Four-path interference and uncertainty principle in photodetachment microscopy

Abstract

Abstract. -We study the quantal motion of electrons emitted by a pointlike monochromatic isotropic source into parallel uniform electric and magnetic fields. The two-path interference pattern in the emerging electron wave due to the electric force is modified by the magnetic lens effect which periodically focuses the beam into narrow filaments along the symmetry axis. There, four classical paths interfere. With increasing electron energy, the current distribution changes from a quantum regime governed by the uncertainty principle, to an intricate spatial pattern that yields to a semiclassical analysis.Introduction. -Two-path interference along classical trajectories has a long-standing tradition as a textbook showpiece of quantum mechanics [1]. A fascinating realization of two-path interference on a macroscopic scale has recently been achieved in near threshold photodetachment microscopy [2][3][4]. Here, electrons are released from negative ions by irradiation with a laser beam in the presence of a uniform external electric field which subsequently governs their motion. Blondel et al. [2][3][4] recorded field-induced interference fringes in configuration space for O − and several other ionic species. In their experiment, the motion of the electrons can be considered as the quantum analogue of throwing a classical particle at constant energy in a uniform gravitational field. Within the classically allowed "shot-put"-range two distinct trajectories will link the electron source with a given destination, causing "double slit" interference in the quantal case. Using detectors with high spatial resolution, images of the resulting fringes were obtained by Blondel et al. [3,4] at large distances (0.5 m) from the electron-emitting negative ions, there extending to the millimeter scale (Fig. 1). In this way, their "photodetachment microscope" demonstrated the nodes and antinodes of the electronic wave function, allowing precise determination of the electron affinity [4]. In order to advance the understanding of the imaging mechanism of their device, this letter serves to point out the intricate spatial properties of the photoelectronic current distribution in parallel electric and magnetic fields. (The integrated photocurrent, i. e. the total photodetachment cross section in parallel fields has been addressed theoretically by several authors [5,6]. Their results are implicitly contained in the following developments.)

Electron dynamics in parallel electric and magnetic fields

“…We infer that the ratio of forces η presents a convenient control parameter that adjusts the degeneracy of the solution set. We also note that no matter how small we choose the magnetic field B, in the vicinity of the "focal points" z = k 2 π 2 /η always a region with at least four coexisting trajectories exists [35]. Figure 1 shows an example with up to eight interfering trajectories.…”

“…(η 1 crit ≈ 2.331 is defined in Eq. (35).) The top of the first caustic (k = 1) is a smooth dome, and the curve is parametrized in the way of the late variant [Sl].…”

“…For a point source, the wave function is proportional to the energy Green function G(r; E) for an electron in the external potential. This function can be expanded into a rapidly convergent series over Landau states, the eigenstates of the cyclotron motion [15,35]. This expansion yields both the total detachment cross section and the spatial electron distribution.…”

Propagation of charged particle waves in a uniform magnetic field

“…Unlike in the presence of an electric field E [40,42], where continuous acceleration tends to align the current profile with the density profile far from the source, the charge and current distributions retain their individual character in a purely magnetic environment, so the vector field j qm (r) warrants a detailed study. Given the symmetry of the problem, it is convenient to extract the radial component j ρ and a component j z parallel to the field:…”

Observed photodetachment in parallel electric and magnetic fields

“…However, the modification is of little practical relevance here [44].) An additional parallel electric field removes the singularity and replaces the root by the square of an Airy function [19,27,35,36]:…”