Directional quantum walks of two bosons on the Hatano-Nelson lattice

We theoretically investigate the interplay of interactions and non-Hermiticity in the dynamics of two bosons on the one-dimensional Hatano-Nelson lattice with non-reciprocal tunneling. We find that the non-reciprocity in the tunneling leads to the formation of an asymmetric density cone during the time-evolution of the system; the degree of asymmetry can be tuned by tuning the non-reciprocity parameter, $\delta$. Next, we analyze the dynamics of this system in the presence of a static external force and demonstrate that non-Hermiticity leads to asymmetric two-particle Bloch oscillations. Interestingly, when $F=0$ ($F \ne 0$), strong interactions leads to the formation of an inner density-cone (density-hourglass) structure; this inner structure also becomes asymmetric in the presence of non-Hermiticity. We further analyze the spatial correlations and establish that the system exhibits non-reciprocal bunching (anti-bunching) in the presence of weak (strong) interactions. Finally, we examine the growth of the Quantum Fisher Information, $F_Q$, with time, and demonstrate that $F_Q \propto t^{\alpha}$ where $\alpha \sim 3$. This feature persists for both one- and two-particle walks, thereby demonstrating that this system can be employed as a quantum-enhanced sensor for detecting weak forces.