Nonlinear dressed states at the miscibility-immiscibility threshold

The dynamical evolution of spatial patterns in a complex system can reveal the underlying structure and stability of stationary states. As a model system we employ a two-component Bose-Einstein condensate at the transition from miscible to immiscible with the additional control of linear interconversion. Excellent agreement is found between the detailed experimental time evolution and the corresponding numerical mean-field computations. Analyzing the dynamics of the system, we find clear indications of stationary states that we term nonlinear dressed states. A steady-state bifurcation analysis reveals a smooth connection of these states with dark-bright soliton solutions of the integrable two-component Manakov model.

Figure 1

Comparison of observed and numerically calculated time dynamics of an elongated two-component condensate in the presence of a dressing field of different amplitudes $\mathrm{\Omega }$. The asymmetry in the intraspecies scattering lengths pushes component $|1\rangle$ to the wings of the trap for $\mathrm{\Omega }=0$. The trend reverses for $\mathrm{\Omega }=2\pi \times 6\text{Hz}$. As $\mathrm{\Omega }$ increases further, the amplitude of the oscillatory dynamics decreases, and in the large coupling limit the system is well approximated by stationary dressed states.

Figure 2

Stationary states in the presence of a linear coupling field, exhibiting an intriguing cascade of branches. The right columns show the theoretically obtained normalized density profiles for the two components $|1\rangle$ (blue) and $|2\rangle$ (green) for five characteristic values corresponding to the parameters of Fig. 1.

Figure 3

Detailed comparison of the time-averaged density difference profiles with the stationary states. (a) We extract the time-averaged density profiles from the experimentally observed dynamics shown in Fig. 1, averaging over full oscillation periods for each value of $\mathrm{\Omega }$ and normalizing the averaged profile (see Appendix), as shown in the left column. The difference of the time-averaged density profiles in the experiment is shown in the right column. (b) The same procedure is repeated for the numerical simulations of the NPSE and in (c) compared to the stationary states from Fig. 2 for the respective values of $\mathrm{\Omega }$, i.e., the NDS. The qualitative agreement reveals that the time-averaged profiles are accurately captured by the corresponding stationary nonlinear dressed states. On a quantitative level, differences remain, as the dissipative mechanisms are not accounted for in the determination of the NDS.

Figure 4

The bifurcation loop structure for the stationary states. The left graph shows the chemical potential of the stationary states as a function of the linear coupling. The states are stable only along the linear parts, e.g., between markers 2 and 3, while the loops are unstable. The graphs at the right illustrate the spatial structure of the probability amplitudes of the corresponding stationary states. Following one specific loop we find that the nonlinear dressed states are smoothly connected to dark-bright soliton trains at $\mathrm{\Omega }=0$ characterized by zero crossings of one component and a corresponding amplitude maximum for the other component. The location of one of the dark-bright solitons in the state identified by marker 5 is indicated by the vertical line. Within each branch two additional topological excitations are added as the energy increases.