Modulation instability and solitary-wave formation in two-component Bose-Einstein condensates

We investigate nonlinear dynamics induced by the modulation instability of a two-component mixture in an atomic Bose-Einstein condensate. The nonlinear dynamics is examined using numerical simulations of the time-dependent coupled Gross-Pitaevskii equations. The unstable modulation grows from initially miscible condensates into various types of vector solitary waves, depending on the combinations of the sign of the coupling constants (intracomponent and intercomponent). We discuss the detailed features of the modulation instability, dynamics of solitary wave formation, and an analogy with the collapsing dynamics in a single-component condensate with attractive interactions.

Figure 1

The modulationally unstable region in ${\tilde{k}}_1\text{−}{\tilde{k}}_2$ space for the combinations (a) and (b) in Table . The unstable region lies below the thick line boundary given by ${\tilde{k}}_2=\sqrt{{\gamma }_{12}^2∕({\tilde{k}}_1^2+1)-{\gamma }_2}$. The magnitude of the gain spectra $G=\mathrm{Im}\Omega$ (arbitrary unit) is shown by a contour plot in the unstable region.

Figure 2

The modulationally unstable region in ${\tilde{k}}_1\text{−}{\tilde{k}}_2$ space for cases (c) and (d) in Table . The unstable region lies below the thick line given by ${\tilde{k}}_2=\sqrt{{\gamma }_{12}^2∕({\tilde{k}}_1^2+1)-{\gamma }_2}$. The boundary curve crossing the origin (bottom left) is given by ${\Lambda }_1+{\Lambda }_2=0$; the right (left) region from the boundary represents the region (i) [(ii)] (see the text). The contour plot shows the magnitude of the gain spectra $G=\mathrm{Im}\Omega$ in arbitrary unit.

Figure 3

(Color online) Numerical results of Region (a) for $N=5\times {10}^3$. (a) Time development of condensate density $\mid {\psi }_1{\mid }^2$ (upper panel) and $\mid {\psi }_2{\mid }^2$ (lower panel) at $z=0$ for $a_{12}=5.6\mathrm{nm}$, $5.8\mathrm{nm}$, and $6.0\mathrm{nm}$. (b) The contour plots of the density of both components with respect to time ([0,1200], horizontal axis) and $z$ ($[-\mathrm{100,100}]$, vertical axis) for $a_{12}=6.0\mathrm{nm}$. (c) The density profiles of $\mid {\psi }_1{\mid }^2$ (solid curve), $\mid {\psi }_2{\mid }^2$ (dashed curve) and total density $n_T=\mid {\psi }_1{\mid }^2+\mid {\psi }_2{\mid }^2$ (dotted curve) for $a_{12}=6.0\mathrm{nm}$ at $t=240$, $t=580$, and $t=960$. The unit of time is ${\omega }_{\perp }^{-1}$.

Figure 4

(Color online) Numerical results of region (a) for $N=5\times {10}^4$. (a) Time development of condensate density $\mid {\psi }_1{\mid }^2$ (upper panel) and $\mid {\psi }_2{\mid }^2$ (lower panel) at $z=0$ for $N=5\times {10}^4$ and $a_{12}=5.6\mathrm{nm}$, $5.8\mathrm{nm}$, and $6.0\mathrm{nm}$. (b) Contour plots of the density of both components with respect to time ([0,1200], horizontal axis) and $z$ ($[-\mathrm{160,160}]$, vertical axis) for $a_{12}=6.0\mathrm{nm}$. (c) The density profiles of $\mid {\psi }_1{\mid }^2$ (solid curve), $\mid {\psi }_2{\mid }^2$ (dashed curve) and total density $n_T=\mid {\psi }_1{\mid }^2+\mid {\psi }_2{\mid }^2$ (dotted curve) for $a_{12}=6.0\mathrm{nm}$ at $t=200$, $t=600$, and $t=1000$. The unit of time is ${\omega }_{\perp }^{-1}$.

Figure 5

(Color online) Numerical results of region (b) for $N=5\times {10}^3$, where the condensates are trapped by a harmonic potential. (a) Time development of condensate density $\mid {\psi }_1{\mid }^2$ at $z=0$ for $a_{12}=-5.6$, $-5.8$, and $-6.0\mathrm{nm}$. The insets show the details of the evolution during the first and third focusing. The development of ${\psi }_2$ component is similar to what is seen in (a) because the modulation develops in-phase. (b) The contour plots of the density of both components with respect to time ([0, 750], horizontal axis) and $z$ ($[-\mathrm{100,100}]$, vertical axis) for $a_{12}=-6.0\mathrm{nm}$. (c) The density profiles of $\mid {\psi }_1{\mid }^2$ (solid curve) and $\mid {\psi }_2{\mid }^2$ (dashed curve) for $a_{12}=-6.0\mathrm{nm}$ at $t=150$, $t=375$, and $t=600$. The unit of time is ${\omega }_{\perp }^{-1}$.

Figure 6

(Color online) Condensate dynamics in an expulsive potential. (a) The contour plots of the density of both components with respect to time ([0, 600], horizontal axis) and $z$ ([−200, 200], vertical axis) for $N=5\times {10}^4$ and $a_{12}=-6.0\mathrm{nm}$. (c) The density profiles of $\mid {\psi }_1{\mid }^2$ (solid curve) and $\mid {\psi }_2{\mid }^2$ (dashed curve) at $t=150$, $t=300$, and $t=450$. The unit of time is ${\omega }_{\perp }^{-1}$.

Figure 7

(Color online) Numerical results of region (c) for $N=5\times {10}^3$, where the condensates are trapped by a harmonic potential. (a) Time development of condensate density $\mid {\psi }_1{\mid }^2$ (upper panel) and $\mid {\psi }_2{\mid }^2$ (lower panel) at $z=0$ for $a_{12}=0\mathrm{nm}$ and $0.5\mathrm{nm}$. (b) The contour plots of the density of both components with respect to time ([0, 600], horizontal axis) and $z$ ($[-\mathrm{100,100}]$, vertical axis) for $a_{12}=0.5\mathrm{nm}$. (c) The density profiles of $\mid {\psi }_1{\mid }^2$ (solid curve), $\mid {\psi }_2{\mid }^2$ (dashed curve) for $a_{12}=0.5\mathrm{nm}$ at $t=78$, $t=120$, and $t=300$. The unit of time is ${\omega }_{\perp }^{-1}$.

Figure 8

(Color online) Contour plots of the density of the attractive ${\psi }_2$ component with respect to time ([0, 500], horizontal axis) and $z$ ([−200, 200], vertical axis) for $N=5\times {10}^4$, $a_{12}=0.5\mathrm{nm}$, and (a) ${\lambda }_2=0$, (b) ${\lambda }_2=0.3\lambda$, and (c) ${\lambda }_2=0.6\lambda$ $(\lambda =0.02)$. (d) The density profiles of $\mid {\psi }_1{\mid }^2$ (solid curve), $\mid {\psi }_2{\mid }^2$ (dashed curve) for the plot (b) at $t=150$, $t=300$, and $t=450$. The unit of time is ${\omega }_{\perp }^{-1}$.

Figure 9

(Color online) Numerical results of region (d) for $N=5\times {10}^3$, where the condensates are trapped by a harmonic potential. (a) Time development of condensate density $\mid {\psi }_1{\mid }^2$ (upper panel) and $\mid {\psi }_2{\mid }^2$ (lower panel) at $z=0$ for $N=5\times {10}^3$ and $a_{12}=0\mathrm{nm}$ and $-0.5\mathrm{nm}$. (b) Contour plots of the density of both components with respect to time ([0, 600], horizontal axis) and $z$ ([−100, 100], vertical axis) for $a_{12}=-0.5\mathrm{nm}$. (c) The density profiles of $\mid {\psi }_1{\mid }^2$ (solid curve) and $\mid {\psi }_2{\mid }^2$ (dashed curve) for $a_{12}=-0.5\mathrm{nm}$ at $t=80$, $t=120$, and $t=300$. The unit of time is ${\omega }_{\perp }^{-1}$.

Figure 10

(Color online) Simulation for $N=5\times {10}^4$ and $a_{12}=-1.0\mathrm{nm}$, in the case without a trapping potential. (a) The contour plots of the density of both components with respect to time ([0, 500], horizontal axis) and $z$ ([−200, 200], vertical axis). (c) The density profiles of $\mid {\psi }_1{\mid }^2$ (solid curve) and $\mid {\psi }_2{\mid }^2$ (dashed curve) at $t=140$, $t=250$, and $t=375$. The unit of time is ${\omega }_{\perp }^{-1}$.