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0\) does not affect the chimera patterns qualitatively. However, for uniform initial conditions in the amplitude, the fluctuations in the amplitude can decrease when P decreases as shown in Fig. S3 in SM./p>1000\) spiral rotations). This observation suggests that if a random phase core is used as an initial condition, the chimera core pattern also persists over such long times scale. This is indeed what we observe (see Fig. S6 in SM)./p>

0\) and so the particles can propagate outward. The additional detuning in the far-detuned regime \(|\Delta | \gg |\Omega |\) can ensure the mediating idea is well-defined: The number of particles \(N_{j}=\int d\textbf{r}|\psi _{j}|^{2}\) in the mediating channel \(N_{2}\ll N_{1}\approx N\) can be neglected. Note that this model is not captured by the framework of nonlocal diffusive coupling58. It is explicitly constructed to always preserve the conservation properties of the underlying Hamiltonian system, even when adiabatic elimination is applied./p>0\) is required for the solution of confined hopping kernels (see the form of \(\psi _2\) in Fig. 6a), while \(\Delta <0\) leads to wave-like solution. Substituting this solution back into Eq. (4a), we can get the continuum NLHM:/p>0\) which is typical for atomic systems. Note that when \(|\psi _i|^2\) is small, the nonlinear effect can be ignored. It can be achieved by decreasing the density, which is one of the main technique used in the analysis of real systems below./p>

0\) is considered here as illustrated in Fig. 6c./p>1}\) are occupied. This is because high energy states do not evolve slowly compared to the mediating component. To avoid occupying higher energy levels, we can confine the system to local ground states \(\phi ({\textbf {r}})\) with energy \(\epsilon _{1}\) and prevent excitation by choosing a suitable detuning such that \(\epsilon _{2}-\epsilon _{1} \gg \hbar \Delta \gg \hbar |\Omega |\) (see Fig. 6c). Under these constraints, along with adiabatic elimination, we can show (Sect. S2 in SM) that Eqs. (10a) and (10b) reduce to the exact form of Eq. (2) with \(U=g_{11}\int |\phi |^4\), \(P=\hbar \Omega ^{2}/\Delta\), hopping kernel \(G_{D}(r)\) in Table 1, and/p>d\) must be satisfied. An example of Rubidium atoms is shown in Fig. 6d with \(d=395\) nm and a deep trap \(s=40\) (expressing \(V_{0}=sE_{R}\) in recoil energy \(E_{R}=\hbar \kappa k^{2}\)). With such a large s, as studied before52, the overlap between wavefuncion of neighboring cell is very small, the direct hopping is weak, and the system becomes a Mott insulator in the quantum regime. Nevertheless, mediated hopping can completely replace the direct hopping (with order \(R\sim d\), see Fig. 6d) and allow real time control. Since \(\Omega\), \(\Delta\), and U can be easily adjusted in experiments, there seems to be no upper bound on R. From a practical point of view, however, it is limited by the lifetime \(\tau\) and experimental duration. A simple estimation of \(\tau \sim 1\)s gives a maximum \(R\sim 30d\) as shown in Fig. 6d./p>