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Journal of Physics B Atomic Molecular and Optical Physics

Elastic and inelastic atomic scattering in a standing-wave laser field

Leonard Rosenberg

Published Feb 13, 2002• 0 citations
Publisher

Abstract

The motion of a two-level atom in a standing-wave laser field is studied using an approximation to the resolvent operator in the form of a truncated eigenfunction expansion, with the field treated in the occupation-number representation. The formalism provides expressions for the time-dependent S -matrix for transitions into both ground and excited atomic states. The effective Hamiltonian for purely elastic scattering is constructed, allowing for the direct study of scattering in the Bragg regime. Contact with standard treatments of Bragg scattering in terms of Mathieu functions is obtained by imposition of additional approximations. Comparison of numerical results obtained with and without imposition of these approximations provides an indication of their range of validity. The dependence of excitation probabilities on field strength and detuning is studied. These numerical illustrations provide evidence that in cases of physical interest excitation probabilities can be significant and can be reliably estimated for a broad range of field parameters using the present formalism.

DOI
10.1088/0953-4075/35/4/325

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Journal of Experimental and Theoretical Physics•Feb 1, 1998•1 citation
The motion of a two-level atom in a standing-wave laser field is studied using an approximation to the resolvent operator in the form of a truncated eigenfunction expansion, with the field treated in the occupation-number representation. The formalism provides expressions for the time-dependent S -matrix for transitions into both ground and excited atomic states. The effective Hamiltonian for purely elastic scattering is constructed, allowing for the direct study of scattering in the Bragg regime. Contact with standard treatments of Bragg scattering in terms of Mathieu functions is obtained by imposition of additional approximations. Comparison of numerical results obtained with and without imposition of these approximations provides an indication of their range of validity. The dependence of excitation probabilities on field strength and detuning is studied. These numerical illustrations provide evidence that in cases of physical interest excitation probabilities can be significant and can be reliably estimated for a broad range of field parameters using the present formalism.
A modified perturbation theory is developed for the calculation of the time-dependent wave function describing the propagation of an atom in a two-mode laser field. With the mode frequencies nearly equal, a first approximation is obtained that reproduces and extends known results, for both resonant and nonresonant transitions. This lowest-order solution sums terms that describe the absorption of a photon from one mode and emission into the other, leaving the total number of photons unchanged and shifting the momentum of the atom accordingly. Corrections are generated by expansion of a resolvent from which the nearly degenerate states have been projected out. These approximate solutions are used to construct asymptotic states in a formulation of atom-atom scattering in the two-mode field. In close analogy with the soft-photon approximation, a ``soft-pair'' approximation is developed based on the dominance of those asymptotic interactions in which the transfer of energy from the field to the atom is small, allowing one to relate the scattering amplitude in the presence of the field to the field-free amplitude. When the atomic ground state and an excited state are closely coupled by the field, the asymptotic wave function can have mixed parity allowing for long-range contributions to the effective potential. In addition to an enhancement of forward-scattering probabilities, this leads to a threshold anomaly that is analyzed here, in the resonant case, with the aid of a modified effective-range theory.
Dressed states of an atom in a classical laser field of frequency close to an atomic transition frequency are obtained, which are a considerable improvement over the usual two-level rotating-wave approximation. These wave functions are used in a calculation of the cross section for elastic electron-atom scattering in a laser field. Numerical calculations for electron-hydrogen scattering demonstrate that scattering involving absorption or emission of photons can be strongly influenced by coupling to the nonresonant states, regardless of the field strength. Inclusion of the counter-rotating terms in the calculation is shown to be essential for extending the applicability of the theory to scattering in nonresonant fields, where a first-order perturbation-theoretic treatment of the atom-field interaction would be expected to be quite justified.
The standard theoretical treatment of the Kapitza-Dirac effect---that is, the scattering of an electron passing through a standing-wave laser field---is extended here through the use of the Pauli equation to account for the interaction of the electron spin with the magnetic field of the standing wave. Prescriptions for determining unitarity-preserving approximations for the transition probabilities for scattering both with and without rotation of the electron spin direction are provided. This formalism is used to develop a perturbation theory for the spin-flip probability which, in the strong-field limit of interest here, reduces to a fairly simple relation between SSS-matrix elements for scattering with and without change in spin orientation, each expressed in terms of a Bessel function. A similar perturbative procedure is applied to estimate corrections to the standard theory for scattering in the absence of spin-flip processes, in which interactions that change the net number of photons in the field are ignored.
A set of coupled equations for the particle scattering by a two level atom in a strong near resonant radiation field is derived from the time dependent Schrodinger equation. This set of equations is discussed in a first and second order approximation. The result can be visualized in terms of a scheme in which the two well separated (nondegenerate) atomic states split, each into two energy levels. The splitting depends on the strength of the field and on the difference between the resonance frequency and the frequency of the incident light and is typically a few meV. In this scheme all possible elastic, quasielastic, inelastic and superelastic transitions between the four levels can be induced by collision processes. In addition, multiphoton processes may occur if the scattered particle remains long enough in the interaction region. Estimates of the two photon gain process for electron and heavy particle scattering by sodium atoms in a dye laser field are given. It is suggested that the superelastic processes may play an important role in laser induced gas breakdown, even if the laser frequency does not coincide with any atomic transition. Spontaneous emission of atoms in laser fields is also treated. The results are similar to the particle scattering processes.
We extend the previous theoretical treatment of standing-wave deflection of two-level atoms to include nonresonant excitation, nonorthogonal laser and atomic beams, and excitation duration upper bounded only by atomic relaxation times. Numerically solving equations analogous to those of abruptly initiated multilevel atomic excitation, we find bounds on the maximum deflection fixed by balancing acquired kinetic energy against interaction energy. We find that frequency detuning or nonorthogonal orientation of laser and atomic beams can enhance the deflection and, given sufficiently long interaction time, one can observe deflections at larger discrete angles. We comment on the connection between these Bragg scatterings and multiphoton resonances.
The scattering of an atom in the field of counterpropagating light waves is studied under conditions such that the state of the atom is a superposition of the ground and excited states. For the case in which this superposition is created by the field of a traveling wave, the momentum distribution function of the atom after scattering by a standing wave is found analytically in the approximation of a short interaction time, when the atom’s motion can be neglected. Longer interactions of the atom with the field are studied numerically. We also consider the case of counterpropagating light waves consisting of Gaussian or supergaussian pulses.