Quadrupole and scissors modes and nonlinear mode coupling in trapped two-component Bose-Einstein condensates

We theoretically investigate quadrupolar collective excitations in two-component Bose-Einstein condensates and their nonlinear dynamics associated with harmonic generation and mode coupling. Under the Thomas-Fermi approximation and the quadratic polynomial ansatz for density fluctuations, the linear analysis of the superfluid hydrodynamic equations predicts excitation frequencies of three normal modes constituted from monopole and quadrupole oscillations, and those of three scissors modes. These six modes are bifurcated into in-phase and out-of-phase modes by the intercomponent interaction, yielding the nonlinear dynamics that are absent in a single-component condensate. We obtain analytically the resonance conditions for the second-harmonic generation in terms of the trap aspect ratio and the strength of intercomponent interaction. The numerical simulation of the coupled Gross-Pitaevskii equations vindicates the validity of the analytical results and reveals the dynamics of the second-harmonic generation and nonlinear mode coupling that lead to nonlinear oscillations of the condensate with damping and recurrence reminiscent of the Fermi-Pasta-Ulam problem.

Figure 1

Schematic representation of the quadrupole and scissors modes of two-component BECs. Oscillations of component 1 are indicated by solid arrows and those of component 2 by dashed arrows. Mode coupling processes listed in Table I are also shown.

Figure 2

$a_{12}∕a_1$ dependences of the 12 collective normal modes obtained from Eq. (13) for the parameter $a_1=a_2$, $\lambda =\sqrt{8}$, and (a) $m_2=m_1$ and (b) $m_2=0.75m_1$. Each mode splits into the in-phase mode (thin curves) and the out-of-phase mode (dashed curves) for $a_{12}\ne 0$. In (a), the numerically obtained frequencies are plotted by open symbols (for in-phase) and filled symbols (for out-of -phase). As explained in Sec. 4B, the reduction to a two-dimensional system changes the frequencies of the $m=0$ low- and high-lying modes; for $\lambda =\sqrt{8}$ their values become small from those of Eqs. (20) by 10% for the low-lying mode and $1\%$ for the high-lying mode.

Figure 3

Conditions of the second-harmonic resonance in the parameter space of $\lambda$ and $\delta a=\left(a_1-a_{12}\right)\left(a_2-a_{12}\right)∕\left(a_{11}{a}_{22}-a_{12}^2\right)$. The mode couplings between two diagonal quadrupole modes are shown in (a), and those between a diagonal quadrupole mode and a scissors mode in (b).

Figure 4

Time development of (a) ${⟨x^2⟩}_{1,2}$ and (b) ${⟨y^2⟩}_{1,2}$, showing the mode-coupling process of (VI). The parameters used are $a_{12}∕a_1=0.665$, $\lambda =\sqrt{8}$, and modulation factor $ϵ=0.05$.

Figure 5

(a) Time development of ${⟨xy⟩}_1$, ${⟨x^2⟩}_1-{⟨x^2⟩}_1^{\left(0\right)}$, and ${⟨y^2⟩}_1-{⟨y^2⟩}_1^{\left(0\right)}$, where ${⟨⟩}^{\left(0\right)}$ represents the initial value, showing the mode-coupling process of (IX). The parameter values are $a_{12}∕a_{11}=0.240$, $\lambda =\sqrt{8}$, and modulation factor $ϵ=0.2$. (b) An enlarged view of (a) for the time interval [0,40].