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arXiv:quant-ph/0306160 · Physical Review A

Population control of 2s−2p transitions in hydrogen

Kh. Kh. Shakov, James H. McGuire

Published Mar 21, 2003• 13 citations
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Abstract

We consider the time evolution of the occupation probabilities for the $2s\ensuremath{-}2p$ transition in a hydrogen atom interacting with an external field $V(t).$ 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.

DOI
10.1103/physreva.67.033405

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We consider the time evolution of the occupation probabilities for the 2s\ensuremath−2p2s\ensuremath{-}2p2s\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 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.
The probabilities of two-photon decay for hydrogen (H) and antihydrogen ( Hˉ \bar H Hˉ ) atoms in the presence and absence of an external electric field are analytically calculated. In particular, the probabilities of the E1E2 and E1M1 transitions between the 2p and 1s levels are calculated for the case when emitted photons are characterized by polarization vectors and wavevectors. It is shown that, in an external electric field, the decay probabilities for 2s and 2p levels differ for H and Hˉ \bar H Hˉ atoms because of interference terms linear in field. Coulomb Green’s function method is used for summing over intermediate states.
The Schrodinger equation for the interaction of a decaying two-level atom with an external oscillating field is separated via a series of matrix transformations into a time independent and a time dependent part; the former yields the radiofrequency Stark shift for decaying states, the latter is normally negligible for atomic beam resonance experiments. In the limit of stable states, or for the case of two levels of equal lifetime, the shift reduces to that of the Bloch-Siegert effect. Exact analytical expressions for the state amplitudes are obtained for the case when the time dependent matrix can be neglected; the resulting transition probabilities are investigated as functions of frequency, time, and interaction strength and are compared with the rotating field theory and the exact numerical solutions.
The control of the formation of dipolar hydrogen atom 2s–2p hybrid states, through varying the relative phase δ of two lasers inducing simultaneous one- and two-photon transitions from the ground state, is discussed by using time-dependent perturbation theory. Numerical examples are presented that illustrate how the populations of the hybrid states can be controlled by varying δ. The perturbative calculations are augmented by various exact finite basis set, and full split operator/fast Fourier transform, solutions to the time-dependent wave equation for the problem.
We consider the time evolution of nearly degenerate two-state systems with an external interaction. Conditions in which full population transfer is possible when the two-states become degenerate are considered. A new variation of the coupled differential equations for the probability amplitudes suggests a possible singularity in the time evolution of the system. The singularity occurs at the point in time where full control in the degenerate limit is achieved. We solve this new variation to derive more information about non-degenerate population control. The new results of numerical calculations for occupational probabilities of non-degenerate systems are presented and interpreted. These results are used to understand how population control breaks down as the photon energy and the energy of the system are changed. Applications are discussed, with an emphasis on interactions of twisted-vortex and plane-wave photons with a variety of targets, including macroscopic gas cells of atoms, of large single crystals, and of molecules.