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

TL;DR

The paper analyzes the two-boson quantum walk on a Hatano-Nelson lattice with non-reciprocal tunneling, uncovering how non-Hermiticity, interparticle interactions, and a dc field sculpt density spreading and correlations. It shows that density evolves into asymmetric cones in the absence of a field and into an asymmetric hourglass pattern when a dc field is present, with the phenomena strongly depending on the initial state and interaction strength. Correlation measures reveal non-reciprocal bunching and anti-bunching governed by $U$ and $\delta$, while the Quantum Fisher Information scales as $F_Q \propto t^3$ for both one- and two-particle walks, even in the non-Hermitian regime. These results establish the system as a platform for quantum-enhanced sensing of weak forces and motivate further studies of non-Hermitian multi-particle dynamics in more complex settings.

Abstract

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, $δ$. 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 $F_Q \propto t^α$ where $α\sim 3$. 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.

Figures (5)

• Figure 1: Transition from symmetric to aysmmetric spreading of the density cone due to non-reciprocal tunneling, in the absence of an external field ($F=0$): (a1) two bosons are initially placed on nearest-neighbur sites with $U=2$ for varying $\delta$, (b1) both bosons are initially on the same site with $U=2$ for varying $\delta$, (a2) two bosons on nearest-neighbur sites with $\delta=0.04$ for varying $U$, (b2) both bosons are initially on the same site with $\delta=0.04$ for varying $\delta$.
• Figure 2: Spatial correlation of bosons in presence of on-site interaction $U$ and non-hermitian parameter $\delta$ in the absence of field ($F=0$): (a1) two bosons are initially placed on nearest-neighbour sites with $U=2$ for varying $\delta$, (b1) bosons are initially placed on the same site with $U=2$ for varying $\delta$, (a2) two bosons on nearest-neighbour sites with $\delta=0.04$ for varying $U$, (b2) bosons are initially placed on the same site with $\delta=0.04$ for varying $\delta$.
• Figure 3: Time evolution of the density in presence of a dc field ($F=0.26$) (a1) two bosons are initially placed on nearest-neighbour sites with $U=2$ for varying $\delta$, (b1) bosons are initially placed on the same site with $U=2$ for varying $\delta$, (a2) two bosons on nearest-neighbour sites with $\delta=0.04$ for varying $U$, (b2) bosons are initially placed on the same site with $\delta=0.04$ for varying $U$.
• Figure 4: The modification of correlations due to the interaction $U$ and the non-Hermitian parameter $\delta$ in the presence of the field($F=0.26)$: (a1) two bosons are initially placed on nearest-neighbour sites with $U=2$ for varying $\delta$, (b1) bosons are initially placed on the same site with $U=2$ for varying $\delta$, (a2) two bosons on nearest-neighbor sites with $\delta=0.04$ for varying $U$, (b2) bosons are initially placed on the same site with $\delta=0.04$ for varying $U$.
• Figure 5: Quantum Fisher Information for one- and two-particle quantum walks for $U=2$ for different $\delta$: (a) One boson at the central site , (b) Two Bosons at neighboring central sites, (c) Both bosons are on the same central site.