Phase coherence and fragmentation of two-component Bose-Einstein condensates loaded in state-dependent optical lattices

A binary mixture of interacting Bose-Einstein condensates (BEC) in the presence of fragmentation-driving external lattice potentials forms two interdependent mean-field lattices made of each component. These effective mean-field lattices, like ordinary optical lattices, can induce additional fragmentation and phase coherence loss of BECs between lattice sites. In this study, we consider the nonequilibrium dynamics of two hyperfine states of one-dimensional Bose-Einstein condensates, subjected to state-dependent optical lattices. Our numerical calculations using the truncated Wigner approximation (TWA) show that phase coherence in a mixture of two-component BECs can be lost not just by optical lattices, but by mean-field lattices gradually formed by other components, and we reveal that such an effect of internal mean-field lattices, however, is limited, contrary to external optical lattices, in regard to phase decoherence.

Figure 1

Phase coherence of component A (left) and component B (right) for various populational fractions $\left(0\le f_B<1\right)$, for $N=5\times {10}^3,{\tau }_{RU}=250/{\omega }_R=11$ ms for ${}^{87}\mathrm{Rb}$. The numerical simulation shows the results from when lattice loading begins. The fractions of component B, $f_B$, and the line colors are as follows (in the large $t$ limit, top to bottom on the left; bottom to top on the right): 0, black (for left side only); 0.1, red; 0.2, green; 0.4, blue; and 0.6, pink.

Figure 2

The profile of combined potentials for component A, $V_{c,A}\left(z\right)=V_{h,A}\left(z\right)+V_{o,A}\left(z\right)$ (black solid line on top left) and $V_{c,A}\left(z\right)=V_{h,A}\left(z\right)+V_{o,A}\left(z\right)+g_{AB}{\left|{\psi }_B\left(z\right)\right|}^2$ (red dashed line on top left) with $s_{max,A}=10$, and for component B, $V_{c,B}\left(z\right)=V_{h,B}\left(z\right)+g_{AB}{\left|{\psi }_A\left(z\right)\right|}^2$ (on bottom). The figure on the top right is the same plot on the top left, but enlarged near the center of the trap. The potentials $V_{c,i}\left(z\right)$ are in units of $E_R$. The populations are $N_A=4\times {10}^3,N_B=1\times {10}^3$.

Figure 3

The on-site interaction energies $U_{ii}$ (solid line) and the widths of single-particle wave functions ${\sigma }_i$ (dashed line) for component A (left) and for component B (right), showing opposite changes as the impurity component B populates increasingly. Here $s_{max,A}$=10, $N_{\mathrm{tot}}=5\times {10}^3$.

Figure 4

The effect of final optical lattice heights on the phase coherence of component A (left) and component B (right). The final lattice height $\left(s_{max,A}\right)$ for each curve is following (top to bottom) the black line (3), the red line (6.5), the green line (10), the blue line (13.5), and the pink line (17).

Figure 5

The on-site interaction energies $U_{ii}$ (solid line) and the widths of single-particle wave functions ${\sigma }_i$ (dashed line) for component A $(U_{AA}$ on the left) and for component B $(U_{BB}$ on the right) for $N_{\mathrm{tot}}=5\times {10}^3$. Both components become more fragmented as the state-dependent optical lattice for only component A has been amplified more.

Figure 6

The effect of localization induced by B on phase coherence on component A (left) and component B (right). The line colors and the final lattice heights for B $\left(s_{max,B}\right)$ are the following (explained below in parentheses for the left figure; top to bottom on the right): the red line, 3 (top); the green line, 6.5 (second to top); the blue line, 10 (one of the two bottom curves); the pink line, 13.5 (the other one of the two bottom curves); the yellow line, 17 (between the bottom and the second to top).

Figure 7

The on-site interaction energies $U_{ii}$ (solid line) and the widths of single-particle wave functions ${\sigma }_i$ (dashed line) for component A $(U_{AA}$ on left) and for component B $(U_{BB}$ on right) as a function of the lattice height for B $\left(s_{max,B}\right)$.