WKB quantization. Thus, closed semiclassical results are obtained for the local density of states of the vari- ous initial preparations, providing useful insights for the associated quantum evolution. Within the framework of mean-field theory (MFT), the dynamics is obtained by evolving a single point in phase space, using the Gross-Pitaevskii (GP) equation, which in this context is better known as the discrete nonlinear Schr¨ odinger (DNLS) equation. By contrast, the trun- cated Wigner phase-space method evolves an ensemble of points according to the DNLS equation, and thereby takes into account the non-linear squeezing or stretching of the distribution. RMS [ 〈 A 〉 $_{t}$] = [ 1 M ∫ ˜ C cl ( ω ) dω ] 1 / 2 . (69) 9 Consider now the RMS of S$_{x}$ ( t ). For TwinFock prepa- ration it follows from Eq.(41) that M ∝ N and hence Eq.(69) implies 1 /N 1 / 2 suppression of the RMS. For co- herent preparations ˜ C cl ( ω ) becomes N dependent too, and consequently from the discussion after Eq. (46) it follows that the RMS is a function of the semiclassical ra- tio Eq. (57), and multiplied by 1 /N 1 / 4 suppression factor that spoils the semi-classical scaling. This is confirmed by the numerics (Fig. 8). If the dynamics is very close to the separatrix the classical fluctuations are O (1) and there- fore the quantum result is RMS [ S$_{x}$ ( t )] ≈ 1 $_{/}$√$_{M}$$_{.}$ The implication for the on-separatrix coherent preparations, Pi versus Edge, is striking: Substitution of Eqs.(45-46) into Eq.(69) leads to In the semiclassical treatment the quantum state in any moment is regarded as a “mixture” of wavefunctions ψ$_{i}$ rather than a single ψ$_{i}$ . It is worth noting that the stationary solutions of the DNLS equation are simply the fixed points of the Hamiltonian flow. The small oscilla- tions obtained by linearization around these fixed points are the so-called Bogoliubov excitations. The typical oscillation frequency of the Bloch vector generally ap- proaches the classical frequency as N is increased keep- ing u fixed. However, in the vicinity of the separatrix convergence to the (vanishing) classical frequency is log- arithmically slow, as found via WKB quantization. Thus, convergence to classicality is far more rapid for the Edge-preparation than it is for the Pi-preparation, even though both lie on the separatrix. With the Pi prepara- tion, even if N is very large (small “ h ”), quantum fluc- tuations still remain pronounced. In fact, from Eq. (45) it follows that the fluctuations in the Pi case are mainly sensitive to the strength u of the interaction. Based on the ratio between the width of the semiclas- sical distribution for SCS and the width of the separatrix phase-space region, we find that the long-time FringeVisi- bility of an initially coherent state in the Josephson inter- action regime, has a u/N dependent value (Fig. 7). The functional dependence on u/N varies according to the preparation. In particular, whereas a Zero relative-phase preparation remains roughly Gaussian (i.e. phase-locked) throughout its motion, thereby justifying the use of MFT for the description of Josephson oscillations around it, a Pi relative-phase SCS squeezes rapidly and its relative- phase information is lost [22, 24]. In contrast, starting from fully separated modes, the phase distribution in the Josephson regime assumes a non-uniform profile, peaked at ϕ = 0, yielding a universal FringeVisibility value of 1 / 3 [41]. Finally, we mention that the analysis of fluctuations above is somewhat related to the discussion of thermal- ization in Ref.[53], and we would like to further connect it with the observation of collapses and quantum revivals as discussed e.g. in Ref.[54]. Relating to the LDOS, as defined in Section V, we note that the collapse time is the semiclassical time which is determined by the the width of the classical envelope, while the revival time is related to the spacing between the spectral lines. The latter can be calculated using the formula Focusing on two types of coherent preparations in the vicinity of the separatrix we find significant differ- ences in their M dependence on ( u ; N ). The Pi SCS preparation (with vanishing population imbalance and a π relative-phase) exhibits u dependent fluctuations, whereas the Edge SCS (having a comparable population imbalance but located elsewhere along the separatrix) ex- hibits N dependent fluctuations. Only in the latter case is the classical limit approached easily by taking large N at fixed u . with the WKB estimate for E$_{ν}$ in Section III, leading in the separtatrix region to ∼ 2 π/ω$_{x}$ . To conclude, we have applied a semiclassical phase- space picture to the analysis of the one-particle coher- ence loss and buildup in the bosonic Josephson junction, described by the two-site BHH. The simplicity of the clas- sical phase space of the dimer allows for its semi-analytic Acknowledgments: We thank Issac Israel for prepar- ing a convenient code for the classical simulations during his visit in BGU. DC and TK acknowledge support from the USA-Israel Binational Science Foundation (Grant No.2006021). AV acknowledges support from the Israel IX. CONCLUSIONS t$_{revival}$ = 2 π [ dE$_{ν}$/dν ] − 1 (71) RMS [ S$_{x}$ ( t )] ∼ { N − 1 / 4 for Edge (log( N )) − 1 / 2 for Pi (70)