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arXiv:physics/9805022 · Journal of Physics B Atomic Molecular and Optical Physics

Resonant three-photon ionization of hydrogenic atoms by a non-monochromatic laser field

V Yakhontov, Robin Santra, Klaus Peter Jungmann

Published Jan 1, 1999• 13 citations
arXivHTMLPDF

Abstract

We present ionization probability and lineshape calculations for the two-step three-photon ionization process, , of the ground state of hydrogenic atoms in a non-monochromatic laser field with a time-dependent amplitude. Within the framework of a three-level model, the AC Stark shifts and non-zero ionization rates of all states involved were taken into account, together with spatial and temporal inhomogeneities of the laser signal. In contrast with the usual perturbative technique, the time evolution of the atomic states was simulated by directly solving the system of coupled time-dependent inhomogeneous differential equations numerically, the equations being equivalent to the appropriate non-stationary Schrödinger equation. Particular numerical results were obtained for typical parameters of the pulsed laser field that are employed in a new experiment to measure the 1S-2S energy separation with muonium at the Rutherford Appleton Laboratory. The shifts and asymmetries of the photoionization lineshapes revealed may be of relevance for ultra-high-precision experiments in hydrogen in CW laser fields.

DOI
10.1088/0953-4075/32/7/006

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We present ionization probability and lineshape calculations for the two-step three-photon ionization process, , of the ground state of hydrogenic atoms in a non-monochromatic laser field with a time-dependent amplitude. Within the framework of a three-level model, the AC Stark shifts and non-zero ionization rates of all states involved were taken into account, together with spatial and temporal inhomogeneities of the laser signal. In contrast with the usual perturbative technique, the time evolution of the atomic states was simulated by directly solving the system of coupled time-dependent inhomogeneous differential equations numerically, the equations being equivalent to the appropriate non-stationary Schrödinger equation. Particular numerical results were obtained for typical parameters of the pulsed laser field that are employed in a new experiment to measure the 1S-2S energy separation with muonium at the Rutherford Appleton Laboratory. The shifts and asymmetries of the photoionization lineshapes revealed may be of relevance for ultra-high-precision experiments in hydrogen in CW laser fields.
Analyses of the resonant multiphoton ionization of atoms require knowledge of ac Stark energy shifts and of multiphoton, bound-to-bound state, transition amplitudes. In this paper, we consider the three-photon photoionization of hydrogen atoms at frequencies that are at and surrounding the two-photon 1s to 2s resonance. AC energy shift sums of both the 1s and 2s states are calculated as a function of the laser frequency along with two-photon 1s ? 2s resonant transition amplitude sums. These quantities are calculated using an extended version of a method, which has often been employed in a variety of ways, of calculating these sums by expressing them in terms of solutions to a variety of differential equations that are derived from the different sums being evaluated. We demonstrate how exact solutions are obtained to these differential equations, which lead to exact evaluations of the corresponding sums. A variety of different cases are analysed, some involving analytic continuation, some involving real number analysis and some involving complex number analysis. A dc Stark sum calculation of the 2s state is carried out to illustrate the case where analytic continuation, pole isolation and pole subtraction are required and where the calculation can be carried out analytically; the 2s state, ac Stark shift sum calculations involve a case where no analytic continuation is required, but where the solution to the differential equation produces complex numbers owing to the finite photoionization lifetime of the 2s state. Results from these calculations are then used to calculate three-photon ionization probabilities of relevance to an analysis of the multiphoton ionization data published by Kyrala and Nichols (1991 Phys. Rev. A 44, R1450).
By numerically solving the three-dimensional time-dependent Schr\"odinger equation, two-photon ionization of hydrogen is investigated at the near-resonant frequencies of the $1s\text{\ensuremath{-}}2p$ transition. Due to the Rabi oscillations between the 1s1s1s and 2p2p2p states, the photoelectron energy spectra exhibit the Autler-Townes (AT) doublets and we focus on the energy spacing and the asymmetry of the doublets. Our results show that the laser frequency for the minimum energy spacing of AT doublets locates at the resonant frequency of the ac-stark-shifted states, while the symmetry of the AT doublets is affected both by the ac-stark shift and the nonresonant ionization pathway. Developing the minimal three-state model including all of the nonresonant (nonessential) states, the effects of the ac-stark shift and the nonresonant ionization pathway on the AT doublets due to the nonresonant states are identified. Furthermore, due to the nonresonant ionization pathway, the photoelectron angular distributions are distinctly different for the lower- and higher-energy peaks in the AT doublets and these angular distributions sensitively depend on the laser intensity and pulse duration. These results are well reproduced by our minimal three-state model.
We derive a set of density matrix equations describing laser photo-excitation and ionization of atomic hydrogen in the presence of an external electric field. The equations are relevant as a description of multiphoton ionization of hydrogen (or deuterium) atoms in laser-produced transient plasmas. The laser wavelength is 243 nm, so that the photoionization is resonant with the 2 s 1/2 level. The electric field causes Stark mixing of the 2 s 1/2 level with the 2 p 1/2 and 2 p 3/2 levels. Because the electric field is taken in the same direction as the linear polarization of the laser, only states with equal m j are coupled, and the atom can be described with a four-level density matrix. The laser bandwidth is taken into account by using the stochastic model of a chaotic laser field introduced by Zoller [1]. We present a few calculations of the probabilities for ionization and excitation of a single hydrogen atom as a function of laser intensity and wavelength.
The two-level model of multiphoton ionization is reviewed within the context of an exact theoretical framework. Parametrization of the complex dressed states arising in a generalized Floquet analysis gives numerical values for the effective two-level parameters for two-, three-, and four-photon ionization of ground-state atomic hydrogen with the (ml=0{m}_{l}=0ml​=0) intermediate resonant states 2p2p2p, 2s2s2s, and 3p3p3p, respectively. Detailed time dependences of the ionization process are given for responses to a linearly polarized, monochromatic, single-mode laser turned on suddenly with respect to near-resonant states, but adiabatically with respect to all others. Appropriate expressions for time-independent ionization rates are obtained during specific time intervals. The KKK index ($K=\frac{\ensuremath{\partial}\mathrm{ln}N}{\ensuremath{\partial}\mathrm{ln}I}$), of interest in recent experiments, is understood within the framework of a simple model. Application to four-photon ionization of the metastable 2S12^{2}S_{\frac{1}{2}}2S21​​ state of hydrogen illustrates the breakdown of the two-level model, as the number of "near-resonant" levels is a function of the field strength.
The influence of ac Stark shifts on the three-photon ionization of cesium in a moderate-intensity laser beam is investigated. The ionization process involves a two-photon resonance with the 8d8d8d fine-structure levels. The work consists of two related parts. In the first part, atomic parameters, such as the two-photon Rabi rate and the ac Stark shift, are calculated in the nonrelativistic Hartree-Fock approximation. In the second part, the atomic parameters thus obtained are used as coefficients in a system of coupled differential equations representing the time evolution of the density matrix. Ionization line shapes are generated for a transform-limited pulsed laser of moderate intensity with a single-mode transverse structure. As the time history of the intensity profile is varied from square to smooth, the ionization line shape becomes asymmetric due to the time dependence of the ac Stark shift. The effects of varying the spatial intensity distribution of the laser light to include transverse centering and longitudinal focusing are also studied.
The process of resonant multiphoton ionization of a hydrogen atom in the ground 1 s state is studied by direct numerical integration of the nonstationary Schrödinger equation for a quantum system in an electromagnetic field. The dependence of photoionization probability on the radiation intensity is found to be nonmonotonic. It is established that the minima of ionization correspond to multiphoton resonances between the ground state and one of the excited (Rydberg) atomic states perturbed by the laser field. It is shown that ionization is suppressed due to rearrangement of Rydberg states in a strong electromagnetic field and is accompanied by efficient Raman Λ transitions, which connect a set of closely lying Rydberg states via the continuum.