2 The quantum evolution of the spin Hamiltonian (4) is given by the unitary operator exp( − i H t ). Its classical limit is obtained by treating ( J$_{x}$, J$_{y}$, J$_{z}$ ) as c -numbers, whose Poisson Brackets (PB) correspond to the SU(2) Lie algebra. The classical equations of motion for any dynamical variable A are derived from ˙ A = − [ H , A ]$_{PB}$. Conservation of J 2 allows for one constraint on the non- canonical set of variables ( J$_{x}$, J$_{y}$, J$_{z}$ ). Due to the spherical phase-space geometry, it is natural to use the non-canonical conjugate variables ( ϕ , θ ), semiclassical quantization semi-analytically [24, 25, 40– 42] and acquire great insight on the ensuing dynamics of the corresponding Wigner distribution [43, 44]. In this work, we consider the Jopheson-regime dynamics of four different preparations, interpreting the results in terms of the semiclassical phase-space structure and its impli- cations on the expansion of each of these initial states in terms of the semiclassical eigenstates. We first explore the phase-sensitivity of phase-diffusion in the Josephson regime [22, 24]. Then we study the buildup of coherence between two initially separated condensates [41], which is somewhat related to the phase-coherence oscillations ob- served in the sudden transition from the Mott insulator to the superfluid regime in optical lattices [45–47]. Then we compare two coherent preparations in the separatrix region of the classical phase space, finding substantial differences in their dynamics due to their different par- ticipation numbers. The narrative of this work is as follows [a]: The two- site Bose-Hubbard model and its classical phase-space are presented in Section II. The semiclassical WKB quantiza- tion is carried out in Section III. In Section IV, we define the initial state preparations of interest and explain their phase-space representation. This is used in Section V to evaluate their expansion in the energy basis (the local density of states), which is the key for studying the dy- namics in Sections VI to VIII. Both the time-averaged dynamics and the fluctuations of the Bloch vector are analyzed. In particular we observe in Section VIII that the fluctuations obeys a remarkable semiclassical scaling relation. Conclusions are given in Section X. where the scaled parameters are where [ J $^{2}$]1 / 2 is a constant of the motion. In the quantum mechanical Wigner picture treatment (see later sections) this constant is identified as [( j +1) j ] 1 / $^{2}$, based on the derivation of section 2.3 of Ref. [44]. For large j we use [ J $^{2}$]1 / 2 ≈ N/ 2. With these new variables the Hamilto- nian takes the form We consider the Bose-Hubbard Hamiltonian for N bosons in a two-site system, The structure of the underlying classical phase space is determined by the dimensionless parameters u and ε . For strong interactions u > 1 the phase space includes a figure-eight shaped separatrix (Fig. 1), provided | ε | < ε$_{c}$ , where [25, 48] This separatrix splits phase space into three inte- grable regions: a ‘sea’ of Rabi-like trajectories and two interaction-dominated nonlinear ‘islands’. For zero bias the separatrix is a symmetric 8 shaped figure that en- closes the two islands, and the relevant energies are [49] where K is the hopping amplitude, U is the interaction, and E = E 2 − E 1 is the bias in the on-site potentials. We use boldface fonts to mark dynamical variables that are important for the semiclassical analysis, and use regu- lar fonts for their values. The total number of particles n$_{1}$ + n$_{2}$ = N is conserved, hence the dimension of the pertinent Hilbert space is N = N + 1. Defining Looking at the phase-space structure as a function of u , the two islands emerge once u > 1. For u > 2, they en- compass the North and the South poles, yielding macro- scopic self-trapping [9], and for very large u ≫ 1 they cover most of the Northern and the Southern hemi- spheres respectively. Note that for u ≫ 1, the expres- sion E$_{+}$ ≈ ( N/ 2) $^{2}$U , reflects the cost of localizing all the particles in one site, compared to equally-populated sites. and eliminating insignificant c -number terms, we can re- write Eq. (1) as a spin Hamiltonian, E$_{−}$ = − (1 / 2) NK = ground energy , (10) E$_{x}$ = +(1 / 2) NK = separatrix , (11) E$_{+}$ = (1 / 4)[ u +(1 /u )] NK = top energy . (12) J$_{z}$ ≡ 1 2 ( n$_{1}$ − n$_{2}$ ) ≡ n , (2) J$_{+}$ ≡ ˆ a † $_{1}$ˆ a$_{2}$ (3) H = UJ 2 z − E J$_{z}$ − KJ$_{x}$, (4) H = ∑ i =1 , 2 [ E $_{i}$n$_{i}$ + U 2 n$_{i}$ ( n$_{i}$ − 1) ] − K 2 (ˆ a † $_{2}$ˆ a$_{1}$ + ˆ a † $_{1}$ˆ a$_{2}$ ) , (1) ε$_{c}$ = ( u 2 3 − 1 ) 3 2 . (9) u ≡ NU K , ε ≡ E K . (8) H ( θ , ϕ ) = NK 2 [ 1 2 u (cos θ ) 2 − ε cos θ − sin θ cos ϕ ] , (7) II. THE TWO-SITE BOSE-HUBBARD MODEL J$_{z}$ ≡ [ J $^{2}$]1 / $^{2}$cos( θ ) (5) J$_{x}$ [ J $^{2}$]1 / $^{2}$sin( θ ) cos( ϕ ) (6)