Fast Dynamics for Atoms in Optical Lattices

Cold atoms in optical lattices allow for accurate studies of many body dynamics. Rapid time-dependent modifications of optical lattice potentials may result in significant excitations in atomic systems. The dynamics in such a case is frequently quite incompletely described by standard applications of tight-binding models (such as, e.g., Bose-Hubbard model or its extensions) that typically neglect the effect of the dynamics on the transformation between the real space and the tight-binding basis. We illustrate the importance of a proper quantum mechanical description using a multiband extended Bose-Hubbard model with time-dependent Wannier functions. We apply it to situations directly related to experiments.

Figure 1

Relevance of different transition amplitudes: (a)—nearest neighbor tunnelings $J_1^{\alpha }$ for different Bloch bands (b)—interaction integrals for $g=1$, $\kappa =2\pi$; the term $U_{iiii}^{0000}$ term present in the Bose-Hubbard Hamiltonian is compared with interaction terms involving excited bands. Panels (c) and (d) show additional amplitudes $T_{i-j}^{\alpha \beta }$ [see Eq. (6)] coming from the time-dependence of Wannier functions [i.e., $\mathcal{W}$ term Eq. (5)].

Figure 2

$\Delta E$, the energy gain (i.e., the excess energy over the corresponding ground state) during a linear quench of a model 1D system from $s_0=12E_R$ to $s_1=40E_R$. For the adiabatic process, $\Delta E=0$. Red, solid (black, dashed) lines correspond to the simulation with (without) the $\mathcal{W}$ term [Eq. (5)] in the time-dependent Hamiltonian. Panels (b) and (c) show time variation of averages of band occupation operators $⟨{\widehat{n}}_1⟩$, $⟨{\widehat{n}}_2⟩$.

Figure 3

Excitation via modulation of the lattice depth with and without the $\mathcal{W}$ term, Eq. (5). The depletion function during the first 10 ms, without (black, solid) and with the $\mathcal{W}$ term (red, dashed). The broadening of the peak around $\omega =18.5$ is a power broadening effect, see discussion in the text.

Figure 4

Floquet spectrum without $\mathcal{W}$ contribution (left) and with the $\mathcal{W}$ term, Eq. (5) (right panel). Broad avoided crossings for the latter are due to the strength of terms omitted within BH description as well as the influence of higher harmonics—see text for discussion. The region of avoided crossings (large curvatures) is highlighted with solid lines.