4 [( j +1) j ] 1 / $^{2}$sin( θ ) cos( ϕ ) [44]. We adopt the convention that ρ$_{W}$ is normalized with respect to the measure d Ω /h , allowing to handle on equal footing a cylindrical phase space upon the re-identification d Ω = dϕdn and h = 2 π . Within this phase-space representation, the Fock states | n 〉 are represented by stripes along constant θ con- tours (see e.g. left panel of Fig. 2). The | n = N 〉 state (all particles in one site) is a Gaussian-like wave packet con- centrated around the NorthPole. From this state, we can obtain a family of spin coherent states (SCS) | θ, ϕ 〉 via rotation. is ω ( E ) given by Eq. (18). In particular, the low energy levels have spacing ω$_{J}$ , while the high energy levels are doubly-degenerate with spacing ω$_{+}$ . In the vicinity of the separatrix we get Using the WKB quantization condition, we find that the level spacing at the vicinity of the separatrix ( E ∼ E$_{x}$ ) is finite and given by the expression In what follows, we explore the dynamics of the follow- ing experimentally-accessible preparations (see Fig. 2), the first being a Fock state, whereas the last three are spin coherent states: • TwinFock preparation: The n =0 Fock preparation. Exactly half of the particles are in each side of the double well. The Wigner function is concentrated along the equator θ = π/ 2. • Zero preparation: Coherent ( θ = π/ 2 , ϕ =0) prepa- ration, located entirely in the (linear) sea region. Both sites are equally populated with definite 0 rel- ative phase. The minimal wave-packet is centered at ( n = 0 , ϕ = 0). • Pi preparation: Coherent ( θ = π/ 2 , ϕ = π ) on- separatrix preparation. The sites are equally pop- ulated with π relative phase. The minimal wave- packet is centered at ( n = 0 , ϕ = π ). • Edge preparation: Coherent ϕ = π on-separatrix preparation. The minimal wave-packet is centered on the separatrix but away from the saddle point on the ϕ =0 side. The Wigner function of an SCS resembles a minimal Gaussian wave-packet, and it should satisfy Our approach for investigating the dynamics of various initial preparations relies on the Wigner-function formal- ism for spin variables, developed in Refs. [43, 44]. Each initial preparation is described as a Wigner distribution function over the spherical phase space. The dynam- ics is deduced from expanding the initial state in terms of the semiclassical eigenstates described in Sec III. In this section we specify the Wigner distribution for the preparations under study whereas the following section presents the eigenstate expansion of each of these four initial wavepackets, evaluated semiclassically. This requirement helps to determine the phase space spread without the need to use the lengthy algebra of Refs. [43, 44]. For the Fock coherent state | n = N 〉 , that is centered at the NorthPole ( θ = 0), one obtains: To recap the phase-space approach to spin [43, 44], the Hilbert space of the BHH has the dimension N = 2 j +1, and the associated space of operators has the dimension- ality N $^{2}$. According to the Stratonovich-Wigner-Weyl correspondence (SWWC) [50], any observable A in this space, as well as the probability matrix of a spin ( j ) entity, can be represented by a real sphere (2 j ) function A$_{W}$ (Ω). The sphere (2 j ) is spanned by the Y $^{ℓm}$(Ω) functions with ℓ ≤ 2 j , and the practical details regarding this formal- ism can be found in Refs. [43, 44]. The SWWC allows to do exact quantum calculation in a classical-like man- ner. A few examples for Wigner functions pertinent to this work, are displayed in Fig. 2. Expectation values are calculated as in classical statistical mechanics: For the coherent states centered around the Equator, it is more convenient to use ( ϕ, n ) coordinates, e.g. the ϕ = 0 coherent state is well approximated as, In particular the Wigner-Weyl representation of the identity operator is 1, and that of J$_{x}$ is as expected with a = 1 $_{/}$√$_{2}$ N and b = √ N / 2. Shifted versions of these expressions describe the ϕ = π and the “Edge” preparations. The Wigner function of a Fock state ψ = | n 〉 is semiclassically approximated as ∫ ρ$_{W}$ ( θ, ϕ ) d Ω h = ∫ [ ρ$_{W}$ ( θ, ϕ )] 2 d Ω h = 1 . (33) ρ ( ψ ) W ( θ, ϕ ) ≈ 2e − N 2 θ 2 . (34) ρ ( ψ ) W ( n, ϕ ) ≈ 1 ab e − ϕ 2 2 a $_{2}$− n 2 2 b 2 , (35) ρ ( ψ ) W ( n, ϕ ) ≈ δ ( n − n ) , (36) trace[ˆ ρ ˆ A ] = ∫ d Ω h ρ$_{W}$ (Ω) A$_{W}$ (Ω) (32) IV. THE INITIAL PREPARATION AND ITS PHASE-SPACE REPRESENTATION - THE WIGNER FUNCTION ω ( E ) ≈ [ 1 π log ∣ NK E − E$_{x}$ ∣ ] − 1 ω$_{J}$. (29)