Group-theoretical formalism of quantum mechanics based on quantum generalization of characteristic functions

Non-adiabatic Mapping Dynamics in the Phase Space of the ${SU}(N)$ Lie Group

We present the rigorous theoretical framework of the generalized spin mapping representation for non-adiabatic dynamics. Our work is based up a new mapping formalism recently introduced by Runeson and Richardson in [J. Chem. Phys. 152, 084110 (2020)], which uses the generators of the su(N) Lie algebra to represent N discrete electronic states, thus preserving the size of the original Hilbert space. Following this interesting idea, the Stratonovich-Weyl transform is used to map an operator in the Hilbert space to a continuous function on the SU(N) Lie group, i.e., a smooth manifold which is a phase space of continuous variables. We further use the Wigner representation to describe the nuclear degrees of freedom, and derived an exact expression of the time-correlation function as well as the exact quantum Liouvillian for the non-adiabatic system. Making the linearization approximation, this exact Liouvillian is reduced to the Liouvillian of several recently proposed methods, and the performance of this Linearized method is tested using non-adiabatic models. We envision that the theoretical work presented here provides a rigorous and unified framework to formally derive non-adiabatic quantum dynamics approaches with continuous variables and connect the previous methods in a clear and concise manner.

Ergodicity and scars of the quantum cat map in the semiclassical regime

We study the quantum localization effect of the cat map manifested in the motions of statistical ensembles. Specifically, the coarse-grained entropy and time-averaged phase space distributions are investigated. For this purpose, an amended version of the Wigner function on the discretized phase torus is presented. We find that the time average of the coarse-grained Wigner function is scarred (antiscarred) along some short periodic orbits, and the heights (depths) of these scars (antiscars) decrease in a linear way with the Planck constant when the semiclassical limit is approached. The relationship between the scars observed here and those exhibited in the quasienergy eigenstates is discussed.