Generation of Dark-Bright Soliton Trains in Superfluid-Superfluid Counterflow

The dynamics of two penetrating superfluids exhibit an intriguing variety of nonlinear effects. Using two distinguishable components of a Bose-Einstein condensate, we investigate the counterflow of two superfluids in a narrow channel. We present the first experimental observation of trains of dark-bright solitons generated by the counterflow. Our observations are theoretically interpreted by three-dimensional numerical simulations for the coupled Gross-Pitaevskii equations and the analysis of a jump in the two relatively flowing components’ densities. Counterflow-induced modulational instability for this miscible system is identified as the central process in the dynamics.

Figure 1

Time evolution of an initial perfectly overlapped mixture without (a)–(c) and with (d) an applied axial magnetic gradient. Images taken after (a) 100 ms, (b) 1 sec, and (c)–(d) 9 sec of in-trap evolution.

Figure 2

Formation of dark-bright soliton trains during superfluid-superfluid counterflow. A mixture with 70% of all atoms in the $|1,1⟩$ state [lower cloud in (a) and (c); black line in (b),(d),(e)] and 30% in the $|2,2⟩$ state (upper cloud and red lines, respectively) is created. An axial gradient is ramped on over 1 sec and held constant thereafter. (a) Experimental images of soliton train formation. Times are measured from start of ramp. (b) Integrated cross sections of 3D numerical simulations, revealing a gradually steepening overlap interface and subsequent soliton formation. (c) Integrated density plot of 3D numerics at 1751 ms. Field of view $627\mu \mathrm{m}\times 17\mu \mathrm{m}$. (d) Zoomed-in view of soliton train in (b) at 1751 ms. The blue line shows the total density. (e) Phase behavior for region shown in (d).

Figure 3

Densities (a),(b) and relative speeds (c),(d) at $t=462$. (e),(f): maximum growth rate and associated wave number of unstable perturbations. Vertical lines correspond to subcritical (dotted) and supercritical (dashed) cases of (a)–(d). See text for further details.

Figure 4

Motion of dark-bright solitons. After creating dark-bright solitons as in Fig. 2a, the axial magnetic gradient is removed and the mixture is allowed to evolve in trap for the evolution times given next to each image. The solitons spread out through the mixture.