Exact Solution of Bogoliubov Equations for Bosons in One-Dimensional Piecewise Constant Potential

TL;DR

This paper provides exact solutions to Bogoliubov equations for one-dimensional bosonic systems with piecewise constant potentials, enabling detailed analysis of excitation transmission, reflection, and phenomena like anomalous tunneling.

Contribution

It introduces a method to solve Bogoliubov equations exactly in one-dimensional piecewise constant potentials and analyzes specific cases like barriers and steps.

Findings

Exact wavefunctions for Bogoliubov excitations derived.

Demonstrates anomalous tunneling effect in specific potentials.

Shows quantum evaporation phenomena in the studied systems.

Abstract

We show that Bogoliubov equations in one-dimensional systems with piecewise constant potentials can be always solved. In particular, we analyze in detail the case where the condensate wavefunction is a real-valued function, and give the explicit expressions for wavefunctions of Bogoliubov excitations. By means of these solutions, we consider transmission and reflection properties of Bogoliubov excitations for two types of potential, namely, a rectangular barrier and a potential step. The results yield simple and exact examples of anomalous tunneling effect and quantum evaporation.