Singularities and internal rotational dynamics of electron beams

We study the internal rotational dynamics of electronic beams in relation to the phase singularities of their wave functions. Given their complex singularity structure, Hermite-Gaussian beams and other superpositions of Laguerre-Gaussian modes are studied here. We show that by inspecting the lowest nonvanishing terms of the wave function near the singularity, it is possible to infer the structure of the Bohmian streamlines. Conversely, starting from a map of the electron's Bohmian velocities, we demonstrate that it is possible to derive the form of the electron's wave function near the singularity. We outline a procedure that could yield an experimental method to determine the main parameters of the electron's wave function close to a singularity.

Figure 1

Rotational dynamics of ${\psi }_{3,3}^{HG}$. A vector field of the kinetic density current is superimposed to the density plot to the probability density ${\rho }_{3,3}^{HG}$. (a) The probability density ${\rho }_{3,3}^{HG}$ at $t=0$. The dashed lines and the (purple) point indicate the position of the singularities $S_{3,3}^{HG}$. (b),(c) The evolution of the probability density at $t=\pi /4\omega$ and $t=\pi /2\omega$, respectively.

Figure 2

Rotational dynamics of ${\psi }_{3,3}^{+HG}$. (a),(b) The time evolution of ${\rho }_{3,3}^{+HG}$ and the vector field ${\mathit{J}}^K$ for $t=0$ and $t=\pi /8\omega$, respectively. (c),(d) The time evolution of ${\rho }_{3,3}^{+HG}$ and the vector field ${\mathit{J}}^C$ for $t=0$ and $t=\pi /8\omega$, respectively. The positions of the different kinds of singularities are indicated with (colored) dots in (a).

Figure 3

Internal rotational dynamics of ${\psi }_{3,3}^{-HG}$. (a),(b) The time evolution of ${\rho }_{3,3}^{-HG}$ and the vector field ${\mathit{J}}^K$ for $t=0$ and $t=\pi /8\omega$, respectively. (c),(d) The time evolution of ${\rho }_{3,3}^{-HG}$ and the vector field ${\mathit{J}}^C$ for $t=0$ and $t=\pi /8\omega$, respectively. The positions of the different kinds of singularities are indicated with (colored) dots in (a).

Figure 4

Position and topological charges of the ${\psi }_{3,3}^{+HG}$ and ${\psi }_{3,3}^{-HG}$ singularities. The position and spinning directions of the main types of singularities are superimposed to the density plot to the probability density ${\rho }_{3,3}^{\pm HG}$. These singularities are also shown in Figs. 2 and 1.

Figure 5

Singularities of ${\psi }_{3,3}^{+HG}$. The positions of the four main types of ${\psi }_{3,3}^{+HG}$ singularities are indicated with (colored) dots. The density plot of the density probability ${\rho }_{3,3}^{+HG}$ along with the corresponding vector field ${\mathit{J}}^C$ are shown. (a)–(d) The vorticity of the $S_{3,3}^{+HG}=+2, S_{3,3}^{+HG}=+1, S_{3,3}^{+HG}=-1$, and $S_{3,3}^{+HG}=0$ topological charges, respectively.

Figure 6

Singularities of ${\psi }_{3,3}^{-HG}$. The positions of the four main types of ${\psi }_{3,3}^{-HG}$ singularities are indicated with (colored) dots. The density plot of the density probability ${\rho }_{3,3}^{-HG}$ along with the corresponding vector field ${\mathit{J}}^C$ are also shown. (a)–(d) show the vorticity of the $S_{3,3}^{-HG}=-2, S_{3,3}^{-HG}=-1, S_{3,3}^{-HG}=+1$, and $S_{3,3}^{-HG}=0$ topological charges, respectively.

Figure 7

${\eta }^K$ and ${\eta }^C$ for ${\psi }_{3,3}^{HG}$ (orange), ${\psi }_{3,3}^{+HG}$ (blue), and ${\psi }_{3,3}^{-HG}$ (green) as functions of $r=\sqrt{x^2+y^2}$. In the negative beam, ${\psi }_{3,3}^{-HG}{\eta }^K$ changes sign and ${\eta }^C$ is symmetric with respect to the positive beam ${\psi }_{3,3}^{+HG}$.

Figure 8

Topological charges of the ${\psi }_{3,3}^{-HG}$ and ${\psi }_{3,3}^{+HG}$ singularities. ${\eta }^K/2\pi \omega l_B^2\rho$ (solid lines) and ${\eta }^C/2\pi \omega l_B^2\rho$ (dashed lines) are plotted as functions of $r$, the distance to the singularity. (a)–(d) show ${\eta }^C/2\pi l_B^2\rho$ and ${\eta }^K/2\pi l_B^2\rho$ corresponding to the $S_{3,3}^{\pm HG}=\pm 2, S_{3,3}^{\pm HG}=\pm 1, S_{3,3}^{\pm HG}=\mp 1$, and $S_{3,3}^{\pm HG}=0$ singularities, respectively. The singularities (a), (c), and (d) are approached along the angle $\theta =\pi /3$, and the one in (b) is approached along $\theta =0$.

Figure 9

Rotational dynamics of ${\psi }_3^{+BEC}$. (a),(b) The time evolution of ${\rho }_3^{+BEC}$ and the vector field ${\mathit{J}}^K$ for $t=0$ and $t=\pi /6\omega$, respectively. (c),(d) The time evolution of ${\rho }_3^{+BEC}$ and the vector field ${\mathit{J}}^C$ for $t=0$ and $t=\pi /6\omega$, respectively. The positions of the three singularities are indicated with (orange) dots in (a).

Figure 10

Rotational dynamics of ${\psi }_3^{-BEC}$. (a),(b) The time evolution of ${\rho }_3^{-BEC}$ and the vector field ${\mathit{J}}^K$ for $t=0$ and $t=\pi /6\omega$, respectively. The singularities clearly revolve around the center in the counterclockwise direction. (c),(d) The time evolution of ${\rho }_3^{-BEC}$ and the vector field ${\mathit{J}}^C$ for $t=0$ and $t=\pi /6\omega$, respectively. The positions of the three singularities are indicated with (orange) dots in (a).

Figure 11

Topological charges and positions of the (a) ${\psi }_3^{+BEC}$ and (b) ${\psi }_3^{-BEC}$ singularities. The position and the kinetic density currents' spinning direction of the singularities are superimposed to the density plot to the probability density ${\rho }_3^{\pm BEC}$. The vector plot indicates the vector field of the kinetic density current ${\mathit{J}}^K$. These singularities are also shown in Figs. 9 and 10.

Figure 12

${\eta }^K$ and ${\eta }^C$ for ${\psi }_3^{BEC}$ (orange), ${\psi }_3^{+BEC}$ (blue), and ${\psi }_3^{-BEC}$ (green) as functions of $r=\sqrt{x^2+y^2}$. In the negative beam ${\psi }_3^{-BEC}, {\eta }^K$ changes sign and ${\eta }^C$ is symmetric with respect to the positive beam ${\psi }_3^{+BEC}$.

Figure 13

Topological charges of the ${\psi }_3^{-BEC}$ and ${\psi }_3^{+BEC}$ singularities. ${\eta }^K/2\pi \omega l_B^2\rho$ (solid lines) and ${\eta }^C/2\pi \omega l_B^2\rho$ (dashed lines) are plotted as functions of $r$, the distance to the singularity. The plot is shown for the $S_3^{\pm BEC}=\pm 1$ singularity located in $z={\left(2\sqrt{3}\right)}^{1/3}exp(i\pi /3)$. The singularity is approached along the angle $\theta =0$.

Figure 14

Rotational dynamics of $\psi$. (a),(b) The time evolution of $\rho ={\left|\psi \right|}^2$ and the vector field ${\mathit{J}}^K$ for $t=0$ and $t=\pi /6\omega$, respectively. (c),(d) The time evolution of $\rho$ and the vector field ${\mathit{J}}^C$ for $t=0$ and $t=\pi /6\omega$, respectively. The positions of the singularity $z_j=3l_B+i{l}_B$ are indicated with an (orange) dot in (a).

Figure 15

Bohmian velocity around the singularity $z_j=3l_B+i{l}_B$ of $\psi$. The position of the singularity is indicated with a dot. The density plot of the density probability $\rho ={\left|\psi \right|}^2$ and the corresponding Bohmian velocity field ${\mathit{v}}^K$ are also shown.

Figure 16

(a) Normalized coordinates ${\mathcal{X}}^{\prime }, {\mathcal{Y}}^{\prime }$ in the standard frame and (b) normalized coordinates $\mathcal{X}, \mathcal{Y}$ in the frame aligned with the singularity's symmetry axes. The symmetry axes form an angle of $\nu =32.{31}^{\circ }=0.5640\mathrm{rad}\approx \pi /6\mathrm{rad}$ with the ${\mathcal{X}}^{\prime }$ axis. In both frames, the fit is indicated with a solid line.

Figure 17

${\mathcal{X}}^2$ vs ${\mathcal{Y}}^2$ and fit ${\mathcal{Y}}^2=-m{\mathcal{X}}^2+c, m=-0.441762, c=1.99527$.