Inducing Transport in a Dissipation-Free Lattice with Super Bloch Oscillations

Particles in a perfect lattice potential perform Bloch oscillations when subject to a constant force, leading to localization and preventing conductivity. For a weakly interacting Bose-Einstein condensate of Cs atoms, we observe giant center-of-mass oscillations in position space with a displacement across hundreds of lattice sites when we add a periodic modulation to the force near the Bloch frequency. We study the dependence of these “super” Bloch oscillations on lattice depth, modulation amplitude, and modulation frequency and show that they provide a means to induce linear transport in a dissipation-free lattice.

Figure 1

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 $\nu$. The resonances correspond to a drastic spreading of the atomic wave packet as a result of modulation-assisted tunneling [19] when $\nu \approx (i/j){\nu }_B$, where $i$, $j$ are integers. The parameters are $F_0=0.096(1)mg$, $\Delta F=0.090(4)mg$, $V=3.0(3)E_R$, and $\tau =2\mathrm{s}$. The dashed line is a guide to the eye.

Figure 2

Observation of super-Bloch oscillations and modulation-driven wave-packet spreading. (a) and (b) In situ absorption images and density profiles for off-resonant modulation ($\Delta \nu =-1\mathrm{Hz}$), 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 modulation ($\Delta \nu =0\mathrm{Hz}$), showing a wave packet that spreads symmetrically (time steps of 100 ms, average of 4 images). The phase $\varphi$ was adjusted to allow for a symmetric spreading, corresponding to a calculated value of $\varphi =\pi /2$. For (a)–(d), the parameters are $F_0=0.062(1)mg$, $\Delta F=0.092(4)mg$, $V=3.0(3)E_R$, $a=11(1)a_0$. (e) Center-of-mass motion for $a=11(1)a_0$ (circles), $a=90(1)a_0$ (diamonds), $a=336(4)a_0$ (squares).

Figure 3

Results from a semiclassical model for SBOs. (a) For a constant force, here $F_0=0.06mg$, the velocity (in units of $\hslash k/m$) exhibits a symmetric, saw-tooth-like time evolution, typical for BOs. (b) Resonant modulation, here with $\Delta F=0.8F_0$, alters the symmetric periodic velocity excursions of normal BOs ($\varphi =0$, solid line, $\varphi =\pi$, dashed line), leading to a net movement, (c), with $\varphi =0$ (i), $\varphi =\pi /2$ (ii), and $\varphi =\pi$ (iii). An additional detuning $\Delta \nu =\pm 0.1{\nu }_B$ 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 $\Delta \nu$, as shown by experimental data in (e), where $F_0=0.096(1)mg$, $\Delta F=0.090(4)mg$, $\Delta \nu =1\mathrm{Hz}$ (circles), $-1\mathrm{Hz}$ (squares).

Figure 4

Quantitative analysis of SBOs. (a) The effect of the detuning $\Delta \nu$ on the oscillation frequency and the amplitude of SBOs, with $\Delta \nu =0.5\mathrm{Hz}$ (circles), 1 Hz (squares), 2 Hz (diamonds). Right: The solid lines are fits with linear and $\Delta {\nu }^{-1}$ dependence, respectively. (b) Dependence of the amplitude of SBOs on $\Delta F/F_0$. The data sets correspond to $\Delta F/F_0=1.52$ (circles), 0.76 (squares), 0.15 (diamonds), 0.08 (stars). Right: The solid line is a fit proportional to $B_1(\Delta F/F_0)$. (c) Amplitude of SBOs as a function of lattice depth, $V=3E_R$ (circles), $4E_R$ (squares), $5E_R$ (diamonds), $7E_R$ (triangles). Right: The solid line is a fit proportional to $J$, for which we omit the first data point for the shallow lattice. If not stated otherwise, the parameters for all measurements shown here are $F_0=0.062(1)mg$, $\Delta F=0.092(4)mg$, $\Delta \nu =-1\mathrm{Hz}$.

Figure 5

Inducing transport. (a) Linear motion for resonant modulation. $\Delta \varphi =0°$ diamonds, 65° circles, 120° triangles, 190° squares. $\Delta \varphi =0°$ and $\Delta \varphi =190°$ were chosen to maximize the speed in opposite directions. The solid lines are linear fits to the data points excluding the first data point. For comparison we plot the linear motion that corresponds to a tunneling rate of $J_{\mathrm{eff}}$, dotted lines. (b) Directed motion for off-resonant modulation. $\Delta \nu$ was switched from $-1$ to 1 Hz after 400 ms. For comparison, (c) shows the oscillatory motion without switching (time steps of 80 ms). The parameters are $F_0=0.096(1)mg$, $\Delta F=0.090(4)mg$.