13 leading to P( E ) = ω ( E ) ∫ ∫ rdrdt 2 π 2 ω$_{0}$ δ ( r 2 − 1 ) 1 ab exp [ − (4 | ϵ | /NK ) (sinh( t )) 2 2 a 2 r 2 − ( | ϵ | /U ) (cosh( t )) 2 2 b 2 r 2 ] (D13) = 1 2 π ( ω ( E ) ω$_{0}$ ) ∫ | t | < 1 π ( ω ( E ) ω 0 ) dt 1 ab exp [ − (4 | ϵ | /NK ) (sinh( t )) 2 2 a 2 − ( | ϵ | /U ) (cosh( t )) 2 2 b 2 ] (D14) = 1 π ( ω ( E ) ω$_{0}$ ) ∫ | t | < 1 π ( ω ( E ) ω 0 ) dt exp [ ± ( 2 | ϵ | K − | ϵ | 2 NU ) − ( 2 | ϵ | K + | ϵ | 2 NU ) cosh(2 t ) ] (D15) = 1 π ( ω ( E ) ω$_{0}$ ) exp [ ± ( 2 K − 1 2 NU ) | ϵ | ] K$_{0}$ [( 2 K + 1 2 NU ) | ϵ | ] . (D16) Note that without the Gaussian the result of the integral should be unity reflecting the proper normalization of the microcanonical state. The upper cutoff of the t integration reflects the finite size of phase space. The cutoff prevents the singularity in the limit ϵ → 0. The Bessel function expression is obviously valid only outside of the cutoff-affected peak. The asymptotic behavior of the Bessel function is K$_{0}$ ( x ) ≈ exp( − x ) $_{/}$√$_{2}$$_{πx}$$_{, and hence we get}$ P( E ) ∼ exp[ −| ϵ | / ( NU )] for ϵ > 0 and P( E ) ∼ exp[ −| ϵ | / ( K/ 4)] for ϵ < 0. One observes that the tails of the LDOS for ϵ < 0 reflect the relatively rapid decay of the Gaussian tails in the ϕ direction. Finally we point out that the calculation of the LDOS for the Edge and for the TwinFock preparations is much easier. In the former case the delta function of ρ ( ν $^{)}$( n, ϕ ) merely replaces the n coordinate in ρ ( ψ $^{)}$( n, ϕ ) by a constant proportional to E − E$_{x}$ , leading to a Gaussian for P( E ). Similarly for the TwinFock preparation: P( E ) = ω ( E ) ∫ ∫ dϕdn 2 π δ ( Un 2 − NK 2 cos( ϕ ) − E ) δ ( n ) ∝ ω ( E ) ( NK/ 2) | sin( ϕ ) $_{|}$] E (D17) The later expression should be calculated for ( NK/ 2) cos( ϕ ) = E leading to Eq.(41) in the main text. Appendix E: Rabi-Josephson oscillations, MFT and semiclassical perspectives but small, then all the trajectories still have the same topology, but ω ( E$_{ν}$ ) is ν dependent. The anharmonic be- havior is due to the spectral stretch Δ ω$_{osc}$ . Accordingly, we distinguish between the harmonic stage ( t < 1 / Δ ω$_{osc}$ ) and the anharmonic stage ( t > 1 / Δ ω$_{osc}$ ) in the time evo- lution. MFT.– For u = 0, the Hamiltonian (4) merely gen- erates rotations in phase space. Consequently, coherent states remain Gaussian-like throughout their motion, re- sulting in Rabi oscillations of the population between the sites. This is the case where MFT gives exact results. For u = 0, MFT maintains the assumption that the sys- tem remains in a coherent state at any instant during its evolution and that propagation only serves to displace this Gaussian distribution on the Bloch sphere. Thus, MFT corresponds to the classical dynamics of a “point” in phase space. In particular, this implies that the occu- pation statistics is binomial at any time. For u > 1, which we call “the Josephson regime”, a separatrix emerges and accordingly there are two types of stable coherent oscillations, depending on the initial population imbalance [9, 11]: Small oscillations around the ground state, lying in the bottom of the sea; and self-trapped oscillations around the top of either island. For u > N 2 which we call “the Fock regime”, the area of the sea becomes less than a Planck cell, and therefore effectively disappears. Our main interest in this work is the unstable motion along the separatrix for 1 ≪ u ≪ N 2 [22–24]. Such motion emerges for the Pi and for the Edge preparations of Fig. 2. Nonlinear effects cannot be neglected once the wave- packet is stretched along the phase-space energy contour lines. Then MFT no longer applies, but the semiclassi- cal approximation still works well for the description of such squeezing. For example, by projecting the evolving semiclassical distribution onto n we obtain the number distribution P$_{t}$ ( n ) [25] whose line shape reflects the phase space structure, with caustics at the borders of forbidden regions (see Fig. 3). Beyond. – Quantum effects that are ignored by the semiclassical approximation are anharmonic beats, long- time recurrences, and the possibility of tunneling between the two islands. Associated with the recurrences are the long-time fluctuations, both in the occupation and in the FringeVisibility, as discussed in Section VIII. Semiclassics. – For u = 0, all trajectories have the same topology and frequency and a Gaussian-like wave- packet that is launched (say) at the NorthPole executes Rabi oscillation between the two wells. If u is non-zero