7 phase-locked, while the coherence of the Pi prepa- ration is partially lost [22, 24, 52], exhibiting huge fluctuations. time average S$_{x}$ . For example, for the initial Twin- Fock preparation (Fig. 4d), it reproduces the univer- sal Josephson-regime FringeVisibility of ∼ 1 / 3, resulting from the dynamical smearing of the Wigner distribution function throughout the linear sea region of phase-space [24]. In the remaining part of this section we quantify these observations by finding the long-time average S$_{x}$ ( t ) based on simple phase-space considerations. In the Joseph- son regime, the value of S$_{x}$ ( t ) for a coherent prepa- ration should be determined by the ratio between its Δ n ≈ √ N/ 2 width and the width of the separatrix re- gion Δ n ≈ √ NK/U , i.e. In what follows we would like to determine the long time average S$_{x}$ and the power spectrum ˜ C ( ω ) = FT[ f ( t )] of the temporal fluctuations f ( t ) = S$_{x}$ ( t ) − S$_{x}$ . (FT denotes Fourier Transform). Characteristic power- spectra for the pertinent preparations are shown in the right panels of Fig. 5. We characterize the fluctuations by their typical frequency ω$_{osc}$ , by their spectral support (discrete or continuous-like), and by their RMS value: The dependence of ω$_{osc}$ , and S$_{x}$ , and RMS[ S$_{x}$ ] on the dimensionless parameters ( u, N ) is illustrated in Figs. 7- 8. It should be realized that the observed frequency of the S$_{x}$ ( t ) oscillations is in fact 2 ω$_{osc}$ due to the mirror symmetry of the observable. This ratio determines the long-time phase-space distribu- tion of the evolving semiclassical cloud: In the case of a TwinFock preparation this cloud fills the entire sea re- gion; in the ”Zero” case it is confined to an ellipse within the sea region; and in the ”Pi” case it stretches along the separatrix and therefore resembles a micro-canonical distribution. The projected phase-distribution P( ϕ ) is determined accordingly. Disregarding a global normal- ization factor we get In the Fock regime, the FringeVisibility of an ini- tial coherent ϕ preparation decays to zero: the ini- tial Gaussian-like distribution located at ( θ = π/ 2 , ϕ ) is stretched along the equator, leading to increased relative- phase uncertainty with fixed population-imbalance. This phase spreading process is known in the literature as ‘phase diffusion’ [14–17, 21]. By contrast, a TwinFock preparation is nearly an eigenstate of the BHH in this regime, and its zero FringeVisibility remains vanishingly small from the beginning. A few words are in order regarding the derivation of the above expressions. The Zero case prepa- ration is represented by the Gaussian of Eq.(35) whose major axis is Δ n ≈ ( N/ 2) 1 / $^{2}$. This Gaussian evolves along the contour lines of the Hamiltonian H ( n, ϕ ) = Un 2 + ( NK/ 2) cos( ϕ ). After sufficiently long time the evolving distribution still has the same Δ n , but because of the spreading its other major axis, as determined by the equation U Δ n 2 = ( NK/ 4)Δ ϕ $^{2}$, be- comes Δ ϕ ≈ (2 U/K ) 1 / 2 leading to Eq.(58). The Twin- Fock preparation is represented by Eq.(36). After suffi- ciently long time the evolving distribution fills the whole sea H ( θ, ϕ ) < E$_{x}$ . The equation that describes this filled sea can be written as n < n$_{x}$ ( ϕ ), where n$_{x}$ ( ϕ ) = √ NK 2 U (1 + cos( ϕ )) . (61) VII. DYNAMICS (II) - FRINGE VISIBILITY IN THE JOSEPHSON REGIME RMS[ S$_{x}$ ] = [ f ( t ) $^{2}$] 1 / 2 = [∫ ˜ C ( ω ) dω ] 1 / 2 , (56) • The Edge preparation exhibits distinct behavior, that neither resembles the Zero nor the Pi prepa- rations, involving sign reversal of S$_{x}$ ( t ). In the Josephson regime the dynamics of the single- particle coherence is more intricate. In this section we discuss the evolution of S ( t ) within the framework of the semiclassical approximation. As a first step, we dis- regard fluctuations and recurrences and address only the time-averaged dynamics. In particular, we determine the long-time average of the FringeVisibility, which for the TwinFock, Pi, and Zero preparations is given by S$_{x}$ ( t ), as noted after Eq. (55). Numerical results for the long-time average and for the RMS of the fluctuations as a function of u/N are pre- sented in Fig. 8 and further discussed below. The main observations regrading the dynamics in the Josephson regime are: The projection of area under n < n$_{x}$ ( ϕ ) is simply P( ϕ ) ∝ n$_{x}$ ( ϕ ), leading to Eq.(60). The Pi case prepa- ration is represented by the Gaussian that is located on the separatrix. After sufficiently long time the evolv- ing distribution is stretched along n ∼ n$_{x}$ ( ϕ ), and looks like δ ( H ( θ, ϕ ) − E$_{x}$ ). If we neglected the finite width of this distribution we would obtain P( ϕ ) ∝ 1 /n$_{x}$ ( ϕ ), which is divergent at ϕ ∼ π . But if we take into account the • The TwinFock preparation of fully-separated con- densates develops phase-locking at ϕ ∼ 0, with FringeVisibility S$_{x}$ ( t ) ≈ 1 / 3 [41]. • Starting from a SCS preparation, phase-diffusion becomes phase sensitive. The Zero preparation is