Figure 4 : Quantum synchronization manifests in momentum space localization for ultracold atoms.

From: Classical synchronization indicates persistent entanglement in isolated quantum systems

Figure 4

(a) In a tilted or accelerated optical lattice, the single-particle eigenstates (blue lines) are localized in a few wells of the lattice potential (bold black line). The overlap with other wells induces a nonlinear coupling between the states described by (b,c) Without coupling , the modes dephase due to the on-site interaction U. The coherences αj,j+1 decay to zero and the momentum density distribution ρ(k, t) spreads over the entire Brillouin zone. (d) For there is no entanglement, W2,3=0. (e,f) For , the nonlinear coupling leads to phase locking at φj+1φjπ/2 at intermediate values of the coherence αj,j+1. This leads to a localization of the momentum space density ρ(k, t) around k=0, which can be readily detected in a time-of-flight image. (g) Synchronization implies strong persistent many-body entanglement quantified by the entanglement parameter W2,3>0 (see ref. 50 and Methods). Parameters are L=4, N=12 and U=1.2/2π in units of ωB=2π/TB. The initial state is a pure Bose-Einstein Condensate (BEC) with zero momentum.