Engineering mesoscopic superpositions of superfluid flow

Modeling strongly correlated atoms demonstrates the possibility to prepare quantum superpositions that are robust against experimental imperfections and temperature. Such superpositions of vortex states are formed by adiabatic manipulation of interacting ultracold atoms confined to a one-dimensional ring trapping potential when stirred by a barrier. Here, we discuss the influence of nonideal experimental procedures and finite temperature. Adiabaticity conditions for changing the stirring rate reveal that superpositions of many atoms are most easily accessed in the strongly interacting, Tonks-Girardeau, regime, which is also the most robust at finite temperature. NOON-type superpositions of weakly interacting atoms are most easily created by adiabatically decreasing the interaction strength by means of a Feshbach resonance. The quantum dynamics of small numbers of particles is simulated and the size of the superpositions is calculated based on their ability to make precision measurements. The experimental creation of strongly correlated and NOON-type superpositions with about 100 atoms seems feasible in the near future.

Figure 1

Ultracold atoms confined to a loop of circumference $L$. The position on the circumference of the loop is given by $x=L\theta$. The barrier is shown by the break in the ring and modeled by a $\delta$ function that moves with a tangential velocity of $v=\hslash \Omega /ML$.

Figure 2

The energy spectrum of five strongly correlated atoms confined to a ring trapping potential and large barrier as a function of $\Omega$ (subsequent results use a smaller barrier height of $b/L=0.008E_0$ for increased numerical accuracy). At $\Omega =\pi$ there is an avoided crossing in the lowest two levels and this is the point where the superposition $|0\rangle +|N\rangle$ is found. The inset shows the probability that the ground state has total angular momentum $K=0$, $P\left(0\right)$, as a full line, and total angular momentum $K=N\hslash$, $P\left(N\right)$, as a dashed line. At $\Omega =\pi$ the probability of the two states is 0.5.

Figure 3

The energy spectrum of atoms confined to a ring trapping potential as a function of interaction strength $g$ with $b/L=0.08E_0$ and $\Omega =\pi$. Energies are given relative to the ground-state energy.

Figure 4

The energy level splitting at the avoided crossing $\Delta E$ is plotted against $N$ for both (a) the NOON state (also found in Ref. [21]) and (b) the Tonks-Girardeau state. Panel (a) shows the analytic result for two modes (solid line) and the numerical result for two modes ($+$), where $b/L=0.008E_0$ and $g/L=4\pi b\sqrt{N}/L(N-1)$. The numerical result breaks down for $N>18$ due to limited numerical accuracy. The analytic result is still valid beyond this point. Panel (b) shows the analytic result for $b/L=0.08E_0$. For large $N$ and a small barrier, $\Delta E$ asymptotes toward $\Delta E=2b/L$.

Figure 5

The inset shows the QFI as a function of $\Omega$ in the Tonks-Girardeau regime. Shown in the inset is the full width half maximum $\Gamma$. In the main figure, $\Gamma$ is plotted on the left axis against the interaction strength $g$ for $b/L=0.08E_0$. The solid line shows the numerical result and the dashed line the semi-analytic result given by Eq. (38). The right axis shows the QFI for different interaction strengths when $\Omega =\pi$ (dotted line).

Figure 6

QFI as a function of temperature and interaction strength. Each line represents a particular interaction strength. The thick blue line shows the analytic result for a Tonks-Girardeau superposition of Eq. (40) and the red crosses show the QFI at $T=0$ from Fig. 5. The barrier height is $b/L=0.08E_0$.