Figure 1

Calculated Floquet band spectra of the system, Eqs. (1, 5, 6), for typical experimental parameters. Only the first eight Floquet bands, maximally overlapping with the Bloch band of the undriven potential, are depicted. The heavy (magenta) lines mark the Floquet states with the largest overlap to the initial atomic state (see text), which here transform themselves into ballistic diabatic bands after passing a first, relatively broad, avoided crossing. A small change in the driving frequency allows switching from a ballistic band with negative velocity (left panel) to the symmetry-related band of positive velocity (right panel). The time-reversal symmetry of the system dynamics implies that the negative part of the Brillouin zone is a mirror image of the presented one. The used parameters are $V_1=0.352\times 16E_r$, $V_2=0.11\times 16E_r$, $A_1=0.66$, $A_2=0.4$. The driving frequency is measured in units of ${\omega }_R=8{\omega }_r$ (see text). The parameters ${\omega }_r$ and $E_r=\hslash {\omega }_r$ denote the recoil frequency and the recoil energy, respectively.Reuse & Permissions

Figure 2

(a) Time-of-flight (TOF) images of the atomic cloud after $t_{\exp }=26T$ periods of modulation for $F=-0.0181E_r/\lambda$ (left), 0 (center), and $0.0181E_r/\lambda$ (right). The visible atomic diffraction orders are $s=-2,\dots ,2$ (from bottom to top). (b) Measured mean velocity of the atomic cloud in the laboratory frame versus applied bias $F$. Note that the overall antisymmetric response behavior; i.e., $\overline{v}(F)=-\overline{v}(-F)$. The mean velocity of the atomic cloud was calculated as $\overline{v}=\overline{p}/M_{\mathrm{Rb}}$ with $\overline{p}=2\hslash k\sum _{s}s|c_s{|}^2$, where $|c_s{|}^2$ denotes the fraction of atoms in the $s$th order momentum state, $|2s\hslash k⟩$, with $s=\pm 1$, $\pm 2$. The error bars show the standard deviation of the mean value. The enlarged (red) data points correspond to the TOF images. The solid lines depict the results of numerical simulations of the theory. The initial state was chosen in the form of a narrow Gaussian packet in the quasimomentum space, with the center at $\kappa =0$ and dispersion ${\sigma }_{\kappa }=0.04\hslash k$ (see the Supplemental Material [33]). The other parameters are the same as in Fig. 1.Reuse & Permissions

Figure 3

(a) Average velocity of the atomic cloud in the laboratory frame as a function of the modulation frequency $\omega$ (blue dots) for a bias strength $F=-0.0181E_r/\lambda$ and exposition time $26T$; (b) Time evolution of the average cloud velocity for driving frequencies $\omega /{\omega }_R=1.05$ (red squares) and $\omega /{\omega }_R=1.076$ (blue dots). The solid lines depict the results of numerical simulations (see the Supplemental Material [33]). The error bars show the standard deviation of the mean. The remaining parameters are the same as in Fig. 1.Reuse & Permissions