Nondispersive Two-Electron Wave Packets in a Helium Atom

We demonstrate the existence of stable nondispersing two-electron wave packets in the helium atom in combined magnetic and circularly polarized microwave fields. These packets follow circular orbits and we show that they can also exist in quantum dots. Classically the two electrons follow trajectories which resemble orbits discovered by Langmuir and which were used in attempts at a Bohr-like quantization of the helium atom. Eigenvalues of a generalized Hessian matrix are computed to investigate the classical stability of these states. Diffusion Monte Carlo simulations demonstrate the quantum stability of these two-electron wave packets in the helium atom and quantum-dot helium with an impurity center.

Figure 1

Possible rotating configurations of two electrons in a helium atom or quantum-dot helium in combined magnetic and CP fields. Type I: Langmuir configuration, type II: transverse configuration, type III: collinear configurations (a and b).

Figure 2

Dependence of the length of the sides of the equilateral triangle described by the three charges in the rotating frame on the strength of the CP field for various rotation frequencies. Note that for ${\omega }_c=-1$ the system is bistable and only the larger solution corresponds to the stable trajectory. When ${\omega }_c=1$ there is no critical field and only a single solution [18].

Figure 3

Stability diagram (scaled units) of the disturbed Langmuir trajectory for anticentrifugal Lorentz force (upper half, ${\omega }_c=-1$) and cocentrifugal (lower half, ${\omega }_c=+1$) as a function of scaled rotation frequency and the electric field. The black regions corresponds to stable trajectories. Note that no magnetic field ${\omega }_c<\omega$ can lead to stability in case $-1$ and magnetic field weaker than rotation ${\omega }_c<\omega$ is permitted to obtain stability in case $+1$. This suggests a maximum of the ZVS and Paul traplike (Trojan) stabilization.

Figure 4

Diffusion Monte Carlo simulations of Langmuir wave packets in the rotating frame for ${\Omega }_c=0.0370\mathrm{a}.\mathrm{u}.$, $\mathcal{E}=0.1235\mathrm{a}.\mathrm{u}.$, and $\Omega ={\Omega }_c/2$. The density of points reflects the wave function itself, not probability density [19, 24]. The wave packets are approximate eigenstates of the Hamiltonian in the rotating frame and follow a circular orbit in the inertial frame.