(e) 0 100 200 300 400 500 600 700 -150 -100 -50 0 50 100 150 position (d) time (T$_{B}$) -1 0 1 0.0 0.5 1.0 1.5 2.0 time (T$_{B}$) momentum ( ħ k/m) (a) 0.0 0.5 1.0 1.5 2.0 time (T$_{B}$) -1 0 1 (c) (b) 0 5 10 15 20 -100 0 100 time (T$_{B}$) (d) 50 0 -50 0 2 4 6 8 10 12 14 16 position (d) time (T$_{B}$) (i) (ii) (iii) (i) (ii) 0 500 1000 1500 2000 2500 0 50 100 150 time (ms) position ( µ m) 0 500 1000 1500 2000 2500 -5 0 5 -50 0 50 100 150 0 1 2 3 0 1 2 3 detuning (Hz) frequency (Hz) position ( µ m) position ( µ m) time (ms) amptlitude ( µ m) 0 500 1000 1500 2000 2500 0 50 100 position ( µ m) time (ms) 3 4 5 6 7 0 20 40 60 lattice depth (E$_{R}$) amptlitude ( µ m) (a) (b) (c) modultion amptlitude (  ∆ F/F$_{0}$) 0 0.5 1.0 1.5 0 20 40 60 amplitude ( µ m) 0 20 40 60 80 3 FIG. 3. (color online) Results from a semi-classical model for sBOs. (a) For a constant force, here F$_{0}$ = 0 . 06 mg , the velocity (in units of ¯ hk / m ) exhibits a symmetric, saw-tooth-like time evolution, typical for BOs. (b) Resonant modulation, here with $_{D}$F = 0 . 8 F$_{0}$ , alters the symmetric periodic velocity excursions of normal BOs ($_{f}$ = 0, solid line,$_{f}$ =$_{p}$ , dashed line), leading to a net-movement, (c), with f = 0 (i), f =$_{p}$ / 2 (ii), and f =$_{p}$ (iii). An additional detuning D $_{n}$= ± 0 . 1$_{n}$$_{B}$ results in a periodically changing phase difference and hence in giant oscillations in position space, (i) and (ii) in (d). On top of the motion, normal BOs can clearly be seen. The phase of sBOs depends on the sign of$_{D}$$_{n}$, as shown by experimental data in (e), where F$_{0}$ = 0 . 096 ( 1 ) mg ,$_{D}$ F = 0 . 090 ( 4 ) mg ,$_{D}$$_{n}$ = 1 Hz (circles), − 1 Hz (squares). FIG. 4. (color online) Quantitative analysis of sBOs. (a) The effect of the detuning$_{D}$$_{n}$ on the oscillation frequency and the amplitude of sBOs, with$_{D}$$_{n}$ = 0 . 5 Hz (circles), 1 Hz (squares), 2 Hz (diamonds). Right: The solid lines are fits with linear and$_{D}$$_{n}$ − $^{1}$-dependence, re- spectively. (b) Dependence of the amplitude of sBOs on$_{D}$ F / F$_{0}$ . The data sets correspond to$_{D}$ F / F$_{0}$ = 1 . 52 (circles), 0 . 76 (squares), 0 . 15 (diamonds), 0 . 08 (stars). Right: The solid line is a fit proportional to B$_{1}$ ($_{D}$ F / F$_{0}$ ) . (c) Amplitude of sBOs as a function of lattice depth, V = 3 E$_{R}$ (circles), 4 E$_{R}$ (squares), 5 E$_{R}$ (diamonds), 7 E$_{R}$ (stars). If not stated otherwise, the parameters for all measurements shown here are F$_{0}$ = 0 . 062 ( 1 ) mg ,$_{D}$ F = 0 . 092 ( 4 ) mg ,$_{D}$$_{n}$ = − 1 Hz. In particular, a change in the sign of$_{D}$$_{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}$$_{n}$ , and its Bessel-function dependence on$_{D}$ F / F$_{0}$ is well reproduced. Given our spatial resolution, we can observe sBOs down to $_{D}$F / F$_{0}$ = 0 . 08 (Fig. 4(b)). Note that sBOs can only be ob- served with sufficient wave function coherence and for well- defined initial conditions, i.e. for sufficient wave packet local- ization in the first Brillouin zone of the lattice. Nevertheless, incoherent atomic samples exhibit a breathing of the spatial distribution [22] as the oscillation period is insensitive to the reaches the edge of the first Brillouin zone, it is Bragg re- flected. This model does not include an effective mass and cannot be used to predict quantitative results. Fig. 3(a)- (d) shows the result of a numerical integration of the time- dependent acceleration a ( t ) = F$_{0}$ / m +$_{D}$ F / m sin ( 2$_{p}$ ($_{n}$$_{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}$.