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# The Gross-Pitaevskii equation and Bogoliubov spectrum

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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

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

Problem: Given the Gross-Pitaevskii equation,

 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.
 Solution References Advances In Atomic Physics: An Overview, Claude Cohen-Tannoudji and David Guery-Odelin, World Scientific (2011), ISBN-10: 9812774963.

 Nonlinear Fiber Optics, Fifth Edition (Optics and Photonics), Govind Agrawal, Academic Press (2012), ISBN-13: 978-0123970237.