Floquet Solitons and Dynamics of Periodically Driven Matter Waves with Negative Effective Mass

We experimentally study the dynamics of weakly interacting Bose-Einstein condensates of cesium atoms in a 1D optical lattice with a periodic driving force. After a sudden start of the driving, we observe the formation of stable wave packets at the center of the first Brillouin zone (BZ) in momentum space, and we interpret these as Floquet solitons in periodically driven systems. The wave packets become unstable when we add a trapping potential along the lattice direction, leading to a redistribution of atoms within the BZ. The concept of a negative effective mass and the resulting changes to the interaction strength and effective trapping potential are used to explain the stability and the time evolution of the wave packets. We expect that similar states of matter waves exist for discrete breathers and other types of lattice solitons in periodically driven systems.

Figure 1

Experimental setup, micromotion, and effective mass. (a) Experimental setup. (b) Effective tunnel coupling $J_{\mathrm{eff}}$ as a function of driving strength $K$ with circles at $K=0$ and $K=2.97$. (c) Dispersion relation for $K=0$ (gray) and $K=2.97$ (red). Dashed lines indicate the effective trapping potential in regions A and B. (d) Absorption images showing the micromotion of atoms for $q_0=0$ (top) and $q_0=q_L$ (middle) with driving force $K=2.97$ and driving period $T$. Red arrows indicate time intervals with the wave packet in region B. (e) Ratio of effective mass $m_{\mathrm{eff}}(K,q_0)$ to real mass $m$ ($V=12E_r$).

Figure 2

Time evolution of the wave packet. (a) Momentum distributions of ground states for $t=0$ (dashed lines) and $t=600\mathrm{ms}$ (solid lines) for $K=1.96$, $q_0=0$ and (b) for $K=2.97$, $q_0=q_L$. (c) Time evolution in momentum space with $m_{\mathrm{eff}}<0$ for suppressed tunneling $J_{\mathrm{eff}}=-0.01J$ ($K=2.43$) and (d) for $J_{\mathrm{eff}}=-0.34J$ ($K=3.26$). Red patches indicated the regions A. (e) Numerical simulation with experimental parameters of (d). (f) Evolution of $N_{\mathrm{tot}}$ and ratio $R$ for the measurement in (d). Solid lines are parabolic and exponential fits ([36], Sec. C). (g) Decay time $t_1$ for increasing trap periods in experiment (red circles) and simulation (blue squares).

Figure 3

Matter-wave Floquet solitons. (a) Evolution of wave packet in momentum space (top) and position space (bottom) for a total initial atom number of 11 000 and (b) of 1600 [$V=12.0(5)E_r$, $a_s=5a_0$, $K=4.2$]. (c) Atom number in region A for 11 000 (red circles) and 1600 (blue triangles) initial atoms. Gray squares provide $N_{\mathrm{tot}}$ for $N_{\mathrm{tot}}(0)=11000$. All measurements are performed without vertical trapping potential.

Figure 4

Matter-wave dispersion. (a) Width of the wave packet in position space in Figs. 3 and 3 for $N_{\mathrm{tot}}(0)=11000$ (red circles) and 1600 (blue triangles) [$V=12.0(5)E_r$, $a_s=5.6a_0$, $K=4.2$]. Lines are linear fits with gradients of 12.7(6) (red line) and $2.8(7)\mu \mathrm{m}/\mathrm{s}$ (blue line). (b) Momentum distribution after 500 ms of driving with for variable driving strength, $N_{\mathrm{tot}}(0)=2000$ [$V=12.0(5)E_r$, $a_s=3a_0$].