Calculation of masses of dark solitons in 1D Bose-Einstein condensates using Gelfand Yaglom method

Abstract

Nonlinear excitations of Bose-Einstein condensates (BEC) play important role in understanding the dynamics of BECs. Solitons, shape preserving wave packets, are the most fundamental nonlinear excitations of BECs. They exhibit particlelike behaviors since their characteristic features do not change during their oscillations and collisons. Moreover, their effective masses are calculated. We are interested in dark solitons which have their density minima at the center. In literature, the mass of dark soliton is obtained with Gross-Pitaevskii approximation. As a result of the contributions of quantum uctuations to the ground state energy, a correction term is added to the effective mass. The dispersion relation of these uctuations are derived from Bogoliubov de Gennes equations. However, with familiar analytical approaches, only a few modes can be taken into account. In order to include all the modes and find an exact expression for ground state energy, we obtain free energy from partition function. The partition function is equivalent to an imaginary-time coherent state Feynman path integral on which periodic boundary conditions are applied. The partition function is in the form of infinite dimensional Gaussian integral, therefore, it is proportional to the determinant of the functional in the integrand. We use Gelfand Yaglom method to calculate the corresponding determinant. Gelfand Yaglom method is a specialized formulation of using zeta functions and contour integrals in calculation of the functional determinant for one-dimensional Schrdinger operators. In this study, we formulate a new technique through this method to calculate ground state energy of stationary dark solitons up to the Bogoliubov order exactly.