$N$-electron giant dipole states in crossed electric and magnetic fields

Multielectron giant dipole resonances of atoms in crossed electric and magnetic fields are investigated. Stationary configurations corresponding to a highly symmetric arrangement of the electrons on a decentered circle are derived, and a normal-mode and stability analysis are performed. A classification of the various modes, which are dominated by the magnetic field or the Coulomb interactions, is provided. Based on the MCTDH approach, we carry out a six-dimensional wave-packet dynamical study for the two-electron resonances, yielding in particular lifetimes of more than $0.1\mu \mathrm{s}$ for strong electric fields.

Figure 1

(Color online) The extremal component $X_{1∕2}$ as a function of $K∕K_{\mathit{cr}}$ for different field strengths $B∕\mathrm{a.u.}$ The critical point $X\left(K_{\mathit{cr}}\right)$ corresponds to the vertex where $X_1$ (lower graph) and $X_2$ (upper graph) meet.

Figure 2

(Color online) A giant-dipole configuration for $N=4$ electrons ($B={10}^{-4}$, $K=2K_{\mathit{cr}}$). The e.c.m. $\mathbf{R}=(X,0,0)$ is decentered, with the relative vectors ${\mathbf{s}}_i$ confined to a circle as indicated. Also shown is the electronic vector relative to the nucleus ${\mathbf{r}}_i=\mathbf{R}+{\mathbf{s}}_i$. A typical example of the cyclotron and the motion due to a Coulomb mode is indicated (see Sec. 4B1).

Figure 3

Eigenmodes ${\left\{{\gamma }_{\rho }\right\}}_{\rho }$ for $N=2$ electrons as a function of the electric field $E\equiv BK∕M$ over the range $K∕K_{\mathit{cr}}∊[1,10]$ (all modes are imaginary). The three plots refer to the magnetic fields $B={10}^{-5},{10}^{-4},{10}^{-3}$ (from left to right). Each top horizontal line represents two almost degenerate cyclotron modes. Below, the two Coulomb modes fall off quickly and intersect the c.m. mode (the nearly horizontal line about four orders below the cyclotron modes).

Figure 4

Eigenmodes $\gamma$ for the cases $N=3$ (a), $N=4$ (b), $N=5$ (c), and $N=10$ (d), as a function of the electric field $E\equiv BK∕M$, plotted over the range $K∕K_{\mathit{cr}}∊[1,10]$. The magnetic field is $B={10}^{-4}$. Imaginary parts appear as solid lines, while points (+) are use d for the real parts. The individual modes are assigned just as for Fig. 3, with an extra class of very small decay modes, which fall off twice as fast as the Coulomb lines.

Figure 5

Visualization of the zero mode, plot of the generalized potential $\mathcal{V}$ for $N=2$ close to the extremum, cut along the $yz$ coordinates of $\mathbf{s}$ ($K∕K_{\mathit{cr}}=2$; $B={10}^{-4}$). There is a circle that continuously connects all minima, referred to as stationary circle.

Figure 6

Upon rotating the circular configuration by an angle $\Delta$, the spectrum is changed in part ($B={10}^{-4}$, $K=2K_{\mathit{cr}}$). The cyclotronic modes $\left(658\mathrm{GHz}\right)$ and c.m. mode $\left(0.17\mathrm{GHz}\right)$ remain constant. The Coulomb modes and, for $N=3$, the decay mode $(\sim {10}^{-2}\mathrm{GHz})$ show some periodic modulation. (For symbol coding, see Fig. 4.)

Figure 7

Spectrum for $N=4$ depending on the rotation angle $\Delta$ ($B={10}^{-4}$, $K=2K_{\mathit{cr}}$); see text.

Figure 8

The motion in $X$ in the case $K∕K_{\mathit{cr}}=1.1$. (a) Snapshots of the motion ${\rho }_X$, reflecting oscillations of both center and shape of the wave packet. (b) The two oscillations as documented in $⟨X⟩\left(t\right),\Delta X\left(t\right)$.

Figure 9

Cuts through the generalized potential near the extremum, $\mathcal{V}(X,Y,0;\mathbf{0})$ (a) and $\mathcal{V}(\mathbf{0};x,y,0)$ (b), exemplified for $K=1.1K_{\mathit{cr}}$. The $\left(XY\right)$ potential is almost harmonic. The $y$ direction virtually coincides with the zero mode, whereas the potential exhibits an instability with respect to $x$.

Figure 10

Cuts through the generalized potential parallel to the magnetic field, (a) $\mathcal{V}(0,0,Z;\mathbf{0})$ is roughly harmonic, (b) $\mathcal{V}(\mathbf{0};0,0,z)$ is strongly flattened on the right-hand region of the minimum. (Note that the extremum is shifted to the origin.)

Figure 11

Snapshots of the wave packet’s motion in the direction of $x$ (a), $y$ (b), and $z$ (c) for $K=1.1K_{\mathit{cr}}$. The plots display the reduced densities ${\rho }_x$, ${\rho }_y$, and ${\rho }_z$ at different times. While the unstable mode $x$ leaks slowly, the zero mode $y$ spreads much more quickly. The wave packet in $z$ is only marginally shifted over time. In (d), the different oscillations in $z$ are illustrated.

Figure 12

The motion of the $Z$ mode in the case $K∕K_{\mathit{cr}}=1.1$. Three snapshots of the reduced density ${\rho }_Z$ are plotted.

Figure 13

The expectation values and uncertainties over selected time periods for the coordinates (a) $Z$ and (b) $z(K∕K_{\mathit{cr}}=2)$.

Figure 14

The relative coordinate ${\mathbf{s}}_{\perp }$ in the case $K∕K_{\mathit{cr}}=2$: Characteristic snapshots of the reduced density ${\rho }_y$.

Figure 15

Snapshots of the wave-packet upon an initial displacement by $Z=z=2000\mathrm{a.u.}$ for $K∕K_{\mathit{cr}}=2$. Plot (a) illustrates the strong deformations in $Z$ in the course of the oscillation; (b) shows how the the $z$ packet is distorted after several thousand picoseconds. The nonstable modes $x,y$ are affected via coupling, see (c) and (d).

Figure 16

(Color online) A generic relaxation state ($K=2K_{\mathit{cr}}$, see text)—plotted is the 2D-reduced density for the plane perpendicular to $B$: the wave packet is driven away from the $x$ axis with increasing basis size.

Figure 17

(Color online) Wave packet of the 2D model $\left(xy\right)$ after $T=160\mathrm{ns}$ ($K=2K_{\mathit{cr}}$, see text).

Figure 18

The time evolution of $⟨z⟩,\Delta z$ (a) and $\Delta y$ (b) for $K∕K_{\mathit{cr}}=10$. Both degrees are altogether stable, but oscillate due to coupling. A marginal broadening can be seen in $y$.

Figure 19

The circular configuration ${\mathbf{s}}^{\left(0\right)}=s_0{(0,\cos {\varphi }_0,\sin {\varphi }_0)}^T$ for $N=2$, depending on the angle ${\varphi }_0$. The arrows refer to ${\mathbf{s}}^{\left(0\right)}$ in the $yz$ plane for different values of ${\varphi }_0$. The $X$ axis points into the page.

Figure 20

The horizontal configuration ${\varphi }_0=\pi (K∕K_{\mathit{cr}}=10)$. The wave packet switches between two positions (snapshots of reduced density for $11500\mathrm{ps}$ and $20000\mathrm{ps}$).

Figure 21

Fourier transform of the autocorrelation function (the spectrum) for $K∕K_{\mathit{cr}}=1.1$.

Figure 22

Spectrum for $K∕K_{\mathit{cr}}=2$. The dotted peak again belongs to the initial state. For the excited (or displaced) initial state, the spectrum shows a sequence of peaks (here almost equidistant).