Geometrically induced complex tunnelings for ultracold atoms carrying orbital angular momentum

We investigate the dynamics of angular momentum states for a single ultracold atom trapped in two-dimensional systems of sided coupled ring potentials. The symmetries of the system show that tunneling amplitudes between different ring states with variation of the winding number are complex. In particular, we demonstrate that in a triangular ring configuration the complex nature of the cross-couplings can be used to geometrically engineer spatial dark states to manipulate the transport of orbital angular momentum states via quantum interference.

Figure 1

(a) Two in-line ring potentials. (b) Sketch of the energy spectrum of the angular momentum eigenstates $|j,m,n\rangle$ for a single atom trapped in each of the two in-line rings, where $j=L,R$ indicates the ring, $m$ the transverse vibrational state, $n=\pm l$ the winding quantum number, and $l$ the orbital angular momentum. The Hamiltonian corresponding to the $(m,l)$ manifold is indicated by ${\widehat{H}}^{m,l}$. (c) Three-ring potentials in an isosceles triangular configuration.

Figure 2

(a) Temporal evolution of the population of each angular momentum state involved in the dynamics, ${\rho }_{j,\pm 1}={\left|\langle \mathrm{\Psi }\left(t\right)|j,0,\pm 1\rangle \right|}^2$, where $j=L,R$, using the numerically integrated 2D SE (points) and the FSH [Eq. (7)] (lines). (b) Atomic probability density (upper plots) and phase distribution (lower plots) of the state of the system at times $A, B, C$ in panel (a). The phase is plotted only where the probability density is non-negligible. For the parameters see text.

Figure 3

(a) Temporal evolution of the population of each angular momentum state involved in the dynamics, ${\rho }_{j,\pm 1}={\left|\langle \mathrm{\Psi }\left(t\right)|j,0,\pm 1\rangle \right|}^2$, where $j=C,L,R$, using the numerically integrated 2D SE (points) and the SSH (lines) when the initial state of the system is $|C,0,1\rangle$. (b) Probability density (upper plots) and phase distribution (lower plots) of the state of the system at times $A$ and $B$ in panel (a). (c) Temporal evolution of the population of the dark state $|D_{+},0\rangle$ using the numerically integrated SE (points) and the SSH (lines) when the system is initialized in this state. Phase is only plotted where the probability density is non-negligible. For the parameters see text.