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Physical Review A

Exactly solvable two-level model with time-dependent coupling, detuning, and loss: Application to strong-field resonance ionization

Ayoub Ait Elarabi, András Csehi

Published Jun 2, 2025• 7 citations
Publisher

Abstract

We present an exactly solvable two-level model with smoothly varying coupling, detuning, and loss functions. Whenever the common temporal profile of these parameter functions is analytically integrable, the complex population amplitudes are found for all times in simple analytic forms. This makes the model suitable for studying the population and phase dynamics of resonance-enhanced ionization processes. Incorporating two-photon Rabi frequency, dynamical Stark shifts, and one-photon ionization rate into this model, the 2 + 1 photon ionization of atomic lithium is studied as a showcase example. Finally, when the intensity profile of the laser pulse has the sine-squared shape, we find the exact analytic expression of the photoelectron spectrum, allowing additional insight into the structure of the Autler-Townes doublet.

DOI
10.1103/ws7n-3z78

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arXiv:0707.3111•Journal of Physics B Atomic Molecular and Optical Physics•Oct 17, 2007•31 citations
We present an exactly solvable two-level model with smoothly varying coupling, detuning, and loss functions. Whenever the common temporal profile of these parameter functions is analytically integrable, the complex population amplitudes are found for all times in simple analytic forms. This makes the model suitable for studying the population and phase dynamics of resonance-enhanced ionization processes. Incorporating two-photon Rabi frequency, dynamical Stark shifts, and one-photon ionization rate into this model, the 2 + 1 photon ionization of atomic lithium is studied as a showcase example. Finally, when the intensity profile of the laser pulse has the sine-squared shape, we find the exact analytic expression of the photoelectron spectrum, allowing additional insight into the structure of the Autler-Townes doublet.
The energy levels of atoms and molecules exposed to intense, short, high-frequency laser pulses undergo time-dependent distortion known as dynamic Stark shift (DSS). These level shifts are induced by the coupling with nonresonant (nonessential) states and follow the temporal intensity profile of the laser pulse. Owing to the different DSSs of the individual atomic levels, transient resonance suppressions and enhancements significantly modify the multiphoton ionization pathway, leaving clear fingerprints (asymmetry, splitting, and shifting) in the energy spectrum of emitted photoelectrons. Here we investigate this phenomenon by solving the time-dependent Schr\"odinger equation of the valence electron of Na and Li in increasing levels of complexity: (i) developing minimal models for two-photon transition and (2+12+12+1)-photon ionization, (ii) applying a spectral method without continuum-continuum couplings, and (iii) propagating the electron wave packet on a large spatial grid accounting for all possible couplings. We show that appropriately detuned transform limited pulses provide a high level of control in strong-field excitation, when the atomic levels are dynamically shifted. Furthermore, we demonstrate the role of DSSs in the multipeak structure of the Autler-Townes doublet, found when several Rabi oscillations are induced during the strong-field resonant ionization.
We study the resonance-enhanced multiphoton ionization (REMPI) of atoms by intense, short, XUV radiation fields. To characterize Rabi oscillations and the resulting Autler-Townes doublet in the photoelectron spectrum, a minimal-state effective description is utilized, incorporating energy-level shifts, multiphoton couplings, and irreversible losses. Extending the semiclassical Rabi model with such higher-order terms, the complex population amplitudes of the Rabi cycling atom are found in simple analytic forms. This allows for the exact analytic evaluation of the photoelectron spectrum under various simultaneous physical effects. The origins of the spectral characteristics, such as the splitting, shifting, asymmetry, and multipeak pattern, are identified through the found spectrum formula. Furthermore, the dependence of these features on the field intensity, frequency detuning, and interaction time are analyzed in detail. Finally, as the simplicity of the spectrum formula is preserved when further higher-order light-matter interaction terms are added, the presented general framework is suitable for studying arbitrary ( n + m ) photon REMPI phenomena.
We present a simple model for calculating strong-field atomic and molecular ionization dominated by Freeman resonances. Our model combines multiphoton coupling between bound states, including dynamic Stark shifts, with coupling to a discretized continuum. The simplicity of the model allows us to interpret pulse-shape-dependent strong-field ionization yields and to demonstrate the relevance of strong-field atomic or molecular phase matching to ionization, as well as bound-state population transfer. Comparison with experimental measurements demonstrates that the calculations capture the essence of the pulse-shape-dependent ionization yields.
For pt.I see ibid., vol.28, p.5057-75 (1995). In the previous paper, a non-perturbative time-dependent theory of resonant multiphoton ionization by both smooth and rectangular pulses was derived, and as a first step, applied to the two-photon ionization of caesium in weak fields. In the present paper, we continue the analysis of the same system, extending the weak-field results and considering the case of stronger fields. At higher intensities the spectra exhibit energy modulations of a different character from those predicted for low power. Here, for stronger fields, the structure arises from Rabi oscillations, which, for the smooth pulses assumed here, are triggered by time-dependent Stark shifts. In order to obtain a better understanding of the energy oscillations, we also look at the photoelectron energy spectrum produced by a rectangular pulse, a problem which is solved exactly. We show that there are other sources of strong-field structure.
Abstract We demonstrate how strong-field multiphoton transitions between dynamically shifted atomic levels can be traced in the energy spectra of emitted photoelectrons. Applying an ultrafast and intense laser pulse, two-photon Rabi oscillations are induced between two bound states of an atom. A third photon from the same pulse directly ionizes the atom, thus the emitted photoelectrons coherently probe the underlying dynamics. As the instantaneous energy of photoelectrons follows the pulse intensity envelope, modulated by the ac Stark shifts, electrons emitted with the same energy but at different times—at the rising and falling edge of the pulse—will interfere leading to pronounced dynamic interference pattern in the spectra. We investigate this phenomenon both numerically and analytically by developing a minimal three-state model that incorporates two-photon coupling and dynamically shifted atomic levels. On the example of atomic lithium (2 s → → 4 s → continuum) we show how the individual ac Stark shifts and the two-photon Rabi frequency are reflected through the asymmetry, shifting and splitting of the interference structure of the computed photopeaks.
We investigate resonant strong field ionization of atomic hydrogen with respect to the 1s–2p transition. By 'strong' we understand that Rabi periods are executed on a femtosecond time scale. Ionization and AC Stark shifts modify the bound state dynamics severely, leading to nonperturbative signatures in the photoelectron spectra. We introduce an analytical model, capable of predicting qualitative features in the photoelectron spectra such as the positions of the Autler–Townes peaks for modest field strengths. Ab initio solutions of the time-dependent Schrödinger equation show a pronounced shift and broadening of the left Autler–Townes peak as the field strength is increased. The right peak remains rather narrow and shifts less. This result is analysed and explained with the help of exact AC Stark shifts and ionization rates obtained from Floquet theory. Finally, it is demonstrated that in the case of finite pulses as short as 20 fs the Autler–Townes duplet can still be resolved. The fourth generation light sources under construction worldwide will provide bright, coherent radiation with photon energies ranging from a tenth of an meV up to tens of keV, hence covering the regime studied in the paper so that measurements of nonperturbative, relative AC Stark shifts should become feasible with these new light sources.