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Journal of Physics B Atomic and Molecular Physics

The shift of atomic states by laser fields

Joel I. Gersten, Marvin H. Mittleman

Published Oct 12, 1976• 118 citations
Publisher

Abstract

An atom or molecule illuminated by a laser has its states modified by the laser. A set of laser-distorted states is defined and the equation for the amplitude of the wavefunction for change of population of these states obtained. The state equation is solved approximately in a novel gauge in the limit where the laser frequency is high, and the utility of this gauge is discussed. The equation is also investigated in the limit of intense fields where the effect of the laser is to make the internal interactions of the atom non-local. The effect of this non-locality is investigated at large distances.

DOI
10.1088/0022-3700/9/15/008

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An atom or molecule illuminated by a laser has its states modified by the laser. A set of laser-distorted states is defined and the equation for the amplitude of the wavefunction for change of population of these states obtained. The state equation is solved approximately in a novel gauge in the limit where the laser frequency is high, and the utility of this gauge is discussed. The equation is also investigated in the limit of intense fields where the effect of the laser is to make the internal interactions of the atom non-local. The effect of this non-locality is investigated at large distances.
As light passes an atom, it exerts forces on the charges, electrons, and nuclei that alter the atomic structure. These may be slight distortions (perturbations) of the electron cloud or they may be more severe, as described in subsequent sections of this monograph. We wish to determine those changes, given the radiation field, or to devise a radiation field that will produce specified changes. The changes to the atomic structure also affect the radiation that subsequently passes the atom. To describe those effects we must consider wave equations for radiation in the presence of altered atomic structure. More generally we must find self-consistent equations for the atoms and the field together, as discussed in Chaps. 21 and 22. Here we consider mathematical descriptions of the influence of coherent radiation on individual atoms, molecules, or other single quantum systems. Individual atoms Traditional sources of emission and absorption spectra, though revealing energy states of the constituent atoms and molecules, are macroscopic samples. One observes averaged characteristics of many individual particles, see Chap. 16. Quantum theory offers the basic formalism for dealing with individual atoms exposed to controlled radiation fields. Several experimental techniques provide acceptable approximations to this ideal. The following paragraphs note some of these examples. Vapors Neutral atoms or molecules in a vapor move freely along straight-line paths, interrupted by brief collisions that redirect the two collision partners. When the kinetic energies of the two partners are small, there can be no transfer of kinetic energy into internal energy of either particle – the collision is elastic .
We study theoretically the quantized motion of previously cooled atoms in a one-dimensional magneto-optical potential under coherent population trapping (CPT) conditions. We consider atoms with a transition interacting with a lin lin laser field and a static uniform magnetic field in the high-intensity limit. It is shown that if the Rabi frequency is greater than the Zeeman splitting, Doppler shift, natural width and detuning, then atoms are captured in the long-lived CPT state while the excited-state population is small. We do not take into account the incoherent photon scattering and describe the atomic motion by the Schrödinger equation. The oscillation frequency, oscillation damping time and localization size are obtained for atoms moving near the bottom of potential wells in a harmonic approximation. Narrowing of the vibrational spectral lines due to the CPT effect is predicted.
A formulation of quantum electrodynamics is given that applies to atoms in a strong laser field by perturbation theory in a non-relativistic regime. Dipole approximation is assumed. The dual Dyson series, here discussed by referring it to the Birkhoff theorem for singularly perturbed linear differential equations, can be applied and a perturbation series obtained transforming the Hamiltonian by a Pauli-Fierz transformation. But, if just few photons are present high-order harmonics cannot be generated. So, it is proven that odd high-order harmonics only appear when the laser field is intense and one can substitute the creation and annihilation operators by the square root of the mean number of photons taken to be huge, the field retaining its coherency property as observed experimentally for harmonics. In this case, the Hamiltonian for perturbation theory comes to the Kramers-Henneberger form. The theory has a dipolar contribution when the free-electron quiver motion amplitude is larger than the atomic radius. For a Coulomb potential one has that the outer electron is periodically kicked, and so a prove is given that the same should happen to Rydberg atoms in intense microwave fields. The distribution representing the kicking has a Fourier series with just odd terms. Using a modified Rayleigh-Schrödinger perturbation theory, it is shown that under the same condition of validity of the quiver motion amplitude to atomic radius ratio, the atomic wave function is only slightly modified by the laser field due to the way the energy levels rearrange themselves. This gives a prove of stabilization in the limit of laser frequency going to infinity. Then, perturbation theory can be applied when the Keldysh parameter becomes smaller with respect to the shifted distance between the energy levels of the atom.
In a related study the equations governing a high-gain laser amplifier have been solved numerically without making the usual slowly-varying-amplitude derivative approximation in Maxwell's equations, and thus the field amplitudes are not restricted to vary negligibly in a distance of one wavelength. The results reveal an instability that involves reflections and oscillatory growth of perturbations away from steady state. In the present study steady-state solutions of the field equations are obtained analytically, and transformations are described for converting the equations to alternative intensity-equation sets. These intensity equations are linearized and solved for the growth rate and oscillation period of the perturbations.