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College Physics

A nonperturbative solution to induced transitions between atomic states

Lin Qiong-gu

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

An introduction is presented to a recently established nonperturbative approach to the transitions between atomic states induced by weak electromagnetic waves. A special case of the general solution is calculated and discussed in detail. Results of the nonperturbative approach reduce under perturbation condition to those obtained by the time- dependent perturbative method,but overcome the difficulty that the perturbative results are obviously unreasonable in the near resonant region. According to the nonperturbative solution,the electric- dipole selection rules are approximately recovered for short time. For large time,however,probabilities for forbidden transitions may become comparable with those for allowed ones.

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An introduction is presented to a recently established nonperturbative approach to the transitions between atomic states induced by weak electromagnetic waves. A special case of the general solution is calculated and discussed in detail. Results of the nonperturbative approach reduce under perturbation condition to those obtained by the time- dependent perturbative method,but overcome the difficulty that the perturbative results are obviously unreasonable in the near resonant region. According to the nonperturbative solution,the electric- dipole selection rules are approximately recovered for short time. For large time,however,probabilities for forbidden transitions may become comparable with those for allowed ones.
The collision-induced absorption and emission coefficients for electric-dipole forbidden atomic transitions are calculated for weak radiation fields. The approximations used are valid for $\ensuremath{\hbar}\ensuremath{\omega}$ near $\ensuremath{\hbar}{\ensuremath{\omega}}_{0}$, the atomic energy differences. The example case of $S\ensuremath{-}D$ transitions induced by a spherically symmetric perturber (e.g., a noble-gas atom) is treated in detail and compared to measurements. The case of "radiative collisions," in which both colliding atoms change their state, is included in the theory and is also compared to experiment.
Excitation of an atomic system subjected to a slowly varying external electromagnetic field is studied using a two-level model. Time evolution of the system is found using an approach which is nonperturbative in the field strength. There is no constraint to small values of the applied field, that is, the field (in appropriate energy units) need not be small compared to the difference in energies of the two levels. Rather, we use the fact that the situation of interest to us is where the frequency of the exciting field is small compared to the frequency associated with the level difference. Transition probabilities and resonance conditions are found which circumscribe both the large- and small-field limits. In the weak-field limit the previous results of high-order perturbation theory are readily recovered. For a monochromatic field the characteristic features of resonance excitation at high harmonic number of the applied field are (a) extremely narrow resonance widths and (b) shifts in resonance positions which are strong functions of field intensity. Because of this sensitivity, we are able to demonstrate that when slow temporal evolution of the field amplitude is taken into account (e.g., due to finite pulse duration) the appropriate mean excitation rate is that due to the uncorrelated contribution of many resonances. The results of this analysis are used to estimate excitation rates in a specific atomic system, Cd12+{\mathrm{Cd}}^{12+}Cd12+, which are then compared to multiphoton-ionization rates. Our calculations suggest that the ionization rate exceeds the excitation rate by several orders of magnitude.
The subject of this chapter is the quantum mechanical analysis of the interaction of electromagnetic radiation with atomic transitions. The analysis is based on the Schrödinger wave equation, and in the first section, the gauge-invariant form of the external electromagnetic field is introduced. The electric dipole interaction and the long-wavelength approximation for the analysis of this interaction are discussed. The perturbative analysis of both single-photon and two-photon electric dipole interactions is presented, and density matrix analysis is introduced. The interaction of radiation with the resonances of atomic hydrogen is then discussed. The analysis is performed for both coupled and uncoupled representations. In the last section of the chapter, the radiative interactions for multielectron atoms are discussed. The Wigner–Eckart theorem and selection rules for transitions between levels characterized by coupling are developed. The effect of hyperfine splitting on radiative transitions is also briefly discussed.
How to deal with time-dependent potentials, what kinds of physical problems having time-dependent potentials can be solved exactly and which are to be solved with the use of perturbation theory, what are different possible forms of time-dependent perturbating fields, and semi-classical treatment of the interaction of fields with matter are the subject matters of this chapter. First order perturbation theory, computation of transition probability, transition rates and Fermi's Golden Rule for harmonic and constant perturbations, absorption, stimulated emission and dipole approximation when electromagnetic fields interact with matter, spontaneous emission and Einstein coefficients, and dipole selection rules are other topics which are dealt with in detail in the chapter. Discussions with properly justified mathematical treatment on time-dependent potentials and on the exactly solvable time-dependent two-state systems are the special foci of this chapter.
This paper presents a nonperturbative treatment of strong-coupling induced effects in atom-field systems which cannot be seen in traditional perturbative treatments invoking compromising assumptions such as the Born-Markov, rotating wave, or Fermi Golden rule. We consider an atom whose internal degrees of freedom are modeled by a harmonic oscillator which is bilinearly coupled to a scalar quantum field, representing one of the two polarizations of an electromagnetic field. Because the whole system is Gaussian we can solve this problem exactly. Using the open quantum system conceptual framework and the influence functional formalism we derive the dynamics of the reduced density matrix for the atom which enables the calculation of atomic transition probability and other relevant physical quantities. Finding an exact solution to this problem has the distinct advantage of enabling one to capture fully the strong coupling regime and discover interesting effects such as spontaneous ground state excitation [R. Passante, T. Petrosky, and I. Prigogine, Long-time behaviour of self-dressing and indirect spectroscopy, Physica (Amsterdam) 218A, 437 (1995).] which is unfathomable in perturbative treatments. The conventional description of atomic-optical activities is predicated on the assumption that the state of the total atom-field system is a product state of the atomic excitations and the photon number states, an assumption which is valid only for vanishingly weak coupling. The correct energy eigenfunctions to use should be that of the Hamiltonian of the combined atom-field system. Other features associated with finite to strong coupling effects such as resonance peak broadening and transition from a gapped to a gapless spectrum can all be understood from this perspective. Finally, to put the issues in a proper perspective we take the perturbative limit of the exact results and compare them with those from conventional time-dependent perturbation theory (TDPT). This enables one to pin-point where the deficiencies of TDPT lie as one removes the ultraweak coupling assumption.
This chapter discusses the time-dependent perturbation theory applicable for small time-dependent perturbation. The incident electromagnetic wave makes the electric and magnetic potentials to oscillate and the perturbation is usually very small compared with the static Hamiltonian. The theory of time-dependent perturbation was developed by Dirac in 1927. Approximation methods are developed for systems where the time dependence of the Hamiltonian is small, slowly varying or rapidly varying. These correspond to perturbation, adiabatic and sudden approximations. An interesting consequence of sudden approximation is that the system can undergo an adiabatic change due to the sudden approximation. The rate of spontaneous emission is proportional to the number of atoms in the excited state. In the stimulated emission, an incident radiation of appropriate frequency induces an atom in an excited state to emit radiation. In the case of time-dependent perturbations, the system undergoes transition from one state to another during the action of the perturbation.