Dynamic electron correlation in interactions of light with matter formulated in $\vec{b}$ space

Scattering of beams of light and matter from multielectron atomic targets is formulated in the position representation of quantum mechanics. This yields expressions for the probability amplitude $a\left(\vec{b}\right)$ for a wide variety of processes. Here the spatial parameter $\vec{b}$ is the distance of closest approach of incoming particles traveling on a straight line with the center of the atomic target. The correlated probability amplitude $a\left(\vec{b}\right)$ reduces to a relatively simple product of single-electron probability amplitudes in the widely used independent electron approximation limit, where the correlation effects of the Coulomb interactions between the atomic electrons disappear. As an example in which $a\left(\vec{b}\right)$ has an explicit dependence on $\vec{b}$, we consider transversely finite vortex beams of twisted photons that lack the translational invariance of infinite plane-wave beams. Relatively simple calculations, illustrating the $\vec{b}$ dependence in transition probabilities for photon beams interacting with a two-state degenerate single-electron atomic target, are included. Further application for many-electron systems is discussed. Possible practical uses are briefly considered.

Figure 1

Sketch in the $\widehat{b}-\widehat{z}$ plane of an atom interacting with a Gaussian or Gauss-Laguerre vortex beam [18]. The maximum of the Gaussian envelope (shown here) of the beam intensity distribution is along the beam axis, and the envelope is cut off when its intensity falls by a factor of $1/e^2\simeq 0.135$ of its maximum. This defines the waist size $w\left(0\right)$ of the center of the beam. The two other independently variable sizes are the mean radius of the atom, $a_T$, and the wavelength of the light, $\lambda$ (not shown here). If the beam is a twisted photon (or electron) beam, it may carry orbital angular momentum, corresponding to a localized photon (or electron) that passes through $\vec{b}$ as it rotates about the $z$ axis. The origin of $\vec{b}$ is arbitrary: it may be either at the center of the beam or the center of the atom, for example. When the twisted vortex beam carries orbital angular momentum ($\ell \ne 0$), the beam has a more complex geometry [16] and has zero intensity along its axis. The atom shown is in its ground state with $\ell =0$. In order to exchange angular momentum with a twisted vortex beam with $\ell \ne 0$, the atomic electron must be in an excited state with a nonzero value of $\ell$ that matches that of the twisted photon but has an opposite direction. In this case the atom has a more complex structure than that shown here and the electronic wave function has a node at the center of the atom.

Figure 2

Probability $P(\vec{b},t)$ of a transition from state $|1\rangle$ to state $|2\rangle$ in a degenerate two-state atom interacting with a plane-wave photon beam, calculated as a function of time $t$ at a typical value of $\vec{b}$. Here $t$ varies from 0 to $T$, where $T$ is the period of oscillation of the photon's electric field; this pattern repeats for longer times. In this figure the interaction is nonperturbative: $H_{12}\left(\vec{b}\right)T/h=2.718$.

Figure 3

Probability $P(\vec{b},t)$ as a function of time $t$ at special values of $\vec{b}$. When $H_{12}\left(\vec{b}\right)T/h\ge \pi /2$, the interaction is nonperturbative and the probability can reach unity. Dashed curve: $\vec{b}$ is chosen such that $H_{12}\left(\vec{b}\right)T/h=1/2$. Solid curve: $H_{12}\left(\vec{b}\right)T/h=\pi /2$; here the peak is broad in time [33]. Dotted curve: $H_{12}\left(\vec{b}\right)T/h=\pi$; here the probability oscillates regularly, but without the broad maximum in time for the full transfer of the atomic electron population from state $|1\rangle$ to state $|2\rangle$.

Figure 4

Transfer probability $P(\vec{b},t)$ versus $R\left(\vec{b}\right)=H_{12}\left(\vec{b}\right)T/h$ occurring at an impact parameter $\vec{b}$ during an interaction of a Gaussian modified plane-wave photon with a degenerate two-state atom. The calculation is performed at $t=n_{\mathrm{odd}}T/4$, where $P(\vec{b},t)$ can have a broad maximum in time. Complete transfer occurs when $R\left(\vec{b}\right)=m_{\mathrm{odd}}\pi /2$. While the horizontal scale in $R\left(\vec{b}\right)$ decreases from $3\pi /2$ on the left to 0 on the right, the impact parameter grows from $b=0$ to $b=\infty$ since $H_{12}\left(\vec{b}\right)$ generally falls off with $b$.

Figure 5

Intensity distribution of the Gaussian envelope of Fig. 1 as a function of the angle ${\theta }_V\left(b\right)$ of the photon trajectory. ${\theta }_V\left(b\right)$ is the same as the asymptotic cone angle of the beam shown in Fig. 1, and is related to the magnitude of the impact parameter $\vec{b}$ by $tan{\theta }_V\left(b\right)=b/z_R$, where $z_R$ is the Rayleigh range of the beam. When $b=w\left(0\right)$ the intensity of the beam has dropped by a factor of $1/e^2\simeq 0.135$ from the maximum intensity at $b={\theta }_V\left(b\right)=0$. Here $w\left(0\right)$ is the waist size of the beam, i.e., the beam width at the longitudinal center of the beam, and $w\left(0\right)=\sqrt{\lambda z_R/\pi }$ where $\lambda$ is the photon wavelength. Since $w\left(0\right)\ll z_R$ in the paraxial approximation, ${\theta }_V\left(b\right)$ is linearly related to $b$ in the range of interest, and thus the linear horizontal scale in ${\theta }_V\left(b\right)/{\theta }_V(b=w\left(0\right))$ is equivalent to a linear scale in $b/w\left(0\right)$ ranging from 0 to 1.