Phase separation and dynamics of two-component Bose-Einstein condensates

The miscibility of two interacting quantum systems is an important testing ground for the understanding of complex quantum systems. Two-component Bose-Einstein condensates enable the investigation of this scenario in a particularly well controlled setting. In a homogeneous system, the transition between mixed and separated phases is fully characterized by a miscibility parameter based on the ratio of intra- to interspecies interaction strengths. Here we show, however, that this parameter is no longer the optimal one for trapped gases, for which the location of the phase boundary depends critically on atom numbers. We demonstrate how monitoring of damping rates and frequencies of dipole oscillations enables the experimental mapping of the phase diagram by numerical implementation of a fully self-consistent finite-temperature kinetic theory for binary condensates. The change in damping rate is explained in terms of surface oscillation in the immiscible regime, and counterflow instability in the miscible regime, with collisions becoming only important in the long time evolution.

Figure 1

(a)–(d) Ground-state phase diagram of a trapped ${}^{87}\mathrm{Rb}–{}^{39}\mathrm{K}$ mixture at temperature $T=0$ for various total numbers of ${}^{87}\mathrm{Rb}$ ($N_{\mathrm{Rb}}$, species 1) and ${}^{39}\mathrm{K}$ ($N_{\mathrm{K}}$, species 2) atoms. The symmetrical phase is characterized by a normalized trap-center density $\mathrm{\Delta }{n}_{\mathrm{norm}}$ [see Eq. (4)] while the asymmetrical phase is shown in white. Typical density profiles along the $z$ axis are shown in (e) for asymmetrical phase and (f)–(i) for symmetrical phase. White circles in (a) indicate points accessible experimentally using the Feshbach resonances reported in Ref. [14].

Figure 2

Results with approximately $5\times {10}^4{}^{87}\mathrm{Rb}$ and $3\times {10}^4{}^{39}\mathrm{K}$ BEC atoms. Top row: Condensate fractions of a mixture with indicated total atom numbers at (a) $T=100$ nK (noninteracting critical temperatures $T_{c,1}=234$ and $T_{c,2}=287$ nK) and (b) $T=150$ nK ($T_{c,1}=259$ and $T_{c,2}=298$ nK), as a function of miscibility parameter $\mathrm{\Delta }$. Middle row: Axial densities of an immiscible mixture of ${}^{87}\mathrm{Rb}$ (thick red) and ${}^{39}\mathrm{K}$ (thin blue) atoms at different temperatures. Solid (dashed) line represents condensate (thermal cloud). The thermal cloud densities have been amplified by 20 times for clarity. Bottom row: Same as middle panels but for a miscible mixture.

Figure 3

(a) Difference in normalized trap-center density $\mathrm{\Delta }{n}_{\mathrm{norm}}$, (b) damping rate $\gamma$, and (c) oscillation frequency $\nu$ of COM of ${}^{39}\mathrm{K}$ atoms as a function of miscibility $\mathrm{\Delta }$ (bottom axis) or Feshbach magnetic field (top axis) at different temperatures. The simulation parameters are the same as in Fig. 2.

Figure 4

The COM $\langle z\rangle$ of ${}^{87}\mathrm{Rb}$ (thick red) and ${}^{39}\mathrm{K}$ (thin blue) condensates at temperature $T=0\mathrm{nK}$ (top) and $150\mathrm{nK}$ (middle) for different miscibilities. Bottom panels reveal the ${}^{39}\mathrm{K}$ COM with our full model (solid blue line), or the reduced version excluding all collisions (dashed black line) The simulation parameters are the same as Fig. 2.

Figure 5

Snapshots of the column-integrated density profiles of ${}^{87}\mathrm{Rb}$ and ${}^{39}\mathrm{K}$ condensates at $T=0$ nK of immiscible (top) and miscible (bottom) mixtures, showing the different dynamical excitations when the clouds execute dipole oscillation. The white dashes mark the initial trap center. The simulation parameters are the same as Fig. 2. See online Supplemental Material, [52] for videos.

Figure 6

Zero-temperature phase diagram of ${}^{87}\mathrm{Rb}–{}^{39}\mathrm{K}$ mixtures with different total number of atoms ($N_{\mathrm{Rb}}, N_{\mathrm{K}}$) in anisotropic harmonic traps. The trap frequencies are ${\nu }_r=119\mathrm{Hz}$ (178 Hz) and ${\nu }_z=166$ Hz (248 Hz) for ${}^{87}\mathrm{Rb}$ (${}^{39}\mathrm{K}$) atoms. $N_{\mathrm{Rb}}$ increases from left to right while $N_{\mathrm{K}}$ increases from top to bottom. The symmetrical phase is characterized by a normalized trap-center density $\mathrm{\Delta }{n}_{\mathrm{norm}}$ [see Eq. (4)] while the asymmetrical phase is shown in white.

Figure 7

Same as Fig. 6 but for atoms in isotropic harmonic trap. The trap frequencies are $\nu =133$ (199) Hz for ${}^{87}\mathrm{Rb}$ (${}^{39}\mathrm{K}$) atoms.