Superposition states of ultracold bosons in rotating rings with a realistic potential barrier

In a recent paper [Phys. Rev. A 82, 063623 (2010)] Hallwood et al. argued that it is feasible to create large superposition states with strongly interacting bosons in rotating rings. Here we investigate in detail how the superposition states in rotating-ring lattices depend on interaction strength and barrier height. With respect to the latter we find a trade-off between energy gap and quality of the superposition state. Most importantly, we go beyond the $\delta$-function approximation for the barrier potential and show that the energy gap decreases exponentially with the number of particles for weak barrier potentials of finite width. These are crucial issues in the design of experiments to realize superposition states.

Figure 1

Energy spectrum and characterization of the ground-state wave function for $N=3$ particles on $L=5$ sites at the critical phase twist with $V_0/J=0.01$ as a function of the interaction parameter $U/J$: (a) many-body energy spectrum, (b) momentum distribution $\langle {\widehat{n}}_q\rangle$, (c) momentum noise correlations $\langle {\widehat{n}}_q{\widehat{n}}_{q^{\prime }}\rangle -\langle {\widehat{n}}_q\rangle \langle {\widehat{n}}_{q^{\prime }}\rangle$, (d) spectrum of single-particle density matrix $\langle {\widehat{a}}_i^{†}{\widehat{a}}_j\rangle$, (e) distribution of total quasimomentum $P\left(K\right)$ (as defined in the text), and (f) fidelity with respect to the states $|{\psi }_0\rangle$, $|{\psi }_1\rangle$, and $|{\psi }_{\infty }\rangle$ in Eqs. (3), (4), and (5).

Figure 2

Energy gap $\Delta E$ (solid lines) and cat quality $Q=4P\left(0\right)P\left(N\right)$ (dashed lines) vs single-site barrier height $V_0$ for $L=9$ as well as $N=2$ (blue), $N=3$ (green), and $N=4$ (red) at the critical phase twist in the strongly interacting limit $U/J\to \infty$. Insets show the distribution of total quasimomentum $P\left(K\right)$ for $V_0/J={10}^{-4}$ and $V_0/J={10}^{+4}$ for $N=4$.

Figure 3

Energy gap $\Delta E$ vs particle number $N$ for $L=50$, $V_0/J=0.01$, $b=\xi /d=0$ (blue), $b=\xi /d=2/3$ (green), $b=\xi /d=4/3$ (red), and $b=\xi /d=2$ (cyan). Points are the exact single-particle spectrum, and lines are from perturbative expression (6). The inset shows the barrier potential $V_j$ for $L=50$, $V_0/J=0.01$, and $b=\xi /d=2$.