Photoionization microscopy in terms of local-frame-transformation theory

P. Giannakeas, F. Robicheaux, and Chris H. Greene

Phys. Rev. A 91, 043424 – Published 30 April 2015

Abstract

Two-photon ionization of an alkali-metal atom in the presence of a uniform electric field is investigated using a standardized form of the local-frame-transformation and generalized quantum defect theory. The relevant long-range quantum defect parameters in the combined Coulombic plus Stark potential are calculated with eigenchannel $R$ -matrix theory applied in the downstream parabolic coordinate $η$ . The present formulation permits us to express the corresponding microscopy observables through the local frame transformation, and it gives a critical test of the accuracy of the Harmin-Fano theory.

Received 16 October 2014

DOI:https://doi.org/10.1103/PhysRevA.91.043424

Authors & Affiliations

P. Giannakeas^*, F. Robicheaux^†, and Chris H. Greene^‡

Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA

^*pgiannak@purdue.edu
^†robichf@purdue.edu
^‡chgreene@purdue.edu

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Vol. 91, Iss. 4 — April 2015

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Images

Figure 1
Matrix elements of the local frame transformation $U_{n_{1} ℓ}^{ε F m}$ vs the number of states $n_{1}$ for $m = 1$ where the angular momentum acquires the values $ℓ = 1, 2, 3, and 6$ . The electric-field strength is $F = 640$ V/cm and total collisional energy is $ε = 135.8231 {cm}^{- 1}$ . The vertical dashed lines indicate the sign and the interval of values of the $β_{n_{1}}$ .
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Figure 2
(a) Amplitudes $1 / R_{n_{1}}$ and (b) the matrix elements $U_{n_{1} ℓ}^{ε F m} R_{n_{1}}$ as a function of the number of states $n_{1}$ for $m = 1$ . In panel (b) the angular momentum acquires the values $ℓ = 1, 2, 3, and 6$ indicated by the blue, green, yellow, and purple solid lines, respectively. The electric-field strength is $F = 640$ V/cm and total energy is $ε = - 135.8231 {cm}^{- 1}$ .
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Figure 3
Irregular solutions in spherical coordinates illustrated up to $r = 80$ a.u. where $r = (r, θ = 5 π / 6, ϕ = 0)$ . In all panels the azimuthal quantum number is set to $m = 1$ and the black solid line indicates the irregular Coulomb function in spherical coordinates, namely $\frac{g_{ε ℓ m}^{(C)} (r)}{r}$ . Panel (a) depicts the case of $ℓ = 1$ where $\frac{g_{ε ℓ m}^{(LFT)} (r)}{r}$ denotes the irregular function in spherical coordinates calculated within the local-frame-transformation (LFT) framework, for two different cases of the total amount of $n_{1}$ states, namely $n_{1}^{(tot)} = 60$ (green dashed line) and $n_{1}^{(tot)} = 100$ (red dots). Panel (b) refers to the case of $ℓ = 6$ where $\frac{g_{ε ℓ m}^{(LFT)} (r)}{r}$ is calculated for $n_{1}^{(tot)} = 60$ (green dashed line), $n_{1}^{(tot)} = 100$ (blue diamonds), and $n_{1}^{(tot)} = 230$ (red dots) states. Note that panel (c) is a zoomed-out plot of the curves shown in panel (b). In all panels the energy is $ε = 135.8231 {cm}^{- 1}$ and the field is $F = 640$ V/cm.
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Figure 4
Irregular solutions in spherical coordinates are shown up to $r = 80$ a.u. where $r = (r, θ = 5 π / 6, ϕ = 0)$ . In all panels the azimuthal quantum number is set to be $m = 1$ and the black solid line indicates the analytically known irregular Coulomb function, namely $\frac{g_{ε ℓ m}^{(C)} (r)}{r}$ . Panel (a) depicts the case of $ℓ = 2$ where $\frac{g_{ε ℓ m}^{(LFT)} (r)}{r}$ denotes the irregular function in spherical coordinates calculated within the local-frame-transformation (LFT) framework for $n_{1}^{(tot)} = 100$ states (red dots). Similarly, panel (b) refers to the case of $ℓ = 3$ with $\frac{g_{ε ℓ m}^{(C)} (r)}{r}$ being calculated for $n_{1}^{(tot)} = 100$ (red dots) states. In all panels the energy is $ε = 135.8231 {cm}^{- 1}$ and the field is $F = 640$ V/cm.
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Figure 5
Irregular solutions in spherical coordinates at negative energies, i.e., $ε = - 135.8231 {cm}^{- 1}$ , illustrated for $r = (r, θ = \frac{5 π}{6}, ϕ = 0)$ with field $F = 640$ V/cm. In all panels the azimuthal quantum number is set to be $m = 1$ and the black solid line indicates the analytically known irregular Coulomb function, namely $\frac{g_{ε ℓ m}^{(C)} (r)}{r}$ . Accordingly, the red dots correspond to the LFT calculations of irregular function, namely $\frac{g_{ε ℓ m}^{(LFT)} (r)}{r}$ . Panels (a)–(d) depict the cases of $ℓ = 1, 2, 3, and 6$ , respectively. For all the LFT calculations the total number of $n_{1}$ states is $n_{1}^{(tot)} = 25$ which corresponds to $β_{n_{1}} < 1$ .
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Figure 6
Differential cross section (in arbitrary units) for Na atoms as a function of the cylindrical coordinate $ρ$ . The red solid lines indicate the LFT theory calculations, whereas the black dots denote the velocity mapping results from a direct solution of the two-dimensional inhomogeneous Schrödinger equation. Panels (a) and (b) refer to energy $ε = - 62 {cm}^{- 1}$ for the transitions $m_{int} = 0 \to m_{f} = 0$ and $m_{int} = 1 \to m_{f} = 1$ , respectively. Similarly, panels (c) and (d) refer to energy $ε = - 41 {cm}^{- 1}$ for the transitions $m_{int} = 0 \to m_{f} = 0$ and $m_{int} = 1 \to m_{f} = 1$ , respectively. In all cases the field strength is $F = 3590$ V/cm and the detector is placed at $z_{det} = - 1$ mm.
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