Interacting bosons in a double-well potential : localization regime

TL;DR

This paper investigates the ground state behavior of large bosonic systems in a double-well potential, revealing an interaction-driven transition from delocalized to localized states as the well separation increases.

Contribution

It provides a rigorous analysis of the localization transition in bosonic systems, connecting many-body Schrödinger Hamiltonian to nonlinear Schrödinger functionals in the large particle limit.

Findings

Strong suppression of particle number fluctuations in each well when tunneling is negligible.

Identification of the condensation modes as minimizers of nonlinear Schrödinger-type functionals.

Evidence of an interaction-driven transition between delocalized and localized states.

Abstract

We study the ground state of a large bosonic system trapped in a symmetric double-well potential, letting the distance between the two wells increase to infinity with the number of particles. In this context, one should expect an interaction-driven transition between a delocalized state (particles are independent and all live in both wells) and a localized state (particles are correlated, half of them live in each well). We start from the full many-body Schr{\"o}dinger Hamiltonian in a large-filling situation where the on-site interaction and kinetic energies are comparable. When tunneling is negligible against interaction energy, we prove a localization estimate showing that the particle number fluctuations in each well are strongly suppressed. The modes in which the particles condense are minimizers of nonlinear Schr{\"o}dinger-type functionals.