Figure 1

The spectrum vs Stark force $F$: (a) two-band Wannier-Stark ladders for the single-particle case for ${\Delta }_g=0.796$ (corresponding to lattice parameters $V_0=5$ and $z_0=2.5$, see the Appendix). Avoided crossings appear at the resonances $F_r$ with width ${\Delta }_r^{\min }\ll {\Delta }_g$. (b) and (c) The many-body spectrum for $N/L=5/3$ with no interparticle interaction, revealing the presence of the $M$ manifolds discussed in Sec. 3. The different lines correspond to eigenstates of the type: lower-band-like states ${\left\{\right|N\rangle }_a\otimes {|0\rangle }_b\}$ (black lines), upper-band-like states ${\left\{\right|0\rangle }_a\otimes {|N\rangle }_b\}$ (thick red lines), and mixedlike states ${\left\{\right|N-M\rangle }_a\otimes {|M\rangle }_b\}$ (thin green lines). The remaining parameters are $C_0=-0.095$, $C_1=0.04,C_2=0.004,J_a=0.078$, and $J_b=-0.24$.Reuse & Permissions

Figure 2

Interaction effects: (a) interacting many-body spectra for $N/L=5/3$ [see Fig. 1a]. (b) Zoom around the resonance position revealing the emerging cluster of avoided crossings. (c) Manifold number $M_i$ for all Floquet eigenstates as a function of the ratio ${\Delta }_g/{\omega }_B$. Here the manifold structure is clearly seen before and after the single-particle resonance $F_{r=1}=0.128$, characterized by the bunches of eigenstates with approximately the same upper band occupation number $M$. The parameters are the same as those in Fig. 1 with additional interaction strengths $W_a=0.023,W_b=0.027$, and $W_x=0.025$ [see Eq. (2)].Reuse & Permissions

Figure 3

Manifold mixing: interacting many-body spectra as a function of the filling factor: (a) $N/L=3/13$, (b) $N/L=4/5$, and (c) $N/L=5/4$. Strong manifold mixing occurs as $N/L$ increases due to the high density of avoided crossings in the resonant regime. The arrows in (a) represent: the (red, thin) width of the many-body avoided crossing $\Delta E$ and (blue, thick) the maximal energy splitting of the central manifold $M=1$. The parameters are the same as those in Fig. 2.Reuse & Permissions

Figure 4

Regular-to-chaotic transition: (a) The main panel shows the level spacing distribution $P\left(s\right)$ for $N/L=7/5({\mathcal{N}}_s=2288)$ for the interparticle control parameter (Sec. 3) $g=0.1$ (red, gray histogram, $\eta =0.98$) and $g=1.0$ (black histogram, $\eta =0.056$), and $W_x=W_a=W_b=0.025$. The inset shows the parameter $\eta$ as a function of $g$ where the black line corresponds to an exponential fit. (b) The main panel depicts the level spacing distribution $P\left(s\right)$ for three different filling factors: $N/L=3/25$ (${\mathcal{N}}_s=848$, red, gray histogram), $N/L=4/11$ (${\mathcal{N}}_s=1050$, blue dashed-dotted histogram), and $N/L=6/5$ (${\mathcal{N}}_s=1001$, black thick histogram). The random-matrix theory (RMT) distributions are those in dashed lines in both panels. The inset shows the cumulative distribution $I\left(s\right)$ for the systems: $N/L=3/25$ (red, gray $\square$) and $N/L=6/5$ (black $\circ$). The solid lines represent the RMT prediction for $I\left(s\right)$ Poisson (red, gray) and GOE (black). The other parameters are the same as in the previous figures.Reuse & Permissions

Figure 5

Diffusion: (a) Sketch of the sweeping process. The initial state $\left|\psi \right(0)\rangle$ is prepared at $F=0$ and suddenly is evolved by means of a quench from $F=0$ to $F=F_0(\lambda \gg 1)$. After this process, the state is then further evolved but nonadiabatically ($\lambda \sim 1$) from $F=F_0$ to $F=F_f$ across the many-body AC during a finite time $\Delta T$. The resonance order is $r=1$ with $F_{r=1}=0.045$. (b) Spreading of the evolved state $\left|\psi \right(t)\rangle$ measured by the Shannon entropy $S_{\mathrm{sh}}$ when dynamically crossing the RET regime. The parameters are $\lambda =0.7,{\Delta }_g=0.285,J_a=0.0382,J_b=-0.0417,W_a=0.028,W_b=0.029,W_x=0.029,C_0=-0.096,C_1=0.046$, and $C_2=0.008$.Reuse & Permissions

Figure 6

Diffusion: Temporal evolution of the initial state $\left|\psi \right(0)\rangle$ across the many-body AC represented by the local density of states $P_{\psi }(\varepsilon ,t)$ as defined in Eq. (16). The panels show the respective transit through the energy spectra for (a) $N/L=4/11$ and (b) $N/L=6/5$. The remaining parameters are those of Fig. 5b.Reuse & Permissions

Figure 7

Relaxation: time-averaged inverse participation ratio $\xi$ for $N/L=4/11,6/5$, and 7/4. The straight lines indicate the power-law tendency of the diffusion processes with $t^{-0.78}$ for chaotic spectra before the resonance $F_{r=1}$ and $t^{-0.5}$ for $F>F_r$. The change in the exponents corresponds to a slowing down of the spreading of the evolved state since, in the chaotic case, the maximal delocalization has already occurred. For the not fully chaotic spectra $N/L=4/11$, the tendency to a power law $t^{-0.64}$ is destroyed by the emerging relocalization as explained in Sec. 4b, see cf. $N/L=3/25$.Reuse & Permissions

Figure 8

Manifold mixing: (a) and (b) depict the final time-manifold degree of mixing $\zeta \left(\Delta T\right)$ and manifold creation $M\left(\Delta T\right)$ for different band gaps: ${\Delta }_g=2.53$ (green $\circ$), ${\Delta }_g=1.16$ (red $▵$), ${\Delta }_g=0.556$ (blue $\times$), ${\Delta }_g=0.285$ (black •), and ${\Delta }_g=0.155$ (dark orange $\square$). To this end, more than ${\mathcal{N}}_s/2$ initial states were evolved starting at $F_0=0$ for (a) $N/L=4/11$ and (b) $N/L=6/5$. Panel (c) depicts $\zeta \left(t\right)$ vs $M\left(t\right)$ with $t$ as a parameter for ${\Delta }_g=0.155$. Note that all trajectories in the plane $\zeta \text{−}M$ converge to the equilibrium point $({\zeta }_{\max },M=N/2)$ (see the main text).Reuse & Permissions

Figure 9

Profile of the optical lattice (left) and its respective band structure (right) for $V_0=3$ and $z_0=4$ as a function of the phase difference $\varphi$. (a) $\varphi =0$, (b) $\varphi =\pi /2$, and (c) $\varphi =\pi$.Reuse & Permissions

Figure 10

: (a) Many-body processes of the two-band Bose-Hubbard Hamiltonian for a bichromatic tilted optical lattice. (b) RET condition for the nearest-neighboring double wells, i.e, for a first-order resonance.Reuse & Permissions