Interaction modulation in a long-lived Bose-Einstein condensate by rf coupling

We demonstrate modulation of the effective interaction between the magnetic sublevels of the hyperfine spin $F=1$ in a ${}^{87}\mathrm{Rb}$ Bose-Einstein condensate by Rabi coupling with radio-frequency (rf) field. The use of the $F=1$ manifold enables us to observe the long-term evolution of the system owing to the absence of inelastic collisional losses. We observe that the evolution of the density distribution reflects the change in the effective interaction between atoms due to rf coupling. We also realize a miscibility-to-immiscibility transition in the magnetic sublevels of $m=\pm 1$ by quenching the rf field. Rf-induced interaction modulation in long-lived states as demonstrated here will facilitate the study of out-of-equilibrium quantum systems.

Figure 1

Performance of the two-photon rf transition between the $|1,-1\rangle$ and $|1,+1\rangle$ states. (a) Typical two-photon rf spectrum. The population of the $|1,-1\rangle$ state after applying a Gaussian rf pulse is plotted. The solid line is a Gaussian fit. (b) Observed Rabi oscillation. The population of the $|1,+1\rangle$ state is plotted against the rf pulse width $t$. The solid curve is a sinusoidal fit.

Figure 2

Typical TOF images for holding times of $T=0.2,0.4,...,1$ s. (a), (b) $|1,-1\rangle$ and $|1,+1\rangle$ states without and with rf coupling, respectively. (c), (d) $|1,-1\rangle$ and $|1,0\rangle$ states without and with rf coupling, respectively. The $|1,-1\rangle$ state is displayed in the top row. The horizontal ($x$) and vertical ($y$) dimensions of each image are 150 and $210\mu \mathrm{m}$, respectively.

Figure 3

Time evolution of the radially integrated densities of the $|1,-1\rangle$ and $|1,+1\rangle$ states (a), (c) without and (b), (d) with rf coupling. (a) and (b) are experimental results, while (c) and (d) are numerical simulation results. Results averaged over data from several experiments are shown in (a) and (b). Since the population imbalance varies owing to the Rabi oscillation, the data with population imbalance within 1:3 and 3:1 are averaged in (b) and the data with 1:1 are used in (d).

Figure 4

Time evolution of the characteristic quantities of the dynamics of the $|1,-1\rangle$ and $|1,+1\rangle$ states. (a) The Fourier-space distribution of the density difference $|\widehat{n}\left(k\right)|$ of experiment (top) and numerical simulation (bottom). (b) The separation parameter $s$. (c), (d) The center of mass $x_0$ (c) without and (d) with coupling. The simulation results are indicated by the dotted and dot-dashed lines. (e) The relative position $\mathrm{\Delta }{x}_0$. In (b) and (e), the open circles and dot-dashed line (closed circles and dotted line) represent the experimental and numerical results without (with) coupling, respectively.

Figure 5

Time evolution of the radially integrated densities of the $|1,-1\rangle$ and $|1,0\rangle$ states, (a), (c) without and (b), (d) with rf coupling. (a) and (b) are experimental results and (c) and (d) are numerical simulation results. Results averaged over data from several experiments are shown in (a) and (b). The data with population imbalance within 4:6 and 6:4 are averaged in (b) and the data with 1:1 are used in (d).

Figure 6

The evolution of the characteristic quantities of the dynamics of the $|1,-1\rangle$ and $|1,0\rangle$ states. (a) The Fourier-space distribution of the density difference between the experimental (top) and numerical (bottom) results. (b) The separation parameter $s$ of experimental and numerical results without (with) coupling. (c), (d) The center of mass $x_0$ (c) without and (d) with coupling. The numerical results are shown as dotted and dot-dashed lines. (e) The relative position $\mathrm{\Delta }{x}_0$. In (b) and (e), the open circles and dot-dashed line (closed circles and dotted line) represents the experimental and numerical results without (with) coupling, respectively.

Figure 7

Quench experiment. Time sequence for the (a) fast and (b) slow quench. Typical TOF images and radially integrated densities after the (c) fast and (d) slow quench. The solid blue and dot-dashed red lines represent the column densities of the $|1,-1\rangle$ and $|1,+1\rangle$ states, respectively. The data with population imbalance $\simeq 1:1$ are chosen in (c) and (d).

Figure 8

Numerically obtained density distributions for a weak constant coupling. (a), (b) The density distributions of the $|1,-1\rangle$ (left) and $|1,+1\rangle$ (right) states with the coupling (a) ${\mathrm{\Omega }}_1$ and (b) ${\mathrm{\Omega }}_2$ at $t=0,200,...,1000$ ms. (c), (d) The density distribution along the condensate center axis at $t=1000$ ms in (a) and (b). The $|1,-1\rangle$ and $|1,+1\rangle$ states are represented by the solid and dot-dashed lines, respectively.