A variational approech to stationary and rotating Bose-Einstein condensates

After the experimental demonstration of Bose-Einstein condensation (BEC) in alkali gases [6, 7, 18], the number of theoretical and experimental papers on ultracold atomic physics increased enormously [48]. BEC experiments provide a way to manipulate quantum many-body systems, and measure their properties precisely. Although the theory of BEC is simpler compared to other many-body systems due to strong correlation, a fully analytical treatment is generally not possible. Therefore, variational methods, which give approximate analytical solutions, are widely used. With this motivation, in this thesis we study on BEC in 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 Schr¨odinger equation in nonlinear optics. A simple analytical ansatz, which has been used to describe the intensity profile of the similariton laser [41, 43] having a similar behavior in the limiting cases of nonlinearity with ground state density profile of BECs, is used as the trial wave function to solve the Gross-Pitaevskii equation with variational principle for a wide range of the interaction parameter. The simple form of the ansatz allowed us to modify it for both cylindrically symmetric and completely anisotropic harmonic traps. The resulting ground state wave function and energy are in very good agreement with the analytical solutions in the limiting cases of interaction and numerical solutions for the intermediate regime. In the second part, we consider a rapidly rotating two-component BoseEinstein condensate containing a vortex lattice. We calculate the dispersion relation for small oscillations of vortex positions (Tkachenko modes) in the mean-field quantum Hall regime, taking into account the coupling of these modes with density excitations. Using an analytic form for the density of the vortex lattice, we numerically calculate the elastic constants for different lattice geometries. We also apply this method to the calculation the elastic constant for the single-component triangular lattice. For a two-component BEC, there are two kinds of Tkachenko modes, which we call acoustic and optical in analogy with phonons. For all lattice types, acoustic Tkachenko mode frequencies have quadratic wave-number dependence at long-wavelengths, while the optical Tkachenko modes have linear dependence. For triangular lattices the dispersion of the Tkachenko modes are isotropic, while for other lattice types the dispersion relations show directional dependence consistent with the symmetry of the lattice. Depending on the intercomponent interaction there are five distinct lattice types, and four structural phase transitions between them. Two of these transitions are second-order and are accompanied by the softening of an acoustic Tkachenko mode. The remaining two transitions are first-order and while one of them is accompanied by the softening of an optical mode, the other does not have any dramatic effect on the Tkachenko spectrum. We also find an instability of the vortex lattice when the intercomponent repulsion becomes stronger than the repulsion within the components.