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Physical Review A

Time evolution of two-photon population transfer between the 1s and 2s statesof a hydrogen atom

Martin Dörr, O. Latinne, C. J. Joachain

Published May 1, 1997• 17 citations
Publisher

Abstract

We have solved the time-dependent Schr\"odinger equation to analyze the time evolution of a hydrogen atom, initially in its ground 1s state, interacting with a one- or two-color linearly polarized laser pulse which is tuned close to a two-photon resonance with the 2s metastable state. The two-photon transition therefore does not involve an intermediate resonance state. Using a suitable combination of two laser pulses of different frequencies, population can be transferred from the 1s state to the 2s state via the continuum. In both the single-color and the two-color cases, a maximum amount of 17% of population can be found in the 2s state at the end of the pulse. We compare and interpret our results in terms of the time-independent Floquet eigenvalues. The system can also be modeled by an ``essential states'' two-level atom with decay. Both the Floquet and the time-dependent solutions are compared with the model predictions. The model allows systematic optimization studies for population transfer under a wide range of laser pulse parameters.

DOI
10.1103/physreva.55.3697

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We have solved the time-dependent Schr"odinger equation to analyze the time evolution of a hydrogen atom, initially in its ground 1s state, interacting with a one- or two-color linearly polarized laser pulse which is tuned close to a two-photon resonance with the 2s metastable state. The two-photon transition therefore does not involve an intermediate resonance state. Using a suitable combination of two laser pulses of different frequencies, population can be transferred from the 1s state to the 2s state via the continuum. In both the single-color and the two-color cases, a maximum amount of 17% of population can be found in the 2s state at the end of the pulse. We compare and interpret our results in terms of the time-independent Floquet eigenvalues. The system can also be modeled by an ``essential states'' two-level atom with decay. Both the Floquet and the time-dependent solutions are compared with the model predictions. The model allows systematic optimization studies for population transfer under a wide range of laser pulse parameters.
Theoretical calculations for the 1S → 2S two- photon transition rate in atomic hydrogen are reviewed. They are compared with two experiments via the saturation effects that are observed when three-photon ionization is enhanced by two- photon resonance with the metastable 2S state. There is a significant disagreement between theoretical and experimental results. The disagreement is illustrated by determining the transition rate as a function of intensity for experimental laser profiles: (1) using parameters that give rates in agreement with experimental data, and (2) using theoretical values for the parameters. Finally, experiments that would involve direct observation of the 2S metastable state are considered. It is shown that single-longitudinal-mode laser experiments would provide an accurate measurement of the Rabi frequency for this transition and yield a definitive value for the rate. A Monte Carlo simulation method is used to consider practical effects associated with such an experiment, for example, chirp and ASE noise in the laser. Also, simulations of the multiphoton ionization process in multimode and wide band laser fields are presented.
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.
We consider the time evolution of the occupation probabilities for the $2s\ensuremath{-}2p$ transition in a hydrogen atom interacting with an external field V(t).V(t).V(t). A two-state model and a dipole approximation are used. In the case of degenerate energy levels an analytical solution of the time-dependent Schr\"odinger equation for the probability amplitudes exists. The form of the solution allows one to choose the ratio of the field amplitude to its frequency that leads to temporal trapping of electrons in specific states. The analytic solution is valid when the separation of the energy levels is small compared to the energy of the interacting radiation.
We investigate the coherent control of nonresonant two-photon excitation and (2+12+12+1) photon ionization processes from the time-dependent perspective. To this end, we have solved the time-dependent Schr\"odinger equation for a hydrogen atom interacting with an ultraviolet laser field. Results are obtained for symmetric and antisymmetric phase distributions leading to dark and bright pulses, for which the temporal field distributions consist of many subpulses. For the two-photon excitation process, it is found that at intermediate time between the subpulses, the population in the excited state can exceed the final state population and, in particular, is nonzero for dark pulses. The results for the (2+12+12+1) photon ionization closely follow those for the excitation process, which shows that the ionization of the electron proceeds as a resonant process via the excited state in all cases considered.
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.
In this chapter,we shall analyze the particular case of an atom interacting with a laser pulse whose duration is sufficiently long, so that the evolution of the atom in the laser field is adiabatic. When this condition is fulfilled, the atom can be considered to interact with a monochromatic laser field. As a consequence, the Hamiltonian of the system is periodic in time, and the Floquet theory [1] can be used to solve the time-dependent Schrödinger equation (TDSE) non-perturbatively. We begin in Section 4.1 by considering the Hermitian Floquet theory. We first derive the Floquet theorem for a monochromatic, spatially homogeneous laser field and show that the solutions of the TDSE correspond to dressed states having real quasi-energies , which can be obtained by solving an infinite system of time-independent coupled equations. We then generalize the Floquet theory to multicolor laser fields and to “non-dipole” laser fields which are not spatially homogeneous. In Section 4.2, the Floquet theory is applied to study the dynamics of a model atom having M discrete levels interacting with a monochromatic laser field. In this case, the coupling between the bound and continuum atomic states is neglected.We analyze the relationship between the Floquet theory and the rotating wave approximation, and examine the perturbative limit of the Floquet theory. We also consider the population transfer between Floquet dressed states.