Theoretical analysis of super–Bloch oscillations

Several recent studies have investigated the dynamics of cold atoms in optical lattices subject to ac forcing; the theoretically predicted renormalization of the tunneling amplitudes has been verified experimentally. Recent observations include global motion of the atom cloud, such as giant “super–Bloch oscillations” (SBOs). We show that, in order to understand unexplained features of SBOs, in addition to the renormalization of the tunneling, a new and important phase correction must be included. For fermionic systems with strong attractive interactions, one may engineer different types of collisions and recollisions between bound-pairs and unpaired atoms.

Figure 1

(a) and (b) One-particle solutions of the Hubbard Hamiltonian for $L=30$ lattice sites, showing the field-dependence of the phase of SBOs. Upper/lower panels correspond to $F_{\omega }=1.52$ and $0.15$, respectively; both have $\varphi =\pi$. For illustration purposes, the theoretical amplitudes were equalized by choosing $J$ values that equalize $J_{\mathrm{eff}}\simeq J{\mathcal{J}}_1(F_{\omega })$. (c) and (d) show that good agreement is obtained between experiment and the Hubbard numerics, as well as the analytical formula: Eq. (14) (dashed lines) reproduces well both the amplitude and phase of SBO experimental data (symbols) of Ref. [5], using $\delta \omega =2\pi /1000$ ($\equiv -1$ Hz detuning) $\varphi =\pi$ and $\mathit{Jd}/(\hslash \delta \omega )=90$ $\mu$m. This implies $J/\hslash \approx 1.06$ ms${}^{-1}$. (e) and (f) show one-particle Hubbard numerics contrasting $F_{\omega }=\pi /2$ (upper panel) with $\pi$ (lower panel); parameters used below for collisions with bound-pair states. For $F_{\omega }=\pi$, the wave packet leaves at high speed and will return (and re-collide with any component) at the original position with high speed; for $F_{\omega }=\pi /2$, the wave packet starts with zero speed and returns with zero speed.

Figure 2

Solution of the Hubbard Hamiltonian dynamics of a mixture of bound pairs and unpaired atoms in the SBOs regime for $L=30$ or $50$ lattice sites. The dynamics are qualitatively similar for $N=2$–$4$ atoms; these calculations use $N=3$, with two spin-up and one spin-down atoms. Since $J^{2}d/U\ll 1$, the bound-pair component is almost static and appears as a bright horizontal line; the free atoms ($1/3$ of the total probability, initially) perform SBOs around them. (a) and (b) correspond to $U=-10$, $F_{\omega }=\pi$. The unpaired atoms separate then re-collide with BPs at maximum speed, then attempt to cross the static BP component. At low energy and large effective $\hslash \propto 1/L$ [(a) $J=0.6$, $L=30$], there is a mixture of transmission and reflection; at higher energy and small $\hslash$ [(b) $J=1$, $L=50$], complete reflection and a series half SBOs predominate. (c) and (d) are for $U=-7$, $F_{\omega }=\pi /2$. For $F_{\omega }=\pi /2$, the wave packet starts at zero speed and returns with zero speed for both $L=30$, $J=0.5$ (c) and $L=50$, $J=1$ (d); there is relatively little transmission.