Interactions and collisions of discrete breathers in two-species Bose-Einstein condensates in optical lattices

The dynamics of static and traveling breathers in two-species Bose-Einstein condensates in a one-dimensional optical lattice is modelled within the tight-binding approximation. Two coupled discrete nonlinear Schrödinger equations describe the interaction of the condensates in two cases of relevance: a mixture of two ytterbium isotopes and a mixture of ${}^{87}\mathrm{Rb}$ and ${}^{41}\mathrm{K}$. Depending on their initial separation, interaction between static breathers of different species can lead to the formation of symbiotic structures and transform one of the breathers from a static into a traveling one. Collisions between traveling and static discrete breathers composed of different species are separated into four distinct regimes ranging from totally elastic when the interspecies interaction is highly attractive to mutual destruction when the interaction is sufficiently large and repulsive. We provide an explanation of the collision features in terms of the interspecies coupling and the negative effective mass of the discrete breathers.

Figure 1

Density profiles of ${}^{170}\mathrm{Yb}$ (black solid line) + ${}^{168}\mathrm{Yb}$ (blue dashed line) mixture. (a) Initial density profile of ${}^{170}\mathrm{Yb}+{}^{168}\mathrm{Yb}$ breathers with $D=0$. Inset displays the logarithmic profile of this, which shows the exponential tails of the breathers. (b) Density profile of ${}^{170}\mathrm{Yb}+{}^{168}\mathrm{Yb}$ breathers at $\tau =1000$ after ${\Lambda }_{1,2}$ is switched on for $D=0$. Note that the density profiles have changed when forming the symbiotic breather. The inset shows the logarithmic profile, in which the background of the ${}^{170}\mathrm{Yb}$ species can be seen more clearly. (c) Initial density profile of ${}^{170}\mathrm{Yb}+{}^{168}\mathrm{Yb}$ breathers with $D=2$. The wave functions of the two species still overlap significantly. (d) Density profile of ${}^{170}\mathrm{Yb}+{}^{168}\mathrm{Yb}$ breathers at $\tau =1000$ after ${\Lambda }_{1,2}$ is switched on for $D=2$. Note that the ${}^{170}\mathrm{Yb}$ is smaller and some of the background has become localized to the left of the main breather.

Figure 2

Interaction between initially stationary breathers in the ${}^{170}\mathrm{Yb}$ (black solid line) + ${}^{168}\mathrm{Yb}$ (blue dashed line) mixture for $D=8$. (a) Evolution of ${}^{170}\mathrm{Yb}$ part of the mixture. The majority of ${}^{170}\mathrm{Yb}$ atoms forms a traveling breather while the remaining atomic density is absorbed by the ${}^{168}\mathrm{Yb}$ breather to form a symbiotic breather. We only show the ${}^{170}\mathrm{Yb}$ part of the mixture since the ${}^{168}\mathrm{Yb}$ evolution is rather straightforward, with the breather highly localized in the center. (b) Density profile of the initial condition in a logarithmic scale, showing the overlap at the tails. The inset shows the density profile of the ${}^{170}\mathrm{Yb}$ mixture at $\tau =1000$. Note that the background is significantly higher than in the iniital condition, allowing the breather to travel.

Figure 3

Collision of two “clean” traveling breathers, with minimal sound waves. The initial conditions are ${\Lambda }_1={\Lambda }_2=1.1,{\overline{n}}_1=16,{\overline{n}}_2=112,{\sigma }_1={\sigma }_2=3$, and $cos{p}_1=cos{p}_2=-0.95$ for all panels. Note that $p_1=-p_2$ and therefore the traveling breathers move in opposite directions. ${\Lambda }_{1,2}=0$ is set to (a) 0, (b) $-20$, and (c) 30. In (a), the breathers ignore each other, acting as if the other species was not present. In (b), the breathers collide elastically. In (c), the breathers are destroyed and a new symbiotic breather is created. The color here represents the total density of the two species, unlike the other figures.

Figure 4

Collisions of two breathers in the ${}^{170}\mathrm{Yb}$ (species 1) + ${}^{174}\mathrm{Yb}$ (species 2) mixture characterized by a large negative interspecies scattering length of $-27.3\mathrm{nm}$. The Gaussian parameters for the initial condition in (a) are ${\overline{n}}_1=16,{\overline{n}}_2=112,{\sigma }_1={\sigma }_2=3$, and $cos{p}_1=cos{p}_2=-0.95$. For (b), the physical situation is the same as in Fig. 4, except that ${\overline{n}}_2=64$ and $cos{p}_2=-1$ to make a stationary breather. In (a), two traveling breathers collide elastically. In (b) a traveling breather transfers large part of its (pseudo)momentum to a stationary one and nearly stops.

Figure 5

Inelastic collisions of ytterbium isotopes. In (a), a traveling (${}^{170}\mathrm{Yb}$, species 1) and stationary breather (${}^{168}\mathrm{Yb}$, species 2) collide for a positive interspecies scattering length of 6.2 nm. The initial condition parameters are ${\overline{n}}_1=16,{\overline{n}}_2=64,{\sigma }_1=5,{\sigma }_2=3,cos{p}_1=0.8,cos{p}_2=1.0$. As in Fig. 3, the traveling breather is destroyed and a new symbiotic soliton is created. In (b) we have the same physical situation as in Fig. 4 (a), except that the parameter ${\Lambda }_{1,2}$ describing the interspecies has now been increased to 4, corresponding to a positive interspecies scattering length of 8.9 nm. In this regime, the two breathers tunnel through each other. Note that a small part of each breather is trapped inside the other forming double-species traveling breathers.

Figure 6

A collision of a traveling (${}^{41}\mathrm{K}$, species 1) and a stationary breather (${}^{87}\mathrm{Rb}$, species 2), with the interspecies interaction parameter ${\Lambda }_{1,2}=3$ (a) and ${\Lambda }_{1,2}=-9$ (b). The initial condition parameters are ${\overline{n}}_1=64,{\overline{n}}_2=12,{\sigma }_1=0.5,{\sigma }_2=3,cos{p}_1=1,cos{p}_2=-0.9$. In (a), the traveling breather tunnels almost completely through the self-trapped state, while in (b), the traveling breather bounces elastically from the self-trapped state with only a minor proportion tunneling through.

Figure 7

The collision outcome as a function of the interspecies interaction, ${\Lambda }_{1,2}$. The top panel displays the mean and the peak site per species, the center panel the standard deviation per species, and the bottom panel the FWHM per species as defined in the text. Symplectic simulations corresponding to the ${}^{170}\mathrm{Yb}+{}^{174}\mathrm{Yb}$ mixture.

Figure 8

Collision of two traveling breathers in the continuous case (A1) for ${\beta }_{i,i}=1$. ${\beta }_{1,2}$ is set to (a) $-18$, giving an elastic collision, and (b) 20.4, with the destruction of the breathers and creation of a symbiotic one. These results are similar to those displayed in Figs. 3 and 3 for the DNLS model.