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Physical Review A

Momentum-Translation Approximation outside Its Domain of Applicability

John E. Rogerson, Robert N. De Witt, Howard R. Reiss

Published Jan 1, 1973• 12 citations
Publisher

Abstract

The momentum-translation approximation is valid for electromagnetic transitions in bound systems caused by external radiation of photon energy much less than the transition energy. Its utility is explored here for two types of problems that violate that restriction. One application is to resonant bound-bound multiphoton transitions of low multiphoton order, and results are compared to recent perturbative calculations. The other application is to both the transition rate and the spectrum of two-photon spontaneous emission from the metastable $2S$ state in hydrogen. Predicted transition rates are too low, but the spectrum agrees well with the known results. All momentum-translation results follow from simple closed-form analytical expressions.

DOI
10.1103/physreva.7.21

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The conditions of validity for the momentum-translation approximation of Reiss are found much more stringent than was thought before: multiphotonic transitions induced by one strong field cannot be described by this method. First-order transitions between bound states dressed by a strong low-frequency field can be calculated, but the method gives nothing but a trivial result; only for the ionization of a dressed atom is the approximation found useful.

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Within the framework of time-dependent perturbation theory, general expressions are derived for the cross section of multiphoton processes induced by radiation acting on a hydrogen atom. Bound-bound transitions involving emission of a single quantum and bound-free transitions are calculated for a wide range of photon energies. In the case of resonance, the transition probability is also discussed. The procedure used to evaluate the sums over intermediate states is described. The numerical computation allows one to compare the results obtained for various competing processes.

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Higher-Order Corrections to the Momentum-Translation Approximation

Bhaskar J. Choudhury

A general analytical result has been obtained for the calculation of higher-order corrections to the momentum-translation approximation. Explicit analytical results are given for the calculation of higher-order corrections for bound-bound and bound-free transitions for hydrogenic atoms. Numerical results are given for $1S$ to $2S$ transitions in hydrogen.

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On the inadequacy of the momentum-translation approximation: multiphoton ionization of atomic hydrogen

A. Rachman, G Laplanche

The cross section for multiphoton ionization of hydrogen is computed using the momentum translation method (MTA) in the cases of linearly and circularly polarized light, alternatively. Extended numerical comparison is made with well-established results obtained from the Nth-order time-dependent perturbation theory in the weak intensity limit. The MTA does not reproduce the values given by perturbation theory and considerable differences are observed. Higher order iterations than the lower one are introduced and an analysis of the leading terms of the development is made: at weak intensities the zeroth-order term that defines the MTA cancels with higher order terms and the final result obtained after making (N-1 iterations is none other than the well-known transition amplitude deduced from standard perturbation theory, defined by the remaining terms of the last iteration after mutual cancellation with lower-order ones. According to the results obtained in this work, it is not possible to affirm that the MTA is adequate to calculate atomic multiphoton ionization cross sections.

Journal of Physics B Atomic and Molecular Physics•Apr 21, 1975•2 citations

Renormalised Perturbation Theory of Resonant Multiphoton Ionisation

K. Unnikrishnan

Perturbation theory of multiphoton ionisation due to a classical electromagnetic field is modified to allow for intermediate resonances with bound states. Complex energies, generally associated with resonances, do not enter into this formalism. For a monochromatic field of frequency u, a constant ionisation rate can then be defined unambiguously and only such continuum states are excited which correspond to the absorption of energy in integral multiples of hu. As an application, differential and total cross sections for the two-photon ionisation of hydrogen, for frequencies below the n = 3 resonance region, are obtained in closed form. Existing data for generalised cross sections, calculated numerically using the complex coordinate method, are in good agreement with the present results. Finally, the mean fractional ionisation resulting from a pulse of finite duration is estimated on the basis of the associated power spectrum. For short pulses, the time dependence of ionisation exhibits a departure from that expected of a time-independent rate.

Australian Journal of Physics•Oct 1, 1995•0 citations
The momentum-translation approximation is valid for electromagnetic transitions in bound systems caused by external radiation of photon energy much less than the transition energy. Its utility is explored here for two types of problems that violate that restriction. One application is to resonant bound-bound multiphoton transitions of low multiphoton order, and results are compared to recent perturbative calculations. The other application is to both the transition rate and the spectrum of two-photon spontaneous emission from the metastable 2S2S2S state in hydrogen. Predicted transition rates are too low, but the spectrum agrees well with the known results. All momentum-translation results follow from simple closed-form analytical expressions.
Within the framework of time-dependent perturbation theory, general expressions are derived for the cross section of multiphoton processes induced by radiation acting on a hydrogen atom. Bound-bound transitions involving emission of a single quantum and bound-free transitions are calculated for a wide range of photon energies. In the case of resonance, the transition probability is also discussed. The procedure used to evaluate the sums over intermediate states is described. The numerical computation allows one to compare the results obtained for various competing processes.
A general analytical result has been obtained for the calculation of higher-order corrections to the momentum-translation approximation. Explicit analytical results are given for the calculation of higher-order corrections for bound-bound and bound-free transitions for hydrogenic atoms. Numerical results are given for 1S1S1S to 2S2S2S transitions in hydrogen.
The cross section for multiphoton ionization of hydrogen is computed using the momentum translation method (MTA) in the cases of linearly and circularly polarized light, alternatively. Extended numerical comparison is made with well-established results obtained from the Nth-order time-dependent perturbation theory in the weak intensity limit. The MTA does not reproduce the values given by perturbation theory and considerable differences are observed. Higher order iterations than the lower one are introduced and an analysis of the leading terms of the development is made: at weak intensities the zeroth-order term that defines the MTA cancels with higher order terms and the final result obtained after making (N-1 iterations is none other than the well-known transition amplitude deduced from standard perturbation theory, defined by the remaining terms of the last iteration after mutual cancellation with lower-order ones. According to the results obtained in this work, it is not possible to affirm that the MTA is adequate to calculate atomic multiphoton ionization cross sections.
Perturbation theory of multiphoton ionisation due to a classical electromagnetic field is modified to allow for intermediate resonances with bound states. Complex energies, generally associated with resonances, do not enter into this formalism. For a monochromatic field of frequency u, a constant ionisation rate can then be defined unambiguously and only such continuum states are excited which correspond to the absorption of energy in integral multiples of hu. As an application, differential and total cross sections for the two-photon ionisation of hydrogen, for frequencies below the n = 3 resonance region, are obtained in closed form. Existing data for generalised cross sections, calculated numerically using the complex coordinate method, are in good agreement with the present results. Finally, the mean fractional ionisation resulting from a pulse of finite duration is estimated on the basis of the associated power spectrum. For short pulses, the time dependence of ionisation exhibits a departure from that expected of a time-independent rate.