Analytical study of parameter regions of dynamical instability for two-component Bose-Einstein condensates with coaxial quantized vortices

The dynamical instability of weakly interacting two-component Bose-Einstein condensates with coaxial quantized vortices is analytically investigated in a two-dimensional isotopic harmonic potential. We examine whether complex eigenvalues appear on the Bogoliubov–de Gennes equation, implying dynamical instability. Rather than solving the Bogoliubov–de Gennes equation numerically, we rely on a perturbative expansion with respect to the coupling constant which enables a simple, analytic approach. For each pair of winding numbers and for each magnetic quantum number, the ranges of intercomponent coupling constant where the system is dynamically unstable are exhaustively obtained. Corotating and counter-rotating systems show distinctive behaviors. The latter is much more complicated than the former with respect to dynamical instability, particularly because radial excitations contribute to complex eigenvalues in counter-rotating systems.

Figure 1

Regions with possible double degeneracy between ${\mathit{y}}_{n\ell j}$ and ${\mathit{z}}_{n^{\prime }\ell j^{\prime }}$ in the ${\kappa }_2-\ell$ plane for a fixed ${\kappa }_1>0$. They are (a) ${\mathit{y}}_{0\ell 1}$ and ${\mathit{z}}_{0\ell 1}$, (b) ${\mathit{y}}_{0\ell 2}$ and ${\mathit{z}}_{0\ell 2}$, (c) ${\mathit{y}}_{0\ell 1}$ and ${\mathit{z}}_{0\ell 2}$, and (d) ${\mathit{y}}_{n\ell 2}$ and ${\mathit{z}}_{n^{\prime }\ell 1}$. The number inside each subregion denotes the value of $n+n^{\prime }$.

Figure 2

Subregions with possible multiple degeneracies involving both $\mathit{y}$ and $\mathit{z}$. The boundaries denoted by solid lines or filled circles are included in the subregion, while those denoted by dashed lines or open circles are excluded from the subregion.

Figure 3

All degenerate types for ${\kappa }_1=3$.

Figure 4

Range of ${\beta }_{12}$ for complex eigenvalues in each $({\kappa }_1=3,{\kappa }_2,\ell )$.

Figure 5

Maximum values of the imaginary parts of $E^{\left(1\right)}$ in units of $m\omega F$ for (a) $({\kappa }_1,{\kappa }_2)=(3,1)$ with $\ell =1–3$, (b) $({\kappa }_1,{\kappa }_2)=(3,-1)$ with $\ell =1–4$, and (c) $({\kappa }_1,{\kappa }_2)=(3,-3)$ with $\ell =1–6$.

Figure 6

Maximum values of the imaginary parts of BdG eigenvalues in units of $m\omega F$, which are obtained by solving the BdG equation numerically for a small coupling $gN\sqrt{m/\omega }=1$.

Figure 7

(a) Magnitude relationship among $|{\kappa }_2-\ell |, |{\kappa }_2+\ell |$, and $|{\kappa }_2|$. The region is divided into four areas by the lines ${\kappa }_2=0$ and $\ell =\pm 2{\kappa }_2$. (b) Diagram of the splitting patterns of the vortices. The pattern of the resultant winding numbers after splitting of the initial $({\kappa }_1,{\kappa }_2)$ are classified into three shaded areas.