10 not an equation for Ψ$_{i}$, but an approximated equation for the mean-field Ψ$_{i}$. We further clarify this point in the next paragraph. Within the framework of the semiclassical treatment the quantum state is described as a distribution of points in phase space. This approach goes beyond the con- ventional MFT approximation: MFT essentially assumes that the state of the system remains coherent throughout its evolution. Such a state corresponds to a Gaussian-like distribution is phase space (“minimal wave-packet”) and accordingly it is fully characterized by “center of mass” coordinates ( ϕ$_{i}$, n$_{i}$ ) that are defined through the mean field Science Foundation (Grant 582/07), the USA-Israel Bi- national Science Foundation (Grant No.2008141), and the National Science Foundation through a grant for the Institute for Theoretical Atomic, Molecular, and Optical Physics at Harvard University and the Smithsonian As- trophysical observatory. KSM, MH, and TK acknowledge support of the DFG within the Forschergruppe 760. (representing the center of a wave-packet). To the extent that the MFT assumption can be trusted, the equation of motion for Ψ$_{i}$ is the DNLS / GP. Indeed, if u = 0 there is no nonlinear spreading and the MFT descrip- tion becomes exact. The approximation holds well in the weak-interaction Rabi regime and for some regions of phase-space (e.g. around the ground state) in the strong-interaction Josephson regime, but in the nonlin- ear regions of phase-space MFT becomes too crude to provide an accurate description of the dynamics. A bosonic L site system has formally the same Hilbert space as that of L coupled (harmonic) oscillators, with an additional constant of motion N that counts the to- tal number of quanta (rather than the total energy). The creation operator ˆ a † i corresponds to a raising op- erator and the occupation of the i th site n$_{i}$ = ˆ a † $_{i}$ˆ a$_{i}$ corresponds to the number of quanta stored in the i th mode. The one-particle states of an L site sys- tem form an L dimensional Hilbert space. The set of unitary transformations (i.e. generalized rotations) within this space is the SU ( L ) group. For N parti- cles in L sites, the dimensionality of the Hilbert space is dim( N ) = ( N +1)! / [( L − 1)!( N − L +2)!]. For example for L = 2 we have dim( N ) = N +1 basis states | n$_{1}$, n$_{2}$ 〉 with n$_{1}$ + n$_{2}$ = N . We can rotate the entire system using dim( N ) matrices. Thus we obtain a dim( N ) representa- tion of the SU ( L ) group. By definition these rotations can be expressed as a linear combination of the SU ( L ) generators J$_{µ}$ , and they all commute with the conserved particle number operator ˆ N . The many-body Hamilto- nian H may contain “nonlinear” terms such as J 2 µ that correspond to interactions between the particles. Accord- ingly, for an interacting system, H is not merely a rota- tion. However, H still commutes with ˆ N , maintaining the fixed-number dim( N ) subspace. The energy contours in a representative case are dis- played in Fig. 1. Considering a section along the big circle ϕ = 0 , π , it is convenient to take − π < θ < + π instead of 0 < θ < π . Along this section the energy is E ( θ ) = ( NK/ 2) f ( θ ) where In the semiclassical framework the dynamics in phase space is generated by the Hamilton equations of motion for the action-angle variables ˙ n$_{i}$ and ˙ ϕ$_{i}$ . These are the “polar coordinates” that describe each of the oscillators. It is common to define the complex coordinates All extremal points of the Hamiltonian are located along this section and are determined by the equation f $^{′}$( θ ) = 0. The number of solutions depends on u and on the bias ε . (representing a single point in phase space). This is the classical version of the destruction operator ˆ a$_{i}$ . The equa- tion for ˙ Ψ$_{i}$ is the discrete nonlinear Schr¨ odinger (DNLS) equation: For u < 1, there are two fixed points: a minimum at θ$_{−}$ at the bottom of the sea and a maximum at the opposite point θ$_{+}$ . For ε = 0 the two fixed points are: which is the space-discretized version of This looks like the Gross-Pitaevskii (GP) equation, but strictly speaking it is not the GP equation. The GP equa- tion is the outcome of a mean-field theory (MFT): it is Ψ$_{i}$ = Mean Field ≡ 〈 Ψ$_{i}$ 〉 (A4) Appendix A: An L site system with N bosons Appendix B: The structure of phase space f ( θ ) ≡ 1 2 u (cos θ ) 2 − ε cos θ − sin θ (B1) f $^{′}$( θ ) = − 1 2 u sin(2 θ ) + ε sin θ − cos θ, (B2) f $^{′′}$( θ ) = − u cos(2 θ ) + ε cos θ + sin θ . (B3) Ψ$_{i}$ ≡ √ n$_{i}$ e i ϕ i (A1) i d Ψ$_{i}$ dt = ( ϵ$_{i}$ + U | Ψ$_{i}$ | 2 ) Ψ$_{i}$ − K 2 (Ψ$_{i}$$_{+1}$ + Ψ$_{i}$$_{−}$$_{1}$) , (A2) i d Ψ( x ) dt = [ V ( x ) + g$_{s}$ | Ψ( x ) | 2 − 1 2 m ∇ 2 ] Ψ( x ) (A3) θ$_{−}$ = π/ 2 [ ϕ =0 point] , (B4) θ$_{+}$ = − π/ 2 [ ϕ = π point] , (B5)