Semi-Classical Discretization and Long-Time Evolution of Variable Spin Systems

Abstract

We apply the semi-classical limit of the generalized SO(3) map for representation of variable-spin systems in a four-dimensional symplectic manifold and approximate their evolution terms of effective classical dynamics on T*S2. Using the asymptotic form of the star-product, we manage to “quantize” one of the classical dynamic variables and introduce a discretized version of the Truncated Wigner Approximation (TWA). Two emblematic examples of quantum dynamics (rotor in an external field and two coupled spins) are analyzed, and the results of exact, continuous, and discretized versions of TWA are compared.

New phase space formulations and quantum dynamics approaches

“…The expression of eq (130) for the amplitudes of being in different states was already used in Appendix B of ref 133 , while the action-angle version of eq (130) was earlier presented in Meyer and Miller's seminal paper 155 . The   U F constraint coordinate-momentum phase space is diffeomorphic to U( ) / U( 1) F F , which is equivalent to SU( ) / SU( 1) F F 309 because both of them lead to the (2 1) F  -dimensional sphere in 2F dimensional Euclidean space 310,311…”

“…Further developments of Stratonovich's formulation have focused on an SU (2) or SU(F) structure of phase space [103][104][105][106][107][108][109][110][111][112][113][114][115][116][117] , while those on the construction of a discrete phase space are described in References 78,[118][119][120][121][122][123][124][125][126] . Other than the 2-state (or spin 1/2) system, the exact equations of motion (EOMs) of phase variables (expressed by the Moyal-like bracket) involved in these approaches for the finite discrete multi-state system are often tedious and numerically unfavourable 109,[127][128][129][130][131][132] . (See Appendix 3 of the Supporting Information for more discussion.)…”

“…Further developments of Stratonovich's formulation have focused on an SU (2) or SU(F) structure of phase space, [103][104][105][106][107][108][109][110][111][112][113][114][115][116][117] while those on the construction of a discrete phase space are described in References 78,[118][119][120][121][122][123][124][125][126] Other than the 2-state (or spin 1/2) system, the exact equations of motion (EOMs) of phase variables (expressed by the Moyal-like bracket) involved in these approaches for the finite discrete multi-state system are often tedious and numerically unfavourable. 109,[127][128][129][130]299,302 (See Appendix 3 of the Supporting Information for more discussion.) Recent theoretical progress on exactly mapping the finite discrete F-state quantum system onto constraint coordinate-momentum phase space suggests that there exists a novel unified framework to derive comprehensive exact mapping Hamiltonians, 44,57,131,132 of which the quantum EOMs of mapping coordinatemomentum variables are simply linear.…”