Hydrogenic spectra A good starting point is photoexcitation from the ground state of H. The problem naturally divides itself into two regimes: below the energy of classical ionization limit, where the states are for all practical purposes stable against ionization, and above it where the spectrum is continuous. As an example, we consider first the excitation of the n = 15 Stark states from the ground state in a field too low to cause significant ionization of n = 15 states. From Chapter 6 we know the energies of the Stark states, and we now wish to calculate the relative intensities of the transitions to these levels. One approach is to calculate them in parabolic coordinates. This approach is an efficient way to proceed for the excitation of H; however, it is not easily generalized to other atoms. Another, which we adopt here, is to express the n = 15 nn 1 n 2 m Stark states in terms of their nℓm components using Eqs. (6.18) or (6.19) and express the transition dipole moments in terms of the more familiar spherical n ℓ m states. In the excitation of the Stark states of principal quantum number n from the ground state only p state components are accessible via dipole transitions, so the relative intensities for light polarized parallel and perpendicular to the static field, π and σ polarizations, are proportional to the squared transformation coefficients |〈 nn 1 n 2 m | n ℓ m 〉| 2 from the nn 1 n 2 m parabolic states to the n ℓ m states for ℓ = 1 and m = 0 and 1. In Fig. 8.1 we show the relative intensities by means of the squared transformation coefficients |〈15 n 1 n 2 m |15 p m 〉| 2 for m = 0 and 1.
The Stark spectroscopy of the hydrogen atom has been investigated near the ionization threshold in static homogeneous external electric fields from \ensuremath{\sim}1000 to \ensuremath{\sim}8000 V/cm with use of a two-step excitation H(n=1)+\ensuremath{\Elzxh}${\ensuremath{\omega}}_{\mathrm{vuv}\ensuremath{\rightarrow}}$H(n=2)+\ensuremath{\Elzxh}\ensuremath{\omega} $_{\mathrm{uv}\ensuremath{\rightarrow}}$
H∗ with vuv and uv laser radiation, independently tunable and linearly polarized either parallel (\ensuremath{\pi}) or perpendicular (\ensuremath{\sigma}) with respect to the external field. Different from other, nonhydrogenic atoms, the hydrogen atom is a peculiar case insofar as the initial states of H(n=2), from which excitation takes place, are essentially of parabolic character at the field strengths employed in this study.Individual sublevels (
n1,
n2,\ensuremath{\Vert}
ml\ensuremath{\Vert}) have been prepared in the n=2 state with sub-Doppler resolution. Photoionization spectra have been obtained with \ensuremath{\pi}- and \ensuremath{\sigma}-polarized uv radiation from the four possible sublevels of the n=2 Stark manifold: \ensuremath{\Vert}1,0,0〉 \ensuremath{\Vert}\ifmmode\pm\else\textpm\fi{}1/2〉, \ensuremath{\Vert}0,1,0〉 \ensuremath{\Vert}\ifmmode\pm\else\textpm\fi{}1/2〉, \ensuremath{\Vert}0,0,\ifmmode\pm\else\textpm\fi{}1〉 \ensuremath{\Vert}\ifmmode\pm\else\textpm\fi{}1/2〉, and \ensuremath{\Vert}0,0,\ensuremath{\mp}1〉 \ensuremath{\Vert}\ifmmode\pm\else\textpm\fi{}1/2〉. All spectra show sharp line structures of quasistable field-ionizing states superimposed on continua in the energy region between the classical saddle point, ${E}_{\mathrm{SP}=\mathrm{\ensuremath{-}}2\phantom{\rule{0ex}{0ex}}\mathrm{\ensuremath{\surd}}\mathrm{F}}$ a.u., and the zero-field ionization threshold E=0 a.u. Employing existing theoretical treatments, the quasistable spectral structures are quantitatively completely analyzed and accounted for, as long as their field ionization rate is less than \ensuremath{\sim}5\ifmmode\times\else\texttimes\fi{}
1011 ${\mathrm{sec}}^{\mathrm{\ensuremath{-}}1}$. Field-ionization rates of quasistable states have been measured and are compared with theory. Oscillatory resonance structures of the ionization cross section near the zero-field threshold are observed with pronounced modulation degree only in the spectrum from the \ensuremath{\Vert}1,0,0〉 \ensuremath{\Vert}\ifmmode\pm\else\textpm\fi{}1/2〉 sublevel excited with the uv \ensuremath{\pi}-polarized. The energy spacing of the oscillations has been measured as a function of the electric field strength F and found to be nearly proportional to
F3/4, with slight but definite deviations. Theoretical analysis of the oscillations yields quantitative agreement with the experiment.
We have measured certain ratios of transition intensities between excited hydrogen Stark sublevels over a range of electric-field values.
CO2 lasers were used to excite a fast beam of hydrogen atoms. The \ensuremath{\pi}-polarized transitions studied were from extremal Stark sublevels of principal quantum number n=10 to extremal and next extremal sublevels of n=30 and 44, at fields from 2 to 700 V/cm. The observed ratios are compared with those obtained using a non-relativistic zero-field formula attributed to Gordon [Ann. Phys. (Leipzig 2, 1031 (1929)] and with calculations employing matrix diagonalization including fine structure. Even for those relatively high n values, we find at low field significant deviations from results from Gordon's formula due to fine structure in the lower state as well as deviations at high field due to n mixing in the upper state induced by the static electric field. The experimental results and the matrix diagonalization calculations agree to within a few percent, which is the order of magnitude of possible saturation effects.
On the basis of the generalized Sturm expansion of the radial part of the Coulomb Green function, a computational method is proposed and numerical results are presented for the dynamic hyperpolarizability γ and the corrections E (4) (quadratic in the light intensity) to the quasi-energy of the ground and excited states of hydrogen with principal quantum numbers n≤5 in a monochromatic light field. In this approach, the problem is reduced to the summation of well-convergent double series of the hypergeometric kind, which ensures reliable numerical results both for states with a large n, and in a wide range of field frequencies ω, including the above-threshold frequency range of ℏω≪|E n | (|E n | is the ionization potential of the state |nlm〉 under investigation). We consider the frequency dependence of γ and E (4), their differences for the cases of linear and circular polarizations of the field, and the relation between their real and imaginary parts, which determine the laser field-induced corrections to the position and width of energy levels. For n=5, the significant role of mixing the |nlm〉 states with different values of l by a laser field in the region of resonances on intermediate bound states is demonstrated. The linear (in intensity) corrections to the photoionization cross section for excited states are analyzed and the threshold intensity corresponding to the onset of atomic level stabilization is estimated for a number of states with n=3 and n=5.
The Stark splitting of the second excited level of atomic hydrogen (with principal quantum number
n=3) has been measured by pulsed laser spectroscopy in order to determine small electric fields by optical means. Doppler-free excitation with two counterpropagating laser beams at
205nm allows us to measure the
n=3 Stark spectra either by an optogalvanic signal after ionization by a third laser photon or by detection of the Balmer-$\ensuremath{\alpha}$ fluorescence photons emitted by the excited atoms. Three different cases of laser polarization have been investigated in order to evaluate the possibilities and limitations of this one-step laser excitation method for electric field determination, especially at field strengths below
200V∕cm. The method provides the best field sensitivity achieved with lower excited levels that can be reached directly by two-photon absorption from the ground state (in contrast to the more sensitive Rydberg levels). For this purpose, an advanced tunable pulsed uv laser with narrow bandwidth almost at the Fourier limit has been used. The high spectral resolution allowed us to deduce electric field strengths as small as
50V∕cm from the frequency shift between two selected isolated Stark components. As a main advantage, this simple analysis of the measured spectra does not rely on knowledge of the line intensities which may be strongly affected by the measuring conditions.
General equations are presented for determining the field-dependent variation of the wavefunctions, matrix elements and intensities of dipole transitions between Stark levels of a hydrogen atom in a field range where the fine-structure effects may be neglected. The Sturm-series resolution for the reduced Coulomb-Green function is used to derive analytical expressions for the first-order corrections in terms of parabolic quantum numbers of initial and final states. Simple forms of these expressions as functions of the upper-state quantum numbers are presented explicitly for the Lyman and Balmer series. The selective action of the dc electric field on the intensity of Stark components of hydrogen lines is revealed in numerical calculations for both and types of emitted radiation. This selectivity may cause, in particular, enhancement of the high-frequency and damping of the low-frequency Stark -lines of the Lyman series.
We present calculations of the photoionization spectrum of excited hydrogen using \ensuremath{\pi} polarization in the presence of a strong electric field as a function of the spherical (applicable to complex atoms) and parabolic quantum numbers of the excited state. We also present corresponding measurements of the photoionization yield from the individual parabolic states of n=2. Both the calculations and the measurements show an enhancement of the depth of the so-called ``Stark-induced modulation'' in the region E\ensuremath{\ge}0 when the initial excited state is a pure
ml=0 blue state, and disappear almost completely when the initial state is a pure
ml=0 red state. These results are understood using arguments based on the fact that the charge distribution of the Stark-induced states is tremendously extended up field. Because of the excellent signal to noise ratio of the enhanced modulations, we were able to measure the field dependence of the spacings with sufficient accuracy to confirm the (3/4) power law and rule out the recently suggested (2/3) power law.