Condensed Matter & Surface Sciences
COLLOQUIUM
Carlos
Trallero-Giner
“Bose-Einstein condensates:
Soliton formation and analytical methods of solutions
for the Gross-Pitaevskii equation”
Since the unambiguous experimental realization in dilute ultra-cold atom clouds of the Bose-Einstein condensed phase (BEC) a lot of work has been devoted for searching the dynamic and physical properties of the nonlinear matter waves and excitations of the condensate. The dynamics of the order parameter of the system, Ф(x), has been ruled by equations of nonlinear Schrödinger (NLS) type and mainly by the Gross-Pitaevskii equation (GPE). Nowadays, NLS equations with attractive (negative scattering length) and repulsive (positive scattering length) nonlinear interactions have been reported to describe experimental observations of different types of wave solitons. Most of the theoretical work has been devoted to implement numerical solutions of the GPE for the order parameter. To study and to control the physical properties of the condensate it will be very useful to manipulate analytical expressions for the chemical potential and for the function Ф(x) as well. A typical example of the great physical interest is the attention devoted to the collective excitation spectrum of a BEC. In this case we have to deal with the time-dependent GP equation under the linear response approximation.
Nowadays, available numerical methods for solving differential equations are fast and accurate. Nevertheless, if the evaluation of several physical magnitudes is carried out, such as the optical properties among others, or to control the properties of the condensate, this advantage is lost due to cumbersome numerical computational procedures that must be performed at the end of the calculation. In contrast, the most important requirements for analytical solutions are simplicity, flexibility, and the viability to be used in perturbation approaches for the calculations of physical properties.
In
the present work we present different methods of solutions of the time
independent GPE based on the equivalent integral GPE and its relation with the
Green function of the corresponding linear operator, on the soliton solution, and
on a bright soliton-like variational function. This discussion will provide
general analytical expressions for the order parameter and for the chemical
potential in a universal range of the non-linear interaction parameter.
Thursday, September 13, 2007
4:10 p.m. -- Walter Lecture Hall 245