Durağan ve dönen Bose-Einstein yoğuşmalarına varyasyonel yaklaşım

After the experimental demonstration of Bose-Einstein condensation (BEC)in alkali gases [6, 7, 18], the number of theoretical and experimental papers onultracold atomic physics increased enormously [48]. BEC experiments provide away to manipulate quantum many-body systems, and measure their propertiesprecisely. Although the theory of BEC is simpler compared to other many-bodysystems due to strong correlation, a fully analytical treatment is generally notpossible. Therefore, variational methods, which give approximate analytical so-lutions, are widely used. With this motivation, in this thesis we study on BECin stationary and rotating regimes using variational methods.All the atoms in the condensate can be described with a single wave function,and in the dilute regime this wave function satisfies a single nonlinear equation(the Gross-Pitaevskii equation) which resembles the nonlinear Schrüdinger equa-otion in nonlinear optics. A simple analytical ansatz, which has been used todescribe the intensity profile of the similariton laser [41, 43] having a similar be-havior in the limiting cases of nonlinearity with ground state density profile ofBECs, is used as the trial wave function to solve the Gross-Pitaevskii equationwith variational principle for a wide range of the interaction parameter. Thesimple form of the ansatz allowed us to modify it for both cylindrically symmet-ric and completely anisotropic harmonic traps. The resulting ground state wavefunction and energy are in very good agreement with the analytical solutionsin the limiting cases of interaction and numerical solutions for the intermediateregime.iiiivIn the second part, we consider a rapidly rotating two-component Bose-Einstein condensate containing a vortex lattice. We calculate the dispersion rela-tion for small oscillations of vortex positions (Tkachenko modes) in the mean-fieldquantum Hall regime, taking into account the coupling of these modes with den-sity excitations. Using an analytic form for the density of the vortex lattice, wenumerically calculate the elastic constants for different lattice geometries. We alsoapply this method to the calculation the elastic constant for the single-componenttriangular lattice. For a two-component BEC, there are two kinds of Tkachenkomodes, which we call acoustic and optical in analogy with phonons. For all lat-tice types, acoustic Tkachenko mode frequencies have quadratic wave-numberdependence at long-wavelengths, while the optical Tkachenko modes have lineardependence. For triangular lattices the dispersion of the Tkachenko modes areisotropic, while for other lattice types the dispersion relations show directionaldependence consistent with the symmetry of the lattice. Depending on the in-tercomponent interaction there are five distinct lattice types, and four structuralphase transitions between them. Two of these transitions are second-order andare accompanied by the softening of an acoustic Tkachenko mode. The remain-ing two transitions are first-order and while one of them is accompanied by thesoftening of an optical mode, the other does not have any dramatic effect onthe Tkachenko spectrum. We also find an instability of the vortex lattice whenthe intercomponent repulsion becomes stronger than the repulsion within thecomponents.Keywords: Bose-Einstein condensation, Gross-Pitaevskii equation, vortex lattice,Tkachenko modes, structural phase transition, phase separation, optical lattices.