6 whereas single-particle purity is reflected by the mea- sures, For the analysis of the temporal fluctuations it is cru- cial to determine the number of eigenstates that partici- pate in the wave-packet superposition. This is given by the LDOS participation number , See Fig. 6 for numerical results. The participation num- ber can be roughly estimated as M = Δ E/ω$_{osc}$ , where Δ E is the energy width of the wave-packet and ω$_{osc}$ is the mean level spacing. In the case of a Pi preparation M = [ ∑ ν P( E$_{ν}$ ) 2 ] − 1 . (44) M ≈ [ log ( N u )] √$_{u,}$ (45) M ≈ [ log ( N u )] √$_{N.}$ (46) while for an Edge preparation we find In particular the FringeVisibilty, given by the transverse component of the Bloch vector, corresponds to the visi- bility of fringes, averaged over many realizations [19, 20]. Loosely speaking it reflects the phase-uncertainty of the state. Coherent states have maximum OneBodyPurity. Of these, equal-population coherent states have maximal FringeVisibility with the smallest phase-variance. Start- ing from a coherent preparation, single-particle purity can be diminished over time due to nonlinear effects (see below) or due to interaction with environmental degrees of freedom (decoherence). By contrast, Fock states carry no phase information, but interactions may lead to their dynamical phase-locking over time and to the buildup of FringeVisibility (see below). Note that $_{M/}$√$_{N}$ is a function of the semiclassical ra- tio ( N/u ) 1 / 2 between the energy width of the wave- packet and the width of the separatrix region. The ex- pected scaling is confirmed by the numerical results of Fig. 6. The above approximations for M assume u/N < 1 and are useful for the purpose of rough estimates. In the next sections we analyze the temporal fluctua- tions of some observables. The associated Fourier power spectrum (Fig. 5, right panels) is related to the LDOS content of the wave-packet superposition. Both the char- acteristic frequency (Fig. 7) and the spectral spread of the frequencies can be estimated from the ω$_{osc}$ and the M that are implied by the above LDOS analysis. We carry out numerically-exact quantum simulations where the state | ψ 〉 is propagated according to | ψ ( t ) 〉 = exp( − it H ) | ψ 〉 , where the Hamiltonian is given by Eq.(4) which is equivalent to Eq.(1). The evolved state after time t can be visualized using its Wigner function. For example, the time evolution of the initial TwinFock state is illustrated in Fig. 3, with the Wigner function of an evolved state shown in Fig. 3(a). Comparison is made in Fig. 3(b) to the Liouville propagation for the same duration, of the corresponding cloud of points, according to the classical equations, where time has been rescaled ( t := Kt ). Good quantum- to-classical correspondence is observed for short-time simulation (e.g. see Fig. 4). In Fig. 3(c) we plot the resulting occupation statistics, which can be regarded as a projection of the phase space distribution (classical, dash-dotted line) or Wigner function (quantum, solid line), namely P$_{t}$( n ) = |〈 n | ψ ( t ) 〉| 2 = trace( ρ ( n $^{)}$ρ( ψ ( t $^{))}$). Our main interest is in the FringeVisibility, and hence in After describing the semiclassical phase-space picture and using it to determine the spectral structure of the ini- tial preparations, we now turn to the ensuing dynamics. At present, most experiments on the bosonic Josephson system, measure predominantly quantities related to the one-body reduced probability matrix, defined via the ex- pectation values S$_{i}$ = (2 /N ) 〈 J$_{i}$ 〉 as, where S = ( S$_{x}$, S$_{y}$, S$_{z}$ ) is the Bloch vector and ˆ σ is com- posed of Pauli matrices. The population imbalance and relative phase between the two sites, determined by di- rect imaging and the position of interference fringes [11], are given respectively by, The prefactor in the last equality is implied by Eq. (5), and cannot be neglected if the number of particles is small. In Fig. 4 we plot S$_{x}$ ( t ) for the four prepara- tions defined in Sec. IV, comparing semiclassical results (dash-dotted lines) to the numerical full quantum cal- culation (solid lines). For all equatorial preparations (Zero,Pi,TwinFock), S$_{x}$ is in fact the FringeVisibility, be- cause S$_{y}$ ( t ) = 0 identically throughout the evolution. As a general observation, the semiclassical simulation cap- tures well the short-time transient evolution and the long S$_{x}$ ( t ) = 2 N 〈 J$_{x}$ 〉 = [( j +1) j ] 1 / 2 j 〈 sin( θ ) cos( ϕ ) 〉 . (55) ˙ S$_{y}$ = − (1 + uS$_{x}$ ) S$_{z}$ , (53) ˙ S$_{x}$ = uS$_{z}$S$_{y}$ , (52) ˙ S$_{z}$ = S$_{y}$ , (54) ρ [1] ji = 1 N 〈 ˆ a † $_{i}$ˆ a$_{j}$ 〉 = 1 2 ( ˆ 1 + S · ˆ σ )$_{ji}$ , (47) VI. DYNAMICS (I) - THE TIME EVOLUTION OF THE BLOCH VECTOR OccupationDiff = N S$_{z}$, (48) RelativePhase = arctan( S$_{x}$, S$_{y}$ ) , (49)