Physical approach to quantum networks with massive particles

TL;DR

This paper presents a physical analysis of quantum networks with massive particles, revealing localized ground states in X-shaped potential wells that differ from traditional quantum graph models, and introduces a solitonic perspective for network behavior.

Contribution

It provides an analytical and physical approach to quantum networks, explicitly deriving eigenstates and contrasting them with quantum graph predictions, highlighting solitonic localized states.

Findings

Ground state is exponentially localized at the X well center.

Contrasts with Kirchhoff boundary conditions predictions.

Localized solitonic states enable discrete lattice coupling.

Abstract

This paper treats a quantum network from a physical approach, explicitly finds the physical eigenstates and compares them to the quantum-graph description. The basic building block of a quantum network is an X-shaped potential well made by crossing two quantum wires, and we consider a massive particle in such an X well. The system is analyzed using a variational method based on an expansion into modes with fast convergence and it provides a very clear intuition for the physics of the problem. The particle is found to have a ground state that is exponentially localized to the center of the X well, and the other symmetric solutions are formed so to be orthogonal to the ground state. This is in contrast to the predictions of the conventionally used so-called Kirchoff boundary conditions in quantum graph theory that predict a different sequence of symmetric solutions that cannot be physically realized. Numerical methods have previously been the only source of information on the ground-state wave function and our results provide a different perspective with strong analytical insights. The ground-state wave function has the shape of a solitonic solution to the non-linear Schr{\"o}dinger equation, enabling an analytical prediction of the wave number. When combining multiple X wells into a network or grid, each site supports a solitonic localized state. The solitons only couple to each other and are able to jump from one site to another as if they were trapped in a discrete lattice.