Fig. 2: Quantum wave dynamics of guided charged particles at a stable Lagrange point.

a An induced logarithmic, defocusing, and spiraling electrostatic potential when viewed within the stationary frame at \({{{z}}}=0\). This potential profile rotates along z at a constant spatial angular velocity \({\Omega }_{0}\) around the center C \((0,0)\). The corresponding iso-contour potential lines are also shown. b In the co-rotating frame (\({{{u}}},{{{v}}}\)), the effective potential \({{{{\rm{U}}}}}_{\Omega,{{{\rm{eff}}}}}\) now incorporates centrifugal effects and exhibits two Lagrange points, \({{{{\rm{L}}}}}_{{{{\rm{A}}}}}\) and \({{{{\rm{L}}}}}_{{{{\rm{B}}}}}\). \({{{{\rm{L}}}}}_{{{{\rm{B}}}}}\) is a saddle point and hence is unstable. On the other hand, \({{{{\rm{L}}}}}_{{{{\rm{A}}}}}\) (\({{{{u}}}}_{{{{l}}}},{{{{v}}}}_{{{{l}}}}\)) exhibits a maximum, around which dynamics are stabilized through Coriolis effects. For comparison, the positions of \({{{{\rm{L}}}}}_{{{{\rm{A}}}}}\) and \({{{{\rm{L}}}}}_{{{{\rm{B}}}}}\) are also marked in (a). c, e, g Numerically obtained quantum wavefunction probability distributions for the fundamental (c) and next two electron Trojan modes (e, g). The quantum states in (c, e, g) at \({{{{\rm{L}}}}}_{{{{\rm{A}}}}}\) correspond to the potential landscapes in (a, b). In this case, the quantum modes are elliptical in the (\({{{{u}}}}^{{\prime} },{{{{v}}}}^{{\prime} }\)) system where \({{{{u}}}}^{{\prime} }={{{u}}}-{{{{u}}}}_{{{{l}}}}\,,\,{{{{v}}}}^{{\prime} }={{{v}}}-{{{{v}}}}_{{{{l}}}}\). d, f, h Respective phase structures associated with these three electron wavefunction states. In all cases, the phase profile exhibits an X-shaped pattern. i Stable propagation of the quantum Trojan ground state shown in (c), as obtained numerically by solving Eq. (1). The quantum probability function remains invariant along its helical path. The helix pitch in (a–i) is taken to be \(\Lambda=6\,{{{\rm{cm}}}}\) while the electron energy is assumed to be 30 \({{{\rm{keV}}}}\). The normalization factor \({{{{x}}}}_{0}\) in (b–h) is taken to be 1 \({{{\upmu }}}{{{\rm{m}}}}\).