Trojan Wave Packets in the Circularly Polarized and the Magnetic Fields on the multi-layer Langmuir type $(1)$ Helium trajectories with the inter-layer polygon type change and twist

Upon the discovery of the single electron Trojan Wave Packets in an atom of Hydrogen driven by the

Circularly Polarized (C.P.) electromagnetic field [1] we have formulated the general theorem stating that the

number of a distinct classically rotating frozen $N$-electron

configurations in the external magnetic field, the nuclear ion

field and the co-rotating electric field may be at least the

product of all naturally defined foldings of total force surfaces (the maximal number

any straight line may cut the surface which nontrivially is 2) obtained

by putting each of the $3N$-dimensional gradient component of

the Zero Velocity Surface to zero. Consistently with it

we have discovered a several highly symmetric configurations

originated from the Langmuir type $(1)$ ``Hoop Earrings" rotating

Helium-like model trajectories [1] with further small discrete

rotational symmetry $C_{nv}$ distortion by the polarizing Circularly Polarized

electromagnetic field without the loss of its nature.

Those were the multi-electron atom trajectories consisting

of many layers of electrons moving in phase on

circles with electron configurations placed at the

vertexes of angles of identical by type but only scaled

regular polygons parallel in space which additionally

could be also symmetrically twisted between each other

or when the neighboring layer vertex-aligned polygons are

plane-symmetrically of a different kind.

Here we combine the cases to the new situation when the neighboring

layer polygons are not only of a different kind but

also symmetrically twisted to preserve the discrete-rotational invariance

of shape.

For example the hexagon in the central plane may accompany

two twisted triangles placed symmetrically one in the upper

parallel plane and the other in the lower with their vertices being equidistant from

each of two of the closest vertices of this hexagon or the octagon

may similarly neighbor two twisted squares etc. The classical

stabilization of the trajectories by the combination of

fields further leads to the existence of non-dispersing

localized Trojan-like wave packets moving around the trajectories.

The time dependent Hartree simulations of such Wave Packets

as well as those with the Time Dependent Quantum Diffusion

Monte Carlo Method are conducted. [1] M. Kalinski, et al.,

Phys. Rev. Lett. {bf 95}, 103001, (2005).