The GrossPitaevskii equation and Bogoliubov spectrum
Pascal Szriftgiser^{1} and Edgardo S. ChebTerrab^{2}
(1) Laboratoire PhLAM, UMR CNRS 8523, Université Lille 1, F59655, France
(2) Maplesoft, Canada
Departing from the equation for a quantum system of identical boson particles, i.e.the GrossPitaevskii equation (GPE), the dispersion relation for planewave solutions are derived, as well as the Bogoliubov equations and dispersion relations for small perturbations around the GPE stationary solutions.

Stationary and planewave solutions to the GrossPitaevskii equation


Problem: Given the GrossPitaevskii equation,

b) Derive the dispersion relation for planewave solutions as a function of G and .

Background: The GrossPitaevskii equation is particularly useful to describe Bose Einstein condensates (BEC) of cold atomic gases [3, 4, 5]. The temperature of these cold atomic gases is typically in the w100 nanoKelvin range. The atomatom interaction are repulsive for and attractive for , where G is the interaction constant. The GPE is also widely used in nonlinear optics to model the propagation of light in optical fibers.


The Bogoliubov equations and dispersion relations


Problem: Given the GrossPitaevskii equation,

b) Show that the dispersion relations of these equations, known as the Bogoliubov spectrum, are given by


,


References
[1] GrossPitaevskii equation (wiki)
[2] Continuity equation (wiki) [3] Bose–Einstein condensate (wiki)
[4] Dispersion relations (wiki)
[5] Advances In Atomic Physics: An Overview, Claude CohenTannoudji and David GueryOdelin, World Scientific (2011), ISBN10: 9812774963.
[6] Nonlinear Fiber Optics, Fifth Edition (Optics and Photonics), Govind Agrawal, Academic Press (2012), ISBN13: 9780123970237.
