Self-localized state and solitons in a Bose-Einstein-condensate–impurity mixture at finite temperature

We study the properties of a Bose-Einstein-condensate (BEC)–impurity mixture at finite temperature employing the time-dependent Hartree-Fock Bogoliubov (TDHFB) theory which is a set of coupled nonlinear equations of motion for the condensate and its normal and anomalous fluctuations on the one hand and for impurity on the other. The numerical solutions of these equations in the static quasi-one-dimensional regime show that the thermal cloud and the anomalous density are deformed as happens to the condensate and the impurity becomes less localized at nonzero temperatures. Effects of the BEC fluctuations on the self-trapping state are studied in homogeneous weakly interacting BEC-impurity at low temperature. The self-trapping threshold is also determined in such a system. The formation of solitons in the BEC-impurity mixture at finite temperature is investigated.

Figure 1

Condensed (gray lines), noncondensed (red-dashed lines), and impurity (blue-dotted lines) densities as functions of the radial distance for $\beta =0$ (left panel) and for $\beta =1.1$ (right panel) for the above parameters.

Figure 2

Anomalous density as a function of the radial distance for $\beta =1.1$ with the same parameters as in Fig. 1. Solid lines: in the presence of the impurity. Dotted lines: without impurity.

Figure 3

Density profiles for solitons in the BEC-impurity mixture with the same parameters as in Fig. 1. Solid lines: ordinary soliton. Red-dashed lines: impurity soliton. Blue-dotted lines: anomalous soliton.

Figure 4

Evolution of the anomalous soliton for $\beta =0.9$ with the same parameters as in Fig. 3. Here we measure time in units of ${\omega }_{Bx}^{-1}$, the position in units of $a_0$, and the density of $a_0^{-1}$.