Fermionization of two distinguishable fermions

In this work we study a system of two distinguishable fermions in a 1D harmonic potential. This system has the exceptional property that there is an analytic solution for arbitrary values of the interparticle interaction. We tune the interaction strength via a magnetic offset field and compare the measured properties of the system to the theoretical prediction. At the point where the interaction strength diverges, the energy and square of the wave function for two distinguishable particles are the same as for a system of two identical fermions. This is referred to as fermionization. We have observed this phenomenon by directly comparing two distinguishable fermions with diverging interaction strength with two identical fermions in the same potential. We observe good agreement between experiment and theory. By adding one or more particles our system can be used as a quantum simulator for more complex few-body systems where no theoretical solution is available

On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid

Golumbic, Lipshteyn and Stern \cite{Golumbic-epg} proved that every graph can be represented as the edge intersection graph of paths on a grid (EPG graph), i.e., one can associate with each vertex of the graph a nontrivial path on a rectangular grid such that two vertices are adjacent if and only if the corresponding paths share at least one edge of the grid. For a nonnegative integer $k$, $B_k$-EPG graphs are defined as EPG graphs admitting a model in which each path has at most $k$ bends. Circular-arc graphs are intersection graphs of open arcs of a circle. It is easy to see that every circular-arc graph is a $B_4$-EPG graph, by embedding the circle into a rectangle of the grid. In this paper, we prove that every circular-arc graph is $B_3$-EPG, and that there exist circular-arc graphs which are not $B_2$-EPG. If we restrict ourselves to rectangular representations (i.e., the union of the paths used in the model is contained in a rectangle of the grid), we obtain EPR (edge intersection of path in a rectangle) representations. We may define $B_k$-EPR graphs, $k\geq 0$, the same way as $B_k$-EPG graphs. Circular-arc graphs are clearly $B_4$-EPR graphs and we will show that there exist circular-arc graphs that are not $B_3$-EPR graphs. We also show that normal circular-arc graphs are $B_2$-EPR graphs and that there exist normal circular-arc graphs that are not $B_1$-EPR graphs. Finally, we characterize $B_1$-EPR graphs by a family of minimal forbidden induced subgraphs, and show that they form a subclass of normal Helly circular-arc graphs