Dirac particles in transparent quantum graphs: Tunable transport of relativistic quasiparticles in branched structures

We consider the dynamics of relativistic spin-half particles in quantum graphs with transparent branching points. The system is modeled by combining the quantum graph concept with the one of transparent boundary conditions applied to the Dirac equation on metric graphs. Within such an approach, we derive simple constraints, which turn the usual Kirchhoff-type boundary conditions at the vertex equivalent to the transparent ones. Our method is applied to quantum star graph. An extension to more complicated graph topologies is straightforward.

Figure 1

Star graph with $N$ semi-infinite bonds.

Figure 2

The position probability density $|{\varphi }_j{(x,t)|}^2+{\left|{\chi }_j(x,t)\right|}^2$ plotted at different time moments for the regime when the sum rule is fulfilled (no reflection occurred): ${\alpha }_1=\sqrt{2/3}$, ${\alpha }_2=1$, and ${\alpha }_3=\sqrt{2}$.

Figure 3

Time dependence of the partial and total norms for the case shown in Fig. 2.

Figure 4

Dependence of the vertex reflection coefficient $R$ on the parameter ${\alpha }_1$ when the wave-packet splitting time elapses ($t=10$). For fixed ${\alpha }_2=1$ and ${\alpha }_3=\sqrt{2}$, $R=0$ when ${\alpha }_1=\sqrt{2/3}\approx 0.816$ (red dot).