Cs are presented in detail.
Abstract We report the observation and quantitative analysis of Autler–Townes splitting in rubidium atoms excited to Rydberg states using a four-level ladder-type electromagnetically induced transparency (EIT) scheme and resonant radio-frequency (RF) fields. By fitting the EIT spectra to Voigt profiles and calibrating the frequency axis with a semi-empirical Rydberg–Ritz model, we extracted transition dipole matrix elements (TDMEs) for the n D 5 / 2 → ( n + 1 ) P 3 / 2 transitions across five principal quantum numbers: n = 53, 55, 68, 75, and 85. Comparison with theoretical predictions revealed systematic deviations in the Rabi frequencies, attributed to RF delivery inefficiencies and standing-wave enhancement within the vapor cell. To address these effects, we developed two complementary approaches: an empirical quartic fit to measured efficiencies and a physics-based model, capturing both transverse radial variation and dominant longitudinal interference along the cell axis. These models explain the observed scaling behavior, predict an efficiency peak near n ≈ 60, and reduce TDME discrepancies to less than 5% at the validation point. Our analysis highlights the influence of spatial field distribution and cell geometry on precision atom–field interaction measurements and establishes a robust experimental framework for TDME extraction in ladder-type EIT systems.
Ponderomotive energy shifts experienced by Rydberg atoms in optical fields are known to be well approximated by the classical quiver energy of a free electron. We examine such energy shifts quantum mechanically and elucidate how they relate to the ponderomotive shift of a free electron in off-resonant fields. We derive and evaluate corrections to the ponderomotive free-electron polarizability in the length and velocity (transverse or Coulomb) gauges, which agree exactly as mandated by the gauge invariance. We also show how the free electron value emerges from the Dirac equation through summation over the Dirac sea states. We find that the free-electron ac Stark shift comes as an expectation value of a term proportional to the square of the vector potential in the velocity gauge. On the other hand, the same dominant contribution can be obtained to first order via a series expansion of the exact energy shift from the second-order perturbation theory in the length gauge. Finally, we numerically examine the validity of the free-electron approximation. The correction to the free-electron value becomes smaller with increasing principal quantum number, and it is well below a percent for 60sss states of Rb and Sr away from the resonances.
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.