as a basis to describe the quantum dynamics, where θ and ϕ are the angles defining the Bloch vector, with the radius of the Bloch sphere being fixed. The spin coherent states |u can be generalized for a N -level system, denoted as |Ω , through a rigorous procedure with Lie groups and Lie algebras. Mathematically, these generalized spin-coherent states |Ω are introduced by acting a parametrized unitary transformation operator on a given diabatic basis |n , i.e., |Ω = Û (θ, ϕ)|n , where the unitary transformation operator Û (θ, ϕ) is an exponential function to linear order of the generators Ŝ (Eq. 2-Eq. 4) associated with real parameters {θ, ϕ}. The detailed expression of Û (θ, ϕ) can be found in Eq. (2.6) of Ref. 50. This particular expression is referred to as the generalized Euler angle parametrization of su(N ), which gives rise to a continuous phase space. Actually, one can regard Û (θ, ϕ) as a unitary representation of the SU (N ) Lie group. The generalized spin coherence state can be expressed as