Single-atom edgelike states via quantum interference

We demonstrate how quantum interference may lead to the appearance of robust edgelike states of a single ultracold atom in a two-dimensional optical ribbon. We show that these states can be engineered within the manifold of either local ground states of the sites forming the ribbon or states carrying one unit of angular momentum. In the former case, we show that the implementation of edgelike states can be extended to other geometries, such as tilted square lattices. In the latter case, we suggest using the winding number associated to the angular momentum as a synthetic dimension.

Figure 1

Sketch of the considered 2D optical ribbon, with circles representing its sites. The index $i=1,...,n$ refers to individual cells, which are squares with one site laying at each of their corners plus a central site placed at an equal distance $d$ from the outer ones. The sites are labeled with the index $j=1,...,N$, with $N=3n+2$. As indicated by the double arrows, we only consider the central sites to be tunneled coupled to their four nearest-neighbor sites.

Figure 2

(a) Average population of the central sites of a two-cell ribbon over a time $T=1000{\omega }^{-1}$ when choosing as initial state ${|\mathrm{\Psi }\left(\phi \right)\rangle }_{l=0}$ (black) and ${|\mathrm{\Psi }\left(\phi \right)\rangle }_{l=1}$ (red). Solid lines and points correspond to the results obtained with the few-state models and full numerical integration of the 2D Schrödinger equation, respectively. (b) Time evolution of the population of different initial states $|{\mathrm{\Psi }}_0\rangle$ under the action of ${\widehat{H}}_0$ in a ribbon of 100 cells.

Figure 3

(a) Density plots of the atomic wave function in a single cell of the considered ribbon when the atom is located on the central site (A), some population has been transferred to the outer sites (B), and the central site is unpopulated (C). The separation between tunneled-coupled sites is $d=5\sigma$. (b) Time evolution of the populations of the central (red) and outer (blue) sites of the cell during the Rabi-type oscillations. Continuous lines correspond to the full integration of the Schrödinger equation and dots correspond to the results obtained with the few-state Hamiltonian (2). (c) By illuminating two sites with a pulse of area $2\pi$, it is possible to induce $\pi$ phase changes on their local functions. In the leftmost sketch this is done at the instant C, giving rise to an ELS. The other panels show how sequential application of pulses allows us to switch between the different ELS.

Figure 4

Density profile of the two ELS (7) of the $l=1$ manifold restricted to one cell of a ribbon with $d=6\sigma$. For each state, the wave function has a nodal line with the same orientation at all sites, giving rise to global chirality.

Figure 5

(a) ELS constructed from SDS of $l=0$ on a tilted square lattice, with all edge sites but the corners equally populated. As indicated by the alternating orange and green colorings, the atomic states at adjacent sites have $\pi$ relative phases. Double arrows indicate nearest-neighbor tunneling couplings, with the ones involved in the SDS highlighted in black. (b) Schematics of the extension of the optical ribbon into the synthetic dimension given by the winding number in the $l=1$ case. Each red dot can host only one state, and tunneling coupling (represented by the double arrows) can occur through sites in the same or different winding number planes.