content_image/440460.jpg content_image/440461.jpg
<img src='content_image/440460.jpg'>

reaches the edge of the first Brillouin zone, it is Bragg re- flected. This model does not include an effective mass and cannot be used to predict quantitative results. Fig. 3(a)- (d) shows the result of a numerical integration of the time- dependent acceleration a ( t ) = F$_{0}$ / m +$_{D}$ F / m sin ( 2$_{p}$ ($_{n}$$_{B}$ + D $_{n}$) t +$_{f}$) with periodic Bragg reflection. For a constant ac- celeration$_{D}$ F = 0, the wave packet’s velocity shows the well- known saw-tooth-like time evolution that corresponds to BOs. The curve in (a) is symmetric, hence, there is no net move- ment, as indicated by the shaded regions of equal area. If, however, there is additional harmonic modulation at n =$_{n}$$_{B}$ , the velocity excursions will not be symmetric about zero, (b), and result in a net movement for each period, leading to lin- ear motion, (c). Only for f =$_{p}$ / 2 or f = 3$_{p}$ / 2 symmetry is restored and no net movement will occur. Note that, in gen- eral, the velocity of the linear motion depends non-trivially on $_{f}$. Off-resonant modulation with$_{D}$$_{n}$ ≪$_{n}$$_{B}$ induces a slowly- varying phase mismatch between the drive and the original Bloch period. This results in a slow oscillation of the net movement for each Bloch cycle, which finally sums up to a giant oscillation in position space, (d). Evidently, this oscilla- tion is the result of a beat between the drive and the original BO. The initial direction of the motion depends on f and$_{D}$$_{n}$. 

<img src='content_image/440461.jpg'>

In particular, a change in the sign of$_{D}$$_{n}$ at a given$_{f}$ can lead to opposite motion in position space, as verified experimentally in Fig. 3(e) for$_{D}$$_{n}$ = ± 1 Hz. 

A quantitative understanding of sBOs [18] can be obtained from an approach based on Wannier-Stark states [16, 17]. In essence, the harmonic drive is expected to lead to a rescaling of the tunneling rate J → J$_{eff}$ = JB$_{1}$ ($_{D}$ F / F$_{0}$ ) and the force F$_{0}$ → F$_{eff}$ = h$_{D}$$_{n}$ / d for a stationary lattice with tilt. Here, B$_{1}$ is the first Bessel function of the first kind. The ampli- tude of sBOs is thus given by a new Wannier-Stark localiza- tion length L$_{eff}$ = J$_{eff}$ / ( dF$_{eff}$ ) [18]. In this sense, sBOs are rescaled BOs. We quantitatively study the dependence of am- plitude and period of sBOs on$_{D}$$_{n}$,$_{D}$ F / F$_{0}$ , and V . The results are shown in Fig. 4. As expected, the period T is given by 1 /$_{D}$$_{n}$ . Also, the oscillation amplitude scales as 1 /$_{D}$$_{n}$ , and its Bessel-function dependence on$_{D}$ F / F$_{0}$ is well reproduced. Given our spatial resolution, we can observe sBOs down to $_{D}$F / F$_{0}$ = 0 . 08 (Fig. 4(b)). Note that sBOs can only be ob- served with sufficient wave function coherence and for well- defined initial conditions, i.e. for sufficient wave packet local- ization in the first Brillouin zone of the lattice. Nevertheless, incoherent atomic samples exhibit a breathing of the spatial distribution [22] as the oscillation period is insensitive to the 

