A variational approech to stationary and rotating Bose-Einstein condensates
Author
Keçeli, Murat
Advisor
Oktel, Mehmet Özgür
Date
2006Publisher
Bilkent University
Language
English
Type
ThesisItem Usage Stats
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Abstract
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.
Keywords
Bose-Einstein condensationGross-Pitaevskii equation
Vortex lattice
Tkachenko modes
Structural phase transition