Equilibrium solutions for immiscible two-species Bose-Einstein condensates in perturbed harmonic traps

We investigate the mean-field equilibrium solutions for a two-species immiscible Bose-Einstein condensate confined by a harmonic confinement with additional linear perturbations. We observe a range of equilibrium density structures, including ball and shell formations and axially or radially separated states, with a marked sensitivity to the potential perturbations and the relative atom number in each species. Incorporation of linear trap perturbations, albeit weak, are found to be essential to match the range of equilibrium density profiles observed in a recent ${}^{87}$Rb-${}^{133}$Cs Bose-Einstein condensate experiment [McCarron et al., Phys. Rev. A 84, 011603(R) (2011)]. Our analysis of this experiment demonstrates that sensitivity to linear trap perturbations is likely to be an important factor in interpreting the results of similar experiments in the future.

Figure 1

(a) Experimental data for the ${}^{87}$Rb-${}^{133}$Cs BEC experiment of Ref. [5]. Depending on relative atom numbers, three distinct structures are observed, represented here through triangles, squares, and circles (Regions I, II, and III). (b) Experimental integrated axial density profiles corresponding to the filled symbols in (a), observed after time-of-flight expansion [45] and rescaled to the optical depth (OD) maximum. (c) Mean-field cylindrically symmetric ground-state density profiles corresponding to the atom numbers in (b). (d) Integrated axial ground-state profiles under an axial linear potential ${\delta }_z=0.9\mu$m and a transverse linear potential ${\delta }_x=1.0\mu$m. (Solid) red curve: Rb; (dashed) blue curve: Cs.

Figure 2

Integrated 2D density profiles of the ground state as a function of the axial (horizontal) and transverse (vertical) offset from top to bottom (a) $N_{\mathrm{Rb}}=840$ and $N_{\mathrm{Cs}}=8570$, (b) $N_{\mathrm{Rb}}=3680$ and $N_{\mathrm{Cs}}=8510$, and (c) $N_{\mathrm{Rb}}=15100$ and $N_{\mathrm{Cs}}=6470$. Red (black): Rb; Blue (light gray): Cs.

Figure 3

1D density profiles of stationary states of the 1D CGPEs with $N=N_1=N_2$, ${\omega }_1={\omega }_2$, $m_1=m_2$ and $U_{22}=1.01U_{11}$, $U_{12}=1.52U_{11}$. Columns correspond to Gaussian, TF, and homogeneous initial conditions for imaginary time propagation. (a) $U_{11}=6$, $N=200$. (b) $U_{11}=6$, $N=2000$. (c) $U_{11}=1$, $N=200$. (Solid) red curve: species 1; (Dashed) blue curve: species 2.

Figure 4

Number of density domains $n_{\mathrm{domain}}$ in the obtained stationary state as a function of the width of the Gaussian initial condition $\zeta$ (with $\zeta =1$ corresponding to the Gaussian ground harmonic oscillator state), for the system parameters of Fig. 3b.