The Gross-Pitaevskii equation and Bogoliubov spectrum
Pascal Szriftgiser1 and Edgardo S. Cheb-Terrab2
(1) Laboratoire PhLAM, UMR CNRS 8523, Université Lille 1, F-59655, France
(2) Maplesoft, Canada
Departing from the equation for a quantum system of identical boson particles, i.e.the Gross-Pitaevskii equation (GPE), the dispersion relation for plane-wave solutions are derived, as well as the Bogoliubov equations and dispersion relations for small perturbations around the GPE stationary solutions.
Stationary and plane-wave solutions to the Gross-Pitaevskii equation
Problem: Given the Gross-Pitaevskii equation,
a) Derive a relationship between the chemical potential entering the phase of stationary, uniform solutions, the atom-atom interaction constant G and the particle density in the absence of an external field ().
b) Derive the dispersion relation for plane-wave solutions as a function of G and .
Background: The Gross-Pitaevskii 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 nano-Kelvin range. The atom-atom interaction are repulsive for and attractive for , where G is the interaction constant. The GPE is also widely used in non-linear optics to model the propagation of light in optical fibers.
Solution
The Bogoliubov equations and dispersion relations
a) Derive the Bogoliubov equations, that is, equations for elementary excitations and around a GPE stationary solution ,
b) Show that the dispersion relations of these equations, known as the Bogoliubov spectrum, are given by
,
where is the wave number of the considered elementary excitation, its energy or, equivalently, its frequency.
References
[1] Gross-Pitaevskii equation (wiki)
[2] Continuity equation (wiki) [3] Bose–Einstein condensate (wiki)
[4] Dispersion relations (wiki)
[5] Advances In Atomic Physics: An Overview, Claude Cohen-Tannoudji and David Guery-Odelin, World Scientific (2011), ISBN-10: 9812774963.
[6] Nonlinear Fiber Optics, Fifth Edition (Optics and Photonics), Govind Agrawal, Academic Press (2012), ISBN-13: 978-0123970237.