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Physical Review A

Approximate Perturbation Theory for High-Order Electromagnetic Transitions

Howard R. Reiss

Published Aug 1, 1972• 28 citations
Publisher

Abstract

An approximation method is developed which yields very simple and general analytical results for electromagnetic bound-bound or bound-free transitions for any arbitrarily high order of interaction with the electromagnetic field. The approximation is perturbative in the sense that transition probabilities are proportional to the field intensity raised to the power $N$, where $N$ is the order of the interaction. Limiting values of electromagnetic field strength are given within which the approximation is valid. These limits depend upon the binding potential which produces the bound states. A feature of the results is that they show the onset of high-order dominance effects which can occur at high-field intensities.

DOI
10.1103/physreva.6.817

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An approximation method is developed which yields very simple and general analytical results for electromagnetic bound-bound or bound-free transitions for any arbitrarily high order of interaction with the electromagnetic field. The approximation is perturbative in the sense that transition probabilities are proportional to the field intensity raised to the power NNN, where NNN is the order of the interaction. Limiting values of electromagnetic field strength are given within which the approximation is valid. These limits depend upon the binding potential which produces the bound states. A feature of the results is that they show the onset of high-order dominance effects which can occur at high-field intensities.
A general approximation method is developed for the treatment of bound quantum systems in the presence of an intense external electromagnetic field of long wavelength. The method is based upon a unitary transformation which approximately removes the electromagnetic field from the problem. The accuracy of the technique increases with the number of photons involved in any given process; and in the case of large photon number, the results have a simple, concise form resembling first-order perturbation theory. It is possible to handle intense-field problems with small numbers of photons also, but then the formalism takes on a complexity analogous to the corresponding perturbation expressions for the same number of photons. The theory is extended to the case where two electromagnetic field modes are present, one or both of which may be intense. Bound-bound transitions in a hydrogenlike atom are calculated as an illustration of the application of the method. One of the explicit nonlinear effects to appear is an intensity-dependent deviation of the transition probability behavior from In{I}^{n}In for nnn-photon transitions, where III is the field intensity. It is demonstrated that intensity effects are much more probable with bound electrons than with free electrons.
Results valid for arbitrarily high intensity of electromagnetic field are presented for induced emission from an excited state of an atom. Comparison with perturbation theory shows that perturbative calculations fail quantitatively and qualitatively at lower intensity than had been expected. Distinctive intense-field features are discussed such as vanishing of the transition probability for certain intensities and photon multiplicities, and appearance of two extrema of the transition amplitude for multiphoton processes at very high intensities.
For pt.I see ibid., vol.17, p.29-52 (1984). Wavelengths, dipole strengths, oscillator strengths and transition probabilities for electromagnetic transitions between low-lying states of the hydrogen atom embedded in a magnetic field of arbitrary strength (0<B<or=4.70*10 8 T) are calculated with high numerical accuracy for a sufficiently dense mesh of B values using the wavefunctions and energy values determined in paper I by expanding the wavefunctions in terms of spherical harmonics (weak and moderate fields), or in terms of Landau states (strong and very strong fields). Effects caused by the finiteness of the proton mass are taken into account, and the scaling laws are quoted by the aid of which the results presented can be utilised to cover quantitatively the whole hydrogenic sequence in intense magnetic fields; finally, the quality of the results of previous calculations is checked.
The orders of the perturbation approximation to the Schroedinger-picture time evolution operator in powers of the presumed-small perturbation part of the Hamiltonian operator are developed by the iteration of an identity; the possibly confusing switch to the interaction picture isn't needed. Those time-evolution operator approximations are sandwiched between two orthogonal, normalized eigenstates of the unperturbed part of the Hamiltonian operator to produce the transition amplitude approximations, whose calculation is reduced to quadrature when every occurrence of the perturbation part of the Hamiltonian operator in them is expanded in the unperturbed basis. That expansion also reveals the time-dependent parts of those approximations to be multiply nested integrals which in the long-time limit approach simple products of non-singular inverses (principle value plus delta function) of differences of unperturbed energy eigenvalues. Closely related to the long-time limits of transition amplitudes are the long-time averages of transition rates. When the unperturbed basis is that of free-particle states, sums over relevant final states of transition rates from an initial state, divided by the initial state's particle flux, produce cross sections. Here the perturbation approximations to generic quantum transition rates are parlayed to the corresponding approximations to differential cross sections for nonrelativistic-particle potential scattering.
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.
A general perturbation procedure is described for writing down the contributions to any order for the probability of a quantum system described by a time-independent Hamiltonian making a transition from an arbitrary initial state to an arbitrary final state. The results are expressed in terms of certain propagators which involve continued fractions, so that some processes are summed to all orders. The general theory is then applied to a single multilevel atom interacting with any number of modes of the electromagnetic field in the electric dipole approximation. The fundamental theory is considered primarily but a detailed discussion is given of spontaneous and stimulated emission. Expressions for the lineshape are derived incorporating the generalized Lamb shifts and damping constants.