Nonlinear mixing of collective modes in harmonically trapped Bose-Einstein condensates

We study nonlinear mixing effects among quadrupole modes and scissors modes in a harmonically trapped Bose-Einstein condensate. Using a perturbative technique in conjunction with a variational approach with a Gaussian trial wave function for the Gross-Pitaevskii equation, we find that mode mixing occurs selectively. Our perturbative approach is useful in gaining a qualitative understanding of the recent experiment [M. Yamazaki et al., J. Phys. Soc. Jpn. 84, 44001 (2015)], exhibiting a beating phenomenon of the scissors mode as well as a modulation phenomenon of the low-lying quadrupole mode by the high-lying quadrupole mode frequency. Within the second-order treatment of the nonlinear mode coupling terms, our approach predicts all the spectral peaks obtained by the numerical simulation of the Gross-Pitaevskii equation.

Figure 1

The $\lambda$ dependence of frequencies ${\mathrm{\Omega }}_{\mathrm{Q}}$, ${\mathrm{\Omega }}_{\pm }$, ${\mathrm{\Omega }}_{zx}$ as well as $2{\mathrm{\Omega }}_{zx}$ in an axially symmetric trap case ($\kappa =1$). The frequencies ${\mathrm{\Omega }}_{yz}$ and ${\mathrm{\Omega }}_{xy}$ are equal to ${\mathrm{\Omega }}_{zx}$ and ${\mathrm{\Omega }}_{\mathrm{Q}}$, respectively.

Figure 2

Absolute values of the nonlinear mixing weight $G_{\rho \sigma \tau }^{\prime }$ in the matrix form in a cigar-shaped trap case ($\lambda =0.14$). (a) Matrix $G_{2,\sigma ,\tau }^{\prime }$(${\mathrm{\Omega }}_{\mathrm{Q}}$ mode), (b) matrix $G_{4,\sigma ,\tau }^{\prime }$ (${\mathrm{\Omega }}_{-}$ mode), (c) matrix $G_{6,\sigma ,\tau }^{\prime }$ (${\mathrm{\Omega }}_{+}$ mode), (d) matrix $G_{8,\sigma ,\tau }^{\prime }$ (${\mathrm{\Omega }}_{\mathrm{xy}}$ mode), (e) matrix $G_{10,\sigma ,\tau }^{\prime }$ (${\mathrm{\Omega }}_{\mathrm{yz}}$ mode), and (f) matrix $G_{12,\sigma ,\tau }^{\prime }$ (${\mathrm{\Omega }}_{\mathrm{zx}}$ mode). The row and column of the matrix represent the $\sigma$ and $\tau$ indices, respectively. The $(\sigma ,\tau )$ section of each matrix provides the modulation term with the frequency ${\mathrm{\Omega }}_{\sigma }+{\mathrm{\Omega }}_{\tau }$ for the $\rho \mathrm{th}$ mode. For instance, the $(\sigma =1,\tau =12)$ section of each matrix provides the modulation term with ${\mathrm{\Omega }}_1+{\mathrm{\Omega }}_{12}=-{\mathrm{\Omega }}_{\mathrm{Q}}+{\mathrm{\Omega }}_{\mathrm{zx}}$. White elements are exactly 0 for arbitrary $\lambda$, where the nonlinear mixing is absent. The absolute value of $G_{\rho \sigma \tau }^{\prime }$ is equal to that of $G_{\rho +1,\sigma +1,\tau +1}^{\prime }$ when $(\rho ,\sigma ,\tau )$ are odd.

Figure 3

The $\lambda$ dependence of the mixing weight for (a) the ${\mathrm{\Omega }}_{\mathrm{Q}}$ mode, (b) the ${\mathrm{\Omega }}_{-}$ mode, (c) the ${\mathrm{\Omega }}_{+}$ mode, (d) the ${\mathrm{\Omega }}_{\mathrm{xy}}$ mode, (e) the ${\mathrm{\Omega }}_{\mathrm{yz}}$ mode, and (f) the ${\mathrm{\Omega }}_{\mathrm{zx}}$ mode. Each line is the maximum absolute value among each four matrix elements in $G^{\prime }$ for the $(\pm {\mathrm{\Omega }}_j,\pm {\mathrm{\Omega }}_k)$ sections as well as the $(\pm {\mathrm{\Omega }}_j,\mp {\mathrm{\Omega }}_k)$ sections. Elements with a value smaller than those presented here are not shown. For instance, in (a), the line for ${\mathrm{\Omega }}_{-}\times {\mathrm{\Omega }}_{\mathrm{Q}}$ is given by the maximum value among $|G_{213}^{\prime }|,|G_{214}^{\prime }|,|G_{223}^{\prime }|$, and $|G_{224}^{\prime }|$, which correspond to ${\mathrm{\Omega }}_{\mathrm{Q}}\leftarrow -{\mathrm{\Omega }}_{-}-{\mathrm{\Omega }}_{\mathrm{Q}}$, ${\mathrm{\Omega }}_{\mathrm{Q}}\leftarrow -{\mathrm{\Omega }}_{-}+{\mathrm{\Omega }}_{\mathrm{Q}}$, ${\mathrm{\Omega }}_{\mathrm{Q}}\leftarrow +{\mathrm{\Omega }}_{-}-{\mathrm{\Omega }}_{\mathrm{Q}}$, and ${\mathrm{\Omega }}_{\mathrm{Q}}\leftarrow +{\mathrm{\Omega }}_{-}+{\mathrm{\Omega }}_{\mathrm{Q}}$, respectively. In (a)–(c), lines for ${\mathrm{\Omega }}_{yz}\times {\mathrm{\Omega }}_{yz}$, which are not shown here, are the same as those for ${\mathrm{\Omega }}_{zx}\times {\mathrm{\Omega }}_{zx}$, because of an axially symmetric geometry case ($\kappa =1$).

Figure 4

Spectral intensities of (a) the ${\mathrm{\Omega }}_{\mathrm{Q}}$ mode, (b) the ${\mathrm{\Omega }}_{-}$ mode, (c) the ${\mathrm{\Omega }}_{+}$ mode, and (d) the ${\mathrm{\Omega }}_{zx}$ mode, obtained from numerical simulation of the GP equation in axially symmetric trap cases ($\lambda =0.5$, 1.0, 1.5). Arrows with the largest labels in each panel indicate the frequencies ${\mathrm{\Omega }}_{\mathrm{Q},\pm ,zx}$ that are determined by the maximum peak position. Filled-head (open-head) arrows together with a linear combination of the ${\mathrm{\Omega }}_i$ labels represent frequencies predicted by the first-order (second-order) perturbation approximation. (a–c) The spectrum of the quadrupole mode involves another quadrupole-mode peak, indicated by the arrow with the parenthetical label $\left({\mathrm{\Omega }}_i\right)$, because each collective mode is extracted from the linear approximations such as (28) and (29) by using moments $\langle x^2\rangle$, $\langle y^2\rangle$, and $\langle z^2\rangle$ obtained from the nonlinear numerical simulation.

Figure 5

Spectral intensities of modes for (a) the ${\mathrm{\Omega }}_{\mathrm{Q}}$ mode, (b) the ${\mathrm{\Omega }}_{-}$ mode, (c) the ${\mathrm{\Omega }}_{+}$ mode, and (d) the ${\mathrm{\Omega }}_{zx}$ mode, together with arrows pointing to the unperturbed frequencies, (22). The only difference between Fig. 4 and Fig. 5 is the use of renormalized frequencies vs unperturbed frequencies, (22), for arrows.