2  (a)  lattice beam magnetic  levitation gravity x y z 1.0 frequency  ν  ( ν$_{B}$ )  0.1 0.2 0.4 0.3 0.6 2.0 1 / 2 1/3 1/4 1 / 5 1/7 1/9 1 /1 6 1 / 1 2 / 1 2 / 3 2 / 5 100 200 20 40 60 80 width W ( µ m)  (b) dip ole tr ap atoms FIG. 1. (color online) Experimental setup (a) and excitation spectrum (b) for atoms in a tilted periodic potential. The width W is plotted as a function of the drive frequency$_{n}$. The resonances correspond to a drastic spreading of the atomic wave packet as a result of modulation- assisted tunneling [21] when$_{n}$ ≈ i / j ×$_{n}$$_{B}$ , where i , i are integers. The parameters are F$_{0}$ = 0 . 096 ( 1 ) mg ,$_{D}$ F = 0 . 090 ( 4 ) mg , V = 3 . 0 ( 3 ) E$_{R}$ , and$_{t}$ = 2 s. The dashed line is a guide to the eye. FIG. 2. (color online) Observation of super Bloch oscillations and modulation-driven wave packet spreading. (a) and (b) In-situ ab- sorption images and density profiles for off-resonant modulation ($_{D}$$_{n}$ = − 1 Hz), showing giant oscillatory motion across more that 200 sites (time steps of 120 ms, average of 4 images). (c) and (d) In-situ absorption images and density profiles for resonant modula- tion ($_{D}$$_{n}$ = 0 Hz), showing a wave packet that spreads symmetrically (time steps of 100 ms, average of 4 images). For (a)-(d), the pa- rameters are F$_{0}$ = 0 . 062 ( 1 ) mg , $_{D}$F = 0 . 092 ( 4 ) mg , V = 3 . 0 ( 3 ) E$_{R}$ , a = 11 ( 1 ) a$_{0}$. (e) Center-of-mass motion for a = 11 ( 1 ) a$_{0}$ (squares), a = 90 ( 1 ) a$_{0}$ (diamonds), a = 336 ( 4 ) a$_{0}$ (squares). time  τ  (T$_{B}$) center-of-mass position (d)  (e) -200 0 200 400 density (arb.) -400 0 400 position (d) position (d) time force  (a) (b)  (c) (d) time 0 50 100 150 200 250 -50 0 50 100 150 200 density (arb.) BOs by removing the dipole trap confinement in the vertical direction and by reducing the levitation in 1 ms to cause a force that is a small fraction of the gravitational force mg , for which$_{n}$$_{B}$ is near 100 Hz. An additional harmonic modulation of the levitation gradient then results in an oscillating driving force F ( t ) = F$_{0}$ +$_{D}$ F sin ( 2$_{pn}$ t +$_{f}$) , where F$_{0}$ is the constant force offset, $_{D}$F is the amplitude of the modulation, n is the modulation frequency, and$_{f}$ is a phase difference between the BOs and the drive. After a given hold time$_{t}$ we switch off all optical beams and magnetic fields and take in-situ absorption images after a short delay time of 800$_{m}$s. We first determine the excitation spectrum. Fig. 1(b) shows the 1 / √ e -width W of the matter wave after t = 2 s as a func- tion of $_{n}$. A series of narrow resonances at rational multi- ples of n$_{B}$ can clearly be identified. In agreement with re- cent experiments [20, 21], we attribute these resonances to modulation-enhanced tunneling between lattice sites, leading to dramatic spreading of the atomic wave packet. Tunneling between nearest neighbor lattice sites is enhanced when n$_{B}$ is an integer multiple j of n via a j -phonon process [24], while tunneling between lattice sites i lattice units apart is enhanced when n is an integer multiple i of $_{n$_{B}$}$. Even combinations thereof, e.g. i / j = 2 / 3 or 2 / 5, are detectable. for ordinary BOs corresponds to about 4 d = 2 . 1 $_{m}$m. Also the width and higher moments of the distribution show oscil- latory behavior. In Fig. 2(e) we plot the center-of-mass posi- tion as a function of time for D n = − 1 Hz. At a = 11 ( 1 ) a$_{0}$ we typically observe sBOs over the course of several seconds. The dynamics of sBOs strongly depends upon the site-to-site phase evolution of the matter-wave. In fact, stronger inter- actions, e.g. a = 90 ( 1 ) a$_{0}$ , distort the density profile of the driven BEC and alter the BEC’s oscillation frequency and am- plitude. For sufficiently strong interactions, no sBOs are ob- served. We also attribute the wave-packet spreading as seen after one cycle in Fig. 2(b) mostly to interactions. For the measurements above, we intentionally use a large modulation amplitude$_{D}$ F to enhance the amplitude of sBOs. However, all effects equally exist for$_{D}$ F ≪ F$_{0}$ , as we will also demonstrate below in Fig. 4(b). It is useful to develop a simple semi-classical model to ob- tain a qualitative understanding of the origin of sBOs. The only elements of this model are that the wave packet is ac- celerated by the applied force and that, once the wave packet We now investigate the dynamics of the wave packet in more detail. For this, we use the resonance with i = j = 1 and choose n =$_{n}$$_{B}$ +$_{D}$$_{n}$ , where$_{D}$$_{n}$ is the detuning. In Fig. 2(a)- (d) we present absorption images and spatial profiles for the weakly-interacting BEC. The time evolution for the width, shape, and center position of the BEC is dramatic. On res- onance ($_{D}$$_{n}$ = 0), (c) and (d), the atomic ensemble spreads as it develops pronounced edges. We will see below that the center-of-mass motion depends crucially on the phase $_{f}$. Off resonance, (a) and (b), for small detuning D n = − 1 Hz, the wave packet exhibits giant oscillatory motion across hundreds of lattice sites that we denote as “super Bloch oscillations” (sBO). Note that, for the parameters used here, the amplitude