Robust mesoscopic superposition of strongly correlated ultracold atoms

We propose a scheme to create coherent superpositions of annular flow of strongly interacting bosonic atoms in a one-dimensional ring trap. The nonrotating ground state is coupled to a vortex state with mesoscopic angular momentum by means of a narrow potential barrier and an applied phase that originates from either rotation or a synthetic magnetic field. We show that superposition states in the Tonks-Girardeau regime are robust against single-particle loss due to the effects of strong correlations. The coupling between the mesoscopically distinct states scales much more favorably with particle number than in schemes relying on weak interactions, thus making particle numbers of hundreds or thousands feasible. Coherent oscillations induced by time variation of parameters may serve as a “smoking gun” signature for detecting superposition states.

Figure 1

Strongly correlated atoms trapped in a narrow ring with a rotating barrier. The inset shows the energy levels of the supercurrent states with total angular momentum $K=0$ and $K=N\hslash$, respectively, as a function of the rotational phase $\Omega$ (dashed lines), and the lowest energy levels of $N=99$ atoms in the Tonks-Girardeau regime with barrier strength $b/L=0.008E_0$ (solid lines).

Figure 2

The level splitting $\Delta E$ between the ground and first excited states at $\Omega =\pi$ is plotted over interaction strength $g$ for five atoms with $b/L=0.008E_0$. The top horizontal axis shows the Lieb-Liniger parameter, $\gamma =g2{\pi }^2/(E_{0}LN)$. The dashes on the figure margins indicate analytic results for noninteracting ($g=0$) and strongly interacting atoms ($g=\infty$). Subplots show the distribution of the total angular momentum for (a) $g=0$ and (b) $g=\infty$.

Figure 3

The quality of the superposition before and after removal of an atom. The dashed line shows the superposition quality $Q=4P(0)P(N\hslash )$ for the ground state of the rotating system with parameters as in Fig. 2. The solid line shows the average superposition quality ${\mathrm{Q̄}}^{[-1]}$ after the loss of an atom, as described in the text. The crosses in inset (a) show ${\mathrm{Q̄}}^{[-1]}$ vs atom number in the Tonks-Girardeau regime, and the solid line shows the result $1-1/N$ for noninteracting fermions for comparison. The horizontal bar visible for $N=6$ estimates the error in the numerical result from an insufficient number $r$ of single-particle modes by extrapolating from $r=14$ (the reported result) to larger $r$ by adding twice the difference from the $r=12$ result. All other results use $r=20$. Inset (b) shows the total angular momentum distribution after removal of an atom with $k=1\hslash$ in the Tonks-Girardeau regime.

Figure 4

NOON state energy-level splitting as a function of atom number. The line shows the energy-level splitting calculated using Eq. (7), where $b/L=0.008E_0$ and $g/L=4\pi b\sqrt{N}/L(N-1)$. In the regime where the NOON state is formed, it is adequate to describe the system using just the 0 and $\hslash$ angular momentum modes. The results from the numerical simulation are shown by the red crosses. The energy-level splitting decreases rapidly as the number of particles is increased. We see the numerical result break down for $N>11$ due to limited numerical accuracy. The analytic result is still valid beyond this point.