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Nepal Journal of Science and Technology

One, Two and Three Photon Absorption in Atomic Hydrogen

Nilam Shrestha, Jeevan Joyti Nakarmi

Published Jul 23, 2012• 1 citation
Publisher

Abstract

Non relativistic, semi-classical in dipole approximation, based on time dependent Schrodinger equation under the framework of perturbation theory, three different absorption processes in hydrogen atom are studied and calculated transition matrix element from an initial state to final state.DOI: http://dx.doi.org/10.3126/njst.v12i0.6520 Nepal Journal of Science and Technology 12 (2011) 324-329

DOI
10.3126/njst.v12i0.6520

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Non relativistic, semi-classical in dipole approximation, based on time dependent Schrodinger equation under the framework of perturbation theory, three different absorption processes in hydrogen atom are studied and calculated transition matrix element from an initial state to final state.DOI: http://dx.doi.org/10.3126/njst.v12i0.6520 Nepal Journal of Science and Technology 12 (2011) 324-329
We have discussed the problems of non linear interaction between electromagnetic radiations with atoms from semi-classical point of view. Time dependent Schrödinger equation for single electron system is solved by using perturbative technique to obtain transition probability. Higher order perturbation is also discussed which is used in multiple processes, in which two or more quanta are emitted instead of a single photon. The theory is based on assumption that the perturbation is small. From this transition probability ionization rate and absorption cross-section of hydrogen atom is calculated. Its variation with photon energy and field strength is analyzed which agrees very well with experimental observations.The Himalayan PhysicsVol. 3, No. 32012Page : 40-43
Some phenomena cannot be solved using the exact solution of the Schrödinger equation. However, using approximated methods such as perturbation theory we are able to obtain remarkable results considering terms up to second order. The physical phenomena studied here are the absorption and stimulated emission by the classical feature. In addition, we are going to discuss the physical meaning of perturbation theory when we treat the behaviour how initial state ket evolves as time goes on. We will evaluate the transition of probability between states, the rate and the energy involved in the atomic transitions. It will be introduced fundamental concepts of quantum mechanics, such as interaction picture, and a general perturbation theory using constant and sinusoidal time-dependent perturbation term. Concerning to these phenomena, we considered the perturbed term of the Hamiltonian of the interaction between the vector potential and electron momentum.
Within the framework of nonrelativistic quantum mechanics, the analysis of the two-photon absorption by atomic hydrogen is generalized to the case of two incident beams with arbitrary direction and polarization (including circular polarization). In the dipole approximation, the second-order matrix elements responsible for two-photon absorption are transformed into a finite sum consisting of the product of a radial part and an angular part. Exact calculation of the angular part predicts that, for the s–s transition only, transparency is obtained regardless of the frequencies of the two beams with perpendicular polarizations. In addition, circularly polarized light is found to be more efficient (up to a factor of 1.5) for the s–d two-photon transition. The radial parts of the matrix elements are accurately evaluated by using the Coulomb Green’s function technique and an implicit technique of Dalgarno and Lewis. Their calculation indicates that “zeros” exist in the two-photon absorption spectrum, thereby predicting that absorption is not possible at certain frequencies. It should be noted that in the calculated spectrum, near or at resonance points, the linewidths of the intermediate levels must be included in order to obtain accurate results.
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 N2\frac{N}{2}2N​ power of intensity. NNN 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 ppp states.
The general theory of multiphoton ionization of hydrogen atoms by simultaneous absorption of several photons is discussed. In the nonrelativistic dipole approximation, the calculations lead to a recursion relationship concerning angular functions and involve radial matrix elements containing many summations over intermediate states. A general method for exactly evaluating several sums is described in detail and applied to the photo-ionization of atomic hydrogen in the ground state. Numerical results are presented as dispersion curves of the transition rate $\frac{{\ensuremath{\sigma}}_{n}}{{I}^{N\ensuremath{-}1}}$ for a range of wavelengths from 1000 up to 7200 \AA{}.