Browsing by Subject "Path integral"

Now showing 1 - 3 of 3

Open Access
Calculation of masses of dark solitons in 1D Bose-Einstein condensates using Gelfand Yaglom method
(2016-11) Yıldız, Kübra Işık
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.
Open Access
Phonon-mediated electron-electron interaction in confined media: low-dimensional bipolarons
(2000-09) Senger, R. Tuğrul
We study the criterion for the formation of confined large bipolarons and their stability. In order to deal with this specific subject of polaron theory, it is required to adopt some particular approximation methods, because the polaronic systems do not admit exact analytic solutions in general. Those approximation techniques, which are applied to the low-dimensional one-polaron problems, are presented to some extent to form a working basis for our main theme, bipolarons. As the model of confined bipolaron, the electrons are treated as bounded within an external potential while being coupled to one another via the Fröhlich interaction Hamiltonian. Within the framework of the bulk-phonon approximation, the model that we use consists of a pair of electrons immersed in a reservoir of bulk LO phonons and confined within an anisotropic parabolic potential box, the barrier slopes of which can be tuned arbitrarily from zero to infinity. Thus, encompassing the bulk and all low-dimensional geometric configurations of general interest, we obtain an explicit tracking of the critical values of material parameters for the bipolarons to exist in confined media. First, in the limit of strong electron-phonon coupling, we present a unified insight into the stability criterion by applying the Landau-Pekar strong coupling approximation. This crude approximation provides us the condition on the ratio of dielectric constants (η = epsilon substcript infinity/epsilon substrcript 0) for large values of electron-phonon coupling constant α. For more reliable results, we consider the path-integral formulation of the problem adopting the Feynman-polaron model to derive variational results over a wide range of the Coulomb interaction and phonon coupling strengths. It is shown that the critical values of α and η exhibit some non-trivial features as the effective dimensionality is varied, and the path integral results conform to those of strong coupling approximation in the limit of large α.
Open Access
SU(2)-path integral investigation of Holstein dimer
(Elsevier Science Publishers B.V., Amsterdam, Netherlands, 2000) Hakioğlu, A. T.; Ivanov, V. A.; Zhuravlev, M. Ye.
The SU(2) coherent state path integral is used to investigate the partition function of the Holstein dimer. This approach naturally takes into account the symmetry of the model. The ground-state energy and the number of the phonons are calculated as functions of the parameters of the Hamiltonian. The renormalizations of the phonon frequency and electron orbital energies are considered. The destruction of quasiclassical mean-field solution is discussed.