Criticality and spin squeezing in the rotational dynamics of a Bose-Einstein condensate on a ring lattice

We examine the dynamics of circulating modes of a Bose-Einstein condensate confined in a toroidal lattice. Nonlinearity due to interactions leads to criticality that separates oscillatory and self-trapped phases among counterpropagating modes which however share the same physical space. In the mean-field limit, the criticality is found to substantially enhance sensitivity to rotation of the system. Analysis of the quantum dynamics reveals the fluctuations near criticality are significant, which we explain using spin-squeezing formalism visualized on a Bloch sphere. We utilize the squeezing to propose a Ramsey interferometric scheme that suppresses fluctuation in the relevant quadrature sensitive to rotation.

Figure 1

The atoms are trapped in a toroidal geometry with effectively one-dimensional dynamics about the major circle. A circular lattice potential along the ring can be introduced or removed as required.

Figure 2

Time evolution of the sum and the difference of populations ${\left|a\left(t\right)\right|}^2$ and ${\left|b\left(t\right)\right|}^2$ in states $|\pm q\rangle$ is plotted by numerically solving Eq. (14). The constancy of their sum justifies the two-mode model, Eqs. (15), when $u\ll {\hslash }^2{q}^2/2m{R}^2$. Panels (a) and (b) illustrate the noninteracting case; (b) shows that significant fluctuations out of two modes occur only at much higher lattice potentials [$u$ is 2000 times higher in panel (b) compared to panel (a)]. With nonlinearity, there can be both (c) oscillatory and (d) self-trapped regimes.

Figure 3

In the mean-field approximation, starting with ${\left|a\left(0\right)\right|}^2=N$, the time-evolved population $|a\left(t_s\right){|}^2$ in mode $|+q\rangle$ at the fixed time $t_s$ is plotted as a function of the angular velocity $\omega$; here $t_s$ is the time required for a complete swap to state $|-q\rangle ,|b\left(t_s\right){|}^2=N$ in the noninteracting case when $\omega =0$ as seen in panel (a). The remaining panels [panels (b)–(d)] show that nonlinearity sharpens the dependence on the $\omega$, as $u\to u_c=N\chi /4$, the critical value from above with $\mathrm{\Delta }u=100(u-u_c)/u_c$. Although the plots look similar, note the scale of the horizontal axis that indicates a 2 orders of magnitude increase in sensitivity to changes in $\omega$ as criticality is approached.

Figure 4

Phase-space trajectories of the semiclassical dynamical variables $z$ and $\varphi$. Each line represents the time evolution for specific values of $\chi$ and $\omega$. Transition from oscillatory to self-trapped behavior as $\omega$ is varied with the interaction strength $\chi$ fixed (just below the critical point for $\omega =0$).

Figure 5

Comparison of the two-mode quantum time evolution (blue line with error bars) with the mean-field evolution (red lines without error bars). The population difference between the two modes of interest are plotted as a function of time for no rotation $\omega =0$. The transition from self-trapping to oscillatory behavior is seen as the lattice strength $u$ passes through the critical value of $u_c=N\chi /4=5.155$. Notably close to the critical point, the quantum fluctuations are enhanced and there is substantial difference between the two pictures, but not so further from the critical point.

Figure 6

Bloch-sphere representation of the two-mode quantum mechanical time evolution with (a) initially all the population centered at the north pole ($\langle |a\left(0\right){|}^2\rangle =N$). (b) In the absence of interaction, the density distribution undergoes rotation but its shape is maintained. (c) With interaction, in addition to rotation, spin-squeezing distorts the distribution, which eventually destroys its Gaussian profile as well (d). Here we use $N=100$ and $N\chi =2.06$ for illustration, but for $N=10000$, used in our estimates, squeezing is much more pronounced.

Figure 7

Comparison of the two-mode quantum time evolution (blue line with error bars) with the mean-field evolution (red line without error bars). The population difference between the two modes of interest as a function of the rotation rate $\omega$ after fixed time evolution $t_s$, the time for full swap with no interaction and no rotation. Notably close to the critical point $u_c=N\chi /4=5.155$, the quantum fluctuations are enhanced and there is a substantial difference between the two pictures, but not so further from the critical point.

Figure 8

Comparison of the Gaussian approximation (green dashed line with dotted error bars) with the quantum simulation (thick blue line with solid error bars) as well as with the mean-field approximation (thin red line without error bars). Panels (a) and (b) show the time evolution of the modal population difference, corresponding to Figs. 5 and 5, and panels (c) and (d) show its value at time $t_s$ as a function of the rotation rate $\omega$ and correspond to Figs. 7 and 7. The Gaussian approximation generally overestimates the fluctuations more so closer to criticality.

Figure 9

The sequence of steps in the Ramsey interferometric scheme to suppress fluctuations: (a) initial state with mean population entirely in state $a=|+q\rangle$, (b) the state is transformed to be centered on the equator, (c) the state has undergone nonlinear evolution with maximal spin squeezing, (d) the state is transformed to have a long axis parallel to the ${\widehat{J}}_z$ axis, (e) free evolution solely under the influence of the angular velocity $\omega$, and (f) the state is transformed to have a short axis aligned with the quadrature of interest ${\widehat{J}}_z$ for reduced fluctuations. Here we use $N=100$ and $N\chi =10.3$ for illustration; but for $N=10000$ as used in our numerical estimates squeezing is much more pronounced.