Radiative Transitions and Form Factors

James H Scofield

B. J. Feldman, Michael S. Feld

It is well known that the spectral line shape of the narrow change signal induced at one Doppler-broadened transition by an intense laser field resonating with a second one can be extremely complex and nonclassical in form, the exact details depending on such factors as the direction of observation relative to the propagation direction of the laser field and its spatial form (traveling wave or standing wave). Nevertheless, the area under the change signal line shape, i.e. its frequency-integrated intensity, is always the same for a given laser field, and is independent of the coupling between laser and change-signal fields. A proof of this equal area property, based on quantum-mechanical transition-rate theory, is given for a three-level system composed of degenerate or near-degenerate states. The resulting expression is also applicable to line-shape calculations for experiments which study resonance fluorescence induced by intense laser or resonance radiation or other forms of excitation.

Marcel Pont, Daniel Proulx, Robin Shakeshaft

The time-dependent Schrodinger for an atom in a radiation field has a very special form, at least within the dipole approximation: The time-dependence of the Hamiltonian is contained entirely in a c-number which appears as a factor in the interaction, V(t), of the atom with the radiation field. In the velocity gauge this factor is the vector potential, A(t), of the field — we have V(t)= − (e/µc)A (t) • p, where e, µ, and p are the charge, reduced mass, and canonical momentum (in the center of mass frame) of the electron — while in the length gauge this factor is the electric field vector F(t) — we have V(t) = –eF(t) • x, where x is the position coordinate. To our knowledge, this factorization property has not been fully exploited in previous approaches to solving the timedependent Schrodinger equation1. As we show here, in an application to multiphoton ionization of atomic hydrogen, it is an extremely useful property; and although at first sight it would seem to be a unique feature of the atom-radiation system, this is not so, for, as Solov’ev2 has pointed out, the time-dependent Schrödinger equation for colliding atoms or ions can be transformed into a form in which a scaled time-dependence also appears only in c-number factors of the (transformed) interactions.

D. Vrinceanu, M. Raymond Flannery

The classical form factor is deduced from exact correspondence with a phase-space representation of the quantal form factor. Analytical expressions are provided for $n\stackrel{\ensuremath{\rightarrow}}{l}{n}^{\ensuremath{'}}{l}^{\ensuremath{'}},$ $n\stackrel{\ensuremath{\rightarrow}}{l}{n}^{\ensuremath{'}}$, and $\stackrel{\ensuremath{\rightarrow}}{n}{n}^{\ensuremath{'}}$ transitions in hydrogenic systems and for $\stackrel{\ensuremath{\rightarrow}}{n}{n}^{\ensuremath{'}}$ in the one-dimensional harmonic oscillator. An efficient procedure for calculation of quantal form factors as analytical functions of momentum transfer, for arbitrary quantum numbers, is presented. The classical approach has the ability to explain quite succinctly interesting trends and various important aspects which remain hidden within the quantal treatment of form factors. The classical-quantal comparison ranges from being qualitatively good for $n\stackrel{\ensuremath{\rightarrow}}{l}{n}^{\ensuremath{'}}{l}^{\ensuremath{'}}$ transitions to close agreement for $n\stackrel{\ensuremath{\rightarrow}}{l}{n}^{\ensuremath{'}}$ and $\stackrel{\ensuremath{\rightarrow}}{n}{n}^{\ensuremath{'}}$ transitions. Excellent agreement is obtained for the integrated form factor for all transitions.

Richard C. Powell