8 Assume the system is prepared in some state ψ , e.g. a Gaussian-like SCS. We define M as in Eq. (44), im- plying that ψ is roughly a superposition of M energy states. If the energy levels are equally spaced the motion is strictly periodic. Otherwise it is quasi-periodic. Our aim is to trace the non-classical behavior in the RMS of the temporal fluctuations of an observable A , say of the FringeVisibility as defined in Eq. (56). Before proceeding, it should be made clear that the RMS of the fluctuations of any observable A in a classi- cal simulation (i.e. classical propagation of a single tra- jectory) is non-zero and characterized by its power spec- trum ˜ C$_{cl}$ ( ω ). However, the RMS of the fluctuations in the semiclassical evolution (i.e. classical, leading-order prop- agation of a cloud of trajectories emulating the Wigner function) goes to zero due to the ergodic-like spreading of the wave-packet. In contrast, the RMS of the fluctua- tions in the quantum evolution (corresponding to the full propagation to all orders, of the Wigner distribution) de- pends on M . This dependence on M can be figured out by expanding the expectation value in the energy basis Δ n ≈ √ N/ 2 width of the preparation, which is effec- tively like adding ( N/ 2) under the square root of Eq.(61), then we get Eq.(59). As implied by the Wigner-Weyl picture one can get from P( ϕ ) the long time average S$_{x}$ through the integral The calculation is straightforward leading to S$_{x}$ ≈ ∫ cos( ϕ ) P( ϕ ) dϕ . (62) 〈 A 〉 t = ∑ ν,µ ψ ∗ $_{ν}$ψ$_{µ}$ A$_{νµ}$ e i ( E$_{ν}$ − E$_{µ}$ ) $^{t}$, (66) These expressions agree with the numerics of Fig. 8, and confirm the predicted scaling with u/N . VIII. DYNAMICS (III) - LONG TIME FLUCTUATIONS where ψ$_{ν}$ = 〈 E$_{ν}$ | ψ 〉 . The time average of this expectation value is 〈 A 〉 t = ∑ ν p$_{ν}$A$_{ν,ν}$ where p$_{ν}$ = P( E$_{ν}$ ) of Eq. (38). This average has a well-defined h -independent classical limit. But if we first square, and then take the time average we get To complete our phase-space characterization of the Bloch vector dynamics, we need to address the long-time fluctuations f ( t ) from the average value S$_{x}$ . As demon- strated in Fig. 4, the evaluation of S$_{x}$ could be done to excellent accuracy based on the semiclassical propagation of phase-space distributions according to purely classical equations of motion. This corresponds to the truncated Wigner approach of quantum optics [28, 31, 34, 36], re- taining only the leading Liouville term in the equation of motion for the Wigner distribution. Obviously one can not guarantee that the remaining Moyal-bracket terms, which are initially O (1 /N ), will remain small through- out the evolution. This is the source of the fluctuations observed in Fig. 4. While their characterization goes be- yond the lowest-order truncated Wigner semiclassics, we still can estimate them based on the phase-space LDOS expansion, as described below. The matrix elements can be evaluated semiclassically us- ing the well know relation | A$_{ν,µ}$ | 2 = ˜ C$_{cl}$ ( E$_{ν}$ − E$_{µ}$ ) / (2 πϱ ), where ϱ is the mean level spacing (see Eq.(6) of Ref.[51] and references therein). For presentation purpose it is convenient to visualize ˜ C$_{cl}$ ( ω ) as having a rectangular- like lineshape of width ω$_{cl}$ , such that its total area is ˜ C (0) × ω$_{c}$ . It is also convenient to define the dimension- less bandwidth b as the spectral width of ˜ C cl ( ω ) divided by the mean level spacing, namely b = ϱω$_{c}$ . The two-site BHH has essentially one degree of free- dom since both the energy and the number of the bosons is conserved. Therefore, away from the separatrix, level spacing is determined by the classical frequency ω ( E ) with small h dependent corrections. This should be contrasted with higher dimensions d > 1, for which the level spacing ∝ h d − 1 is highly non-classical. For d = 1, the only region where h determines the level spacing ∝ | log( h ) | − 1 is in the vicinity of the separatrix as im- plied by Eq. (30). Using the semiclassical estimate for the matrix ele- ments, we consider the outcome of Eq.(67), in two lim- iting cases. If the energy spread of the wave-packet is smaller than the spectral bandwidth, we can factor out ˜ C$_{cl}$ (0) / (2 πϱ ), and carry out the summation ∑ p$_{ν}$p$_{µ}$ = 1, leading to 〈 A 〉 2 t = ∑ ν,µ p$_{ν}$p$_{µ}$ | A$_{ν,µ}$ | 2 . (67) RMS [ 〈 A 〉 $_{t}$] = [ 1 b ∫ ˜ C cl ( ω ) dω ] 1 / 2 . (68) For integrable one-dimensional systems b ∼ 1 reflects that only nearby levels are coupled. A semiclassically large bandwidth b ∝ ℏ 1 − d is typical for chaotic systems, which is not the case under consideration. Therefore we turn to the other possibility, in which the energy spread of the wave-packet is large compared with the The Heisenberg time is defined as the inverse of the mean spacing of the participating levels. For a d = 1 sys- tem and away from the separatrix, the Heisenberg time is merely the period of classical oscillations. Thus in the case of a Zero preparation we have ω$_{osc}$ = ω$_{J}$ (the ob- served frequency is doubled due to Mirror symmetry). Close to the separatrix, ω$_{osc}$ becomes h dependent as in Eqs.(42-43). This prediction is confirmed by the numer- ics (see Fig. 7), including the non-symmetry related fac- tor 2 that distinguishes the Edge from the Pi preparation.