Josephson plasma oscillations and the Gross-Pitaevskii equation: Bogoliubov approach vs two-mode model

TL;DR

This paper compares the accuracy of the Bogoliubov approach and the two-mode model in calculating Josephson plasma oscillations in a Bose-Einstein condensate, showing the Bogoliubov method's superior precision especially at stronger interactions.

Contribution

It demonstrates that the Bogoliubov approach accurately predicts Josephson plasma frequencies, while the two-mode model is only reliable for weak interactions, and discusses a proper two-mode formulation using Bogoliubov functions.

Findings

Bogoliubov approach yields precise plasma frequency calculations.

Two-mode model is accurate only for weak interactions.

Proper two-mode model with Bogoliubov functions is discussed.

Abstract

We show that the Josephson plasma frequency for a condensate in a double-well potential, whose dynamics is described by the Gross-Pitaevskii (GP) equation, can be obtained with great precision by means of the usual Bogoliubov approach, whereas the two-mode model - commonly constructed by means of a linear combinations of the low-lying states of the GP equation - generally provides accurate results only for weak interactions. A proper two-mode model in terms of the Bogoliubov functions is also discussed, revealing that in general a two-mode approach is formally justified only for not too large interactions, even in the limit of very small amplitude oscillations. Here we consider specifically the case of a one-dimensional system, but the results are expected to be valid in arbitrary dimensions.