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Physical Review A

Semiclassical calculations on one-, two-, three-, and four-photon absorption in truncated models of the hydrogen atom

W. R. Salzman

Published Oct 1, 1977• 19 citations
Publisher

Abstract

The results of semiclassical calculations on one-, two-, three-, and four-photon absorption in truncated models of the hydrogen atom are presented. The temporal behavior near "resonance" is confirmed to be Rabi-like, with a transition probability given by $P(t)={\frac{{\ensuremath{\gamma}}^{2}}{[{(\frac{{\ensuremath{\omega}}_{0}}{N}+\ensuremath{\delta}\ensuremath{-}\ensuremath{\omega})}^{2}+{\ensuremath{\gamma}}^{2}]}}{sin}^{2}{(\frac{\mathrm{Nt}}{2}){[{(\frac{{\ensuremath{\omega}}_{0}}{N}+\ensuremath{\delta}\ensuremath{-}\ensuremath{\omega})}^{2}+{\ensuremath{\gamma}}^{2}]}^{\frac{1}{2}}},$ where the resonance shift $\ensuremath{\delta}$ is proportional to the intensity of the monochromatic driving field with frequency $\ensuremath{\omega}$ and the power broadening $\ensuremath{\gamma}$ is proportional to the $\frac{N}{2}$ power of intensity. $N$ is the number of photons involved and $\mathrm{h}{\ensuremath{\omega}}_{0}$ is the energy separation between initial and final states. Calculated values of $\ensuremath{\delta}$ and $\ensuremath{\gamma}$ are given for several models; also, for the two-photon $1s\ensuremath{\rightarrow}2s$ absorption, $\ensuremath{\delta}$ and $\ensuremath{\gamma}$ are given which include the contributions of all bound $p$ states.

DOI
10.1103/physreva.16.1552

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The results of semiclassical calculations on one-, two-, three-, and four-photon absorption in truncated models of the hydrogen atom are presented. The temporal behavior near "resonance" is confirmed to be Rabi-like, with a transition probability given by P(t)=\ensuremathγ2[(\ensuremathω0N+\ensuremathδ\ensuremath−\ensuremathω)2+\ensuremathγ2]sin2(Nt2)[(\ensuremathω0N+\ensuremathδ\ensuremath−\ensuremathω)2+\ensuremathγ2]12,P(t)={\frac{{\ensuremath{\gamma}}^{2}}{[{(\frac{{\ensuremath{\omega}}_{0}}{N}+\ensuremath{\delta}\ensuremath{-}\ensuremath{\omega})}^{2}+{\ensuremath{\gamma}}^{2}]}}{sin}^{2}{(\frac{\mathrm{Nt}}{2}){[{(\frac{{\ensuremath{\omega}}_{0}}{N}+\ensuremath{\delta}\ensuremath{-}\ensuremath{\omega})}^{2}+{\ensuremath{\gamma}}^{2}]}^{\frac{1}{2}}},P(t)=[(N\ensuremathω0​​+\ensuremathδ\ensuremath−\ensuremathω)2+\ensuremathγ2]\ensuremathγ2​sin2(2Nt​)[(N\ensuremathω0​​+\ensuremathδ\ensuremath−\ensuremathω)2+\ensuremathγ2]21​, where the resonance shift \ensuremathδ\ensuremath{\delta}\ensuremathδ is proportional to the intensity of the monochromatic driving field with frequency \ensuremathω\ensuremath{\omega}\ensuremathω and the power broadening \ensuremathγ\ensuremath{\gamma}\ensuremathγ is proportional to the N2\frac{N}{2}2N​ power of intensity. NNN is the number of photons involved and h\ensuremathω0\mathrm{h}{\ensuremath{\omega}}_{0}h\ensuremathω0​ is the energy separation between initial and final states. Calculated values of \ensuremathδ\ensuremath{\delta}\ensuremathδ and \ensuremathγ\ensuremath{\gamma}\ensuremathγ are given for several models; also, for the two-photon 1s\ensuremath→2s1s\ensuremath{\rightarrow}2s1s\ensuremath→2s absorption, \ensuremathδ\ensuremath{\delta}\ensuremathδ and \ensuremathγ\ensuremath{\gamma}\ensuremathγ are given which include the contributions of all bound ppp states.
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).
The two-photon transition rate for hydrogen is evaluated with the $\stackrel{\ensuremath{\rightarrow}}{\mathrm{E}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}$ and the $\stackrel{\ensuremath{\rightarrow}}{\mathrm{A}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}$ interactions including the continuum. We find that the $\stackrel{\ensuremath{\rightarrow}}{\mathrm{E}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}$ interaction approaches the final result with a small number of intermediate states and consequently is the one to be used in any approximate calculation. Numerical values of the two-photon absorption coefficient as a function of frequency are given and the resonance enhancement is exhibited.
The resonantly enhanced multiphoton ionization of atomic hydrogen is studied for the case of four-photon ionization (three-photon resonant). The time-dependent probabilities, the shape of the resonance profile at different intensities, and the nonlinear index are computed for coherent light at frequencies around the resonance (365 nm). The systematic dependence of the cross section as a function of the principal quantum number of the resonant state (as this tends towards the Rydberg states) is investigated at zero detuning. The probabilities of absorption of one photon more than necessary for the four-photon ionization process (resonantly enhanced above-threshold ionization) is computed and the relevant branching ratios are examined. Then it is shown that the effect of the spatio-temporal structures of the laser pulse is to red shift the maximum of the resonance profile, whereas that of an ideal chaotic light is to blue shift and broaden the resonance peak. The shape, shift, and width of the profile resulting from both these effects are in very fair agreement with the experimental ones.
The coherent interaction of two smoothly varying, near-resonant, two-photon pulses with a three-level system can be described by "two-photon damped Bloch equations" which are analogous to those for a one-photon transition in a two-level system except for the presence of a two-photon coupling and a frequency shift. These equations are solved for the cases ${\ensuremath{\gamma}}_{1}$, ${\ensuremath{\gamma}}_{2}\ensuremath{\ll}\ensuremath{\Omega}$, ${\ensuremath{\gamma}}_{1}={\ensuremath{\gamma}}_{2}$, and $\frac{{\ensuremath{\gamma}}_{2}{k}^{2}{\ensuremath{\epsilon}}^{4}}{{\ensuremath{\Omega}}^{2}}$, ${\ensuremath{\gamma}}_{1}\ensuremath{\ll}\ensuremath{\Omega}$, where ${\ensuremath{\gamma}}_{1}$ and ${\ensuremath{\gamma}}_{2}$ are the atomic energy and phase relaxation widths, respectively, and $\ensuremath{\Omega}$ is the Rabi frequency. The leading contribution to the refractive index is intensity dependent, caused by the level shifts inherent in multiphoton processes; it includes a relaxation dependent part which is important at times shorter than ${\ensuremath{\gamma}}_{1}^{\ensuremath{-}1}$. The second-order contributions depend on the square of the intensity and the time-integrated square of the intensity. The latter contribution, which is relaxation dependent, causes line asymmetry at the long-wavelength wing; it consists of a term proportional to ${\ensuremath{\gamma}}_{2}\ensuremath{-}{\ensuremath{\gamma}}_{1}$ and only important at early times and a term proportional to $2{\ensuremath{\gamma}}_{2}\ensuremath{-}{\ensuremath{\gamma}}_{1}$.
We have measured the resonance-enhanced four-photon ionization of atomic hydrogen. The degenerate four-photon process occurs via a three-photon resonance between the 1s and the 2p levels, with subsequent one-photon ionization near threshold. We have studied the highly asymmetric resonance-enhanced profile, i.e., the photoion yield as a function of laser detuning from three-photon resonance between the 1s and 2p levels. In particular, we have determined the width, shift, peak, and asymmetry of the profile as a function of laser intensity. The experimental results are compared to theoretical models. These models involve both the properties of the atom in an intense near-resonant radiative field, and a detailed model for the multimode laser field, particularly the field fluctuations due to mode beating. The asymmetric resonance-enhanced photoionization profile is well reproduced by both a random-phase and a chaotic-light model.