Perturbation Theory in the Analysis of Quantum Vortices Formed by Impact of Ultrashort Electromagnetic Pulse on Atom

Abstract

Nonstationary perturbation theory is used to study generation of quantum vortices resulting from ionization of hydrogen-like atom by an ultrashort pulse of classical electromagnetic field. It is shown that the vortices are determined by quantum interference effects.

APPENDIX

Cylindrical waves represented in terms of momenta. We use the known expressions for the Bessel function [17]:

$$\begin{gathered} {{J}_{m}}(x) = \frac{{{{{(i)}}^{m}}}}{{2\pi }}\int\limits_0^{2\pi } {\exp ( - ix\cos (\varphi ) + im\varphi )d\varphi ,} \\ {{J}_{m}}(x) = \frac{1}{{2\pi }}\int\limits_0^{2\pi } {\exp ( - ix\sin (\varphi ) + im\varphi )d\varphi ,} \\ \end{gathered} $$

((A.1))

properties

$$\begin{gathered} {{J}_{{ - m}}}(x) = {{( - 1)}^{m}}{{J}_{m}}(x), \\ {{J}_{m}}( - x) = {{( - 1)}^{m}}{{J}_{m}}(x), \\ \end{gathered} $$

((A.2))

and condition (13) to derive the following momentum representation of cylindrical waves:

$$\tilde {\Psi }_{{km}}^{{(0)}}({{k}_{x}}) = {{(i)}^{{ - m}}}{{C}_{m}}\frac{{\delta (k - {{k}_{x}})}}{k},$$

((A.3))

$$\tilde {\Psi }_{{km}}^{{(0)}}({{k}_{y}}) = {{C}_{m}}\frac{{\delta (k - {{k}_{y}})}}{k}.$$

((A.4))

For negative projections, quantity k_x,y is chan-ged by quantity |k_x,y|. The coefficients are C_m = 1 at k_x,y > 0 and m ≥ 0 or k_x,y < 0 and m ≤ 0 and C_m = (–1)^m at k_x, y > 0 and m < 0 or k_x,y < 0 and m > 0.

Radial component of the matrix element of perturbation operator 〈α, |m'||ρ|β, |m|〉 (10) calculated in the free-electron approximation. For the transition of hydrogen atom from the ground state to the state of continuous spectrum, we have

$$\langle k,1{\text{|}}\rho {\text{|}}1,0\rangle = \frac{{6k}}{{{{{(1 + {{k}^{2}})}}^{{5/2}}}}}.$$

((A.5))

When transitions to the final state of continuous spectrum via intermediate states of the same continuous spectrum are considered, we must calculate the following matrix elements 〈k, |m||ρ|k', |m$\mp$ 1|〉. Such calculations can be performed using the known expressions that are satisfied for cylindrical functions [17]

$$\begin{gathered} {{J}_{{\alpha - 1}}}(x) = \left( {\frac{\alpha }{x} + \frac{d}{{dx}}} \right){{J}_{\alpha }}(x), \\ {{J}_{{\alpha + 1}}}(x) = \left( {\frac{\alpha }{x} - \frac{d}{{dx}}} \right){{J}_{\alpha }}(x) \\ \end{gathered} $$

((A.6))

and condition (13). Then, we obtain

$$\langle k,{\text{|}}m{\text{||}}\rho {\text{|}}k',{\text{|}}m - 1{\text{|}}\rangle \, = \,\left\{ \begin{gathered} \left( {\frac{m}{{k'}} + \frac{\partial }{{\partial k'}}} \right)\frac{{\delta (k - k')}}{k},\quad m > 0, \hfill \\ \left( {\frac{{{\text{|}}m{\text{|}}}}{{k'}} - \frac{\partial }{{\partial k'}}} \right)\frac{{\delta (k - k')}}{k},\quad m \leqslant 0 \hfill \\ \end{gathered} \right.$$

((A.7))

and

$$\langle k,{\text{|}}m{\text{||}}\rho {\text{|}}k',{\text{|}}m + 1{\text{|}}\rangle \, = \,\left\{ \begin{gathered} \left( {\frac{m}{{k'}} - \frac{\partial }{{\partial k'}}} \right)\frac{{\delta (k - k')}}{k},\quad m \geqslant 0, \hfill \\ \left( {\frac{{{\text{|}}m{\text{|}}}}{{k'}} + \frac{\partial }{{\partial k'}}} \right)\frac{{\delta (k - k')}}{k},\quad m < 0. \hfill \\ \end{gathered} \right.$$

((A.8))