18 0.01 1 100 u/N 0 0.5 1 1.5 M / N 1/2 0.01 1 100 u/N 0 1 2 3 4 N 1/2 FIG. 6: (Color online) The participation number M as determined from the LDOS for N = 100 ( ◦ ) , 500 ( GLYPH<Box> ) , and 1000 ( ⋄ ) particles. The left panel contains the Zero (lower set in blue) and Pi (upper set in red) preparations, while the Edge preparation is presented in the right panel. Note the different vertical scale. In the crudest approximation we expect in the Edge case M ∼ N 1 / $^{2}$, while in the Pi case M ≪ N 1 / 2 as long as ( u/N ) ≪ 1 0.01 1 100 u/N 1 10 $_{w}$osc / w J FIG. 7: (Color online) The mean frequency of the S$_{x}$ ( t ) oscillations versus u/N for for N = 100 ( ◦ ) , 500 ( GLYPH<Box> ) , and 1000 ( ⋄ ) particles. The preparations are (upper to to lower sets of data points): Zero (blue), Edge (magenta), and Pi (red). The doubled Josephson frequency 2 ω$_{J}$ is marked by a dashed blue line. The theoretical predictions of Eq. (42), doubled due to mirror symmetry, are represented by red and magenta dashed lines, while Eq. (43) for u/N ≫ 1 is represented by black dash-double-dotted line (there is one fitting parameter as explained there). FIG. 8: (Color online) Left: The long-time average of S$_{x}$ ( t ) versus u/N for N = 100 ( ◦ ) , 500 ( GLYPH<Box> ) , and 1000 ( ⋄ ) particles. The preparations are (upper to lower sets of data points): Zero (blue), TwinFock (black), Edge (magenta), and Pi (red). The symbols are used for the quantum results and the dashed lines are the semiclassical prediction for fourty particles. Note that the scaling holds only in the Josephson regime 1 ≪ u ≪ N $^{2}$, and therefore, for a given u/N range, becomes better for large N . Right: The long time RMS of S$_{x}$ ( t ) for the three coherent preparations (lower to upper sets): Zero (blue), Edge (magenta), and Pi (red). The implied N 1 / 4 scaling based on Eq. (69)) is confirmed. In the inset, the RMS of S$_{x}$ ( t ) for Edge ( △ ) and Pi ( ▽ ) preparations is plotted versus N while u = 4 is fixed. The dashed lines are power-law fits that nicely agree with the predictions of Eq. (70). 0.01 1 100 u/N -1 -0.5 0 0.5 1 S x 0.01 1 100 u/N 0.001 0.01 0.1 1 N 1/4  [S x ] RMS 100 1000 N 0.01 0.1 [S x ] RMS ~N -0.26 ~N -0.09