Rotational response of two-component Bose-Einstein condensates in ring traps

We consider a two-component Bose-Einstein condensate in a ring trap in a rotating frame and show how to determine the response of such a configuration to being in a rotating frame via accumulation of a Sagnac phase. This may be accomplished through either population oscillations or the motion of spatial-density fringes. We explicitly include the effect of interactions via a mean-field description and study the fidelity of the dynamics relative to an ideal configuration.

Figure 1

(a) Equipotential surface of an axially symmetric toroidal trapping potential with radius $\rho$ in a frame rotating with angular frequency $\Omega$ about the axis of symmetry ($z$ axis). (b) In the quasi-1D limit, the equipotential surface shrinks to a ring, and the trapped atoms’ spatial dynamics are in terms of the angle $\theta$ only. Spatial fringes in the macroscopic atomic wave function will shift by $\Delta \theta =\Omega$ multiplied by the interrogation time.

Figure 2

(a) State $j$ mean relative population $N_j/N$ as a function of interrogation time $T$ for $j=1$ (red circles) and $j=2$ (blue crosses), following an $S_N(T)$ sequence, with $\ell =1$. (b) Angular density fringes of state $1$, with $\ell =1$, $m=0$, $\Omega =1$, for $T=0$ (black diamonds), $T=0.1$ (black pluses), $T=1$ (green squares), and $T=8.3$ (magenta triangles), following an $S_F(T)$ sequence. The interaction strengths $g_{\mathit{jk}}$ correspond to an ${}^{87}\mathrm{Rb}$ configuration. All quantities are dimensionless.

Figure 3

Component 1 angular position density $|{\psi }_1(\theta ,t)|{}^2$ evolved by Eq. (2) from Eq. (10) for $m=0$, $\ell =1$ $\Omega =0$, $g_{11}=g_{22}=1000$, with (a) $g_{12}=1000$, (b) $g_{12}=750$, (c) $g_{12}=500$, (d) $g_{12}=0$. Note that $|{\psi }_2(\theta )|{}^2$ is identical except for a $\pi$ shift in $\theta$, and finite $\Omega$ causes the densities to precess by $\Delta \theta =\Omega t$. All quantities are dimensionless.

Figure 4

(a) Fidelity $F(t)=|{\int }_0^{2\pi }d\theta {\varphi }^{*}(\theta ,t){\psi }_1(\theta ,t)|{}^2$ of ${\psi }_1(\theta ,t)$ evolved by Eq. (2) from Eq. (10) with $m=0$, $\ell =1$, $g_{11}=g_{22}=1000$, for varying $g_{12}$ [$\varphi (\theta ,t)$ is equivalent, but with $g_{11}=g_{12}=g_{22}$]. (b, c) Time-averaged fidelities $\overline{F}(T)=T^{-1}{\int }_0^T\mathit{dt}F(t)$, when $g_{12}=1000$, $T=10$, for $j=1,2$; ${}^{87}\mathrm{Rb}$ lies within the (white) high-fidelity region. All quantities are dimensionless, and the maximum possible $F(t)$ is $0.5$.