Manifesting the connection between Laguerre-Gaussian modes and hypotrochoid modes from quantum Fock-Darwin system

The relationship between quantum degeneracies and the emergence of classical periodic orbits in the Fock-Darwin (FD) system is systematically explored by using the ladder operators in the Cartesian coordinates. The quantum-classical connection is analytically developed with the time-dependent coherent state that is theoretically verified to correspond to the Gaussian wave packet state with unitary transformation. The time-dependent coherent state is further used to derive the stationary coherent state that can be expressed as a superposition of degenerate eigenstates. More importantly, the stationary coherent state can alternatively be expressed as an integral of the Gaussian wave packet over a periodic orbit. With the integral representation, the quantum vortex structures of the stationary coherent states can be precisely characterized up to extremely high order. The vortex arrays are generally observed when the phase structures are near the crossing points and cusps of the classical trajectories. Finally, the quantum Fourier transform is employed to verify that the circularly symmetric eigenstate physically corresponds to an ensemble of classical hypotrochoid orbits. This work can provide a pedagogical insight into the formation of Laguerre-Gaussian eigenmodes and hypotrochoid geometric modes in degenerate spherical cavities.