Manifesting the connection between topological structures of quantum stationary coherent states and bundles of classical Lissajous orbits

Quantum stationary coherent states with spatial intensities localized on Lissajous orbits are theoretically explored by taking the inverse Fourier transform of the time-dependent coherent state. It is analytically verified that the stationary coherent state can be expressed as an integral of the Gaussian wave packet over the classical periodic orbit. With the derived integral, the phase singularities of the stationary coherent state can be precisely manifested from low-order to extremely high-order states. It is found that the phase singularities near the cross-point of the Lissajous orbits are generally arranged as rhombic vortex arrays. Finally, the stationary coherent states can be used as a basis to manifest the connection between topological structures of quantum eigenstates and bundles of classical Lissajous orbits.