Nonlinear Schr\"odinger Equation for Superconductors

TL;DR

This paper derives a nonlinear Schrödinger equation for superconductors using quantum many-body approximations, revealing fundamental properties like Josephson effects and collective modes.

Contribution

It introduces a novel nonlinear Schrödinger framework for superconductors based on the Hartree-Fock-Bogoliubov and Born-Oppenheimer approximations.

Findings

Derivation of a nonlinear Schrödinger equation for superconductors

Explicit demonstration of Galilean invariance

Extraction of superconductor properties such as Josephson effects and collective modes

Abstract

Using the Hartree-Fock-Bogoliubov factorization of the density matrix and the Born-Oppenheimer approximation we show that the motion of the condensate satisfies a nonlinear Schr\"odinger equation in the zero temperature limit. The Galilean invariance of the equation is explicitly manifested. {}From this equation some general properties of a superconductor, such as Josephson effects, the Magnus force, and the Bogoliubov-Anderson mode can be obtained readily.