We study the splitting of multiply quantized vortices with large quantum numbers in dilute nonrotated Bose-Einstein condensates in the zero-temperature limit. The splitting is observed to result in vortex-free condensate fragments which are separated by vortex sheets. The number of these fragments is found to be equal to the angular-momentum quantum number of the Bogoliubov excitation mode responsible for the splitting, although the formulation of the fragments cannot be described by small-amplitude excitations. Thus, the realization of an isolated giant vortex and the observation of its splitting would provide a means to directly relate the experimental data to discrete theoretical quantities.
We study the splitting of multiply quantized vortices with large quantum numbers in dilute nonrotated Bose-Einstein condensates in the zero-temperature limit. The splitting is observed to result in vortex-free condensate fragments which are separated by vortex sheets. The number of these fragments is found to be equal to the angular-momentum quantum number of the Bogoliubov excitation mode responsible for the splitting, although the formulation of the fragments cannot be described by small-amplitude excitations. Thus, the realization of an isolated giant vortex and the observation of its splitting would provide a means to directly relate the experimental data to discrete theoretical quantities.
We study the splitting dynamics of giant vortices in dilute Bose-Einstein condensates by numerically integrating the three-dimensional Gross-Pitaevskii equation in time. By taking advantage of tetrahedral tiling in the spatial discretization, we decrease the error and increase the reliability of the numerical method. An extensive survey of vortex splitting patterns is presented for different aspect ratios of the harmonic trapping potential. The discrete rotational symmetries of the splitting patterns that emerge in the time evolution are in good agreement with predictions obtained by solving the prevailing dynamical instabilities from the Bogoliubov equations. Furthermore, we observe intertwining of the split vortices in prolate condensates and a split-and-revival phenomenon in a spherical condensate.
Bose-Einstein condensation, first predicted by Bose and Einstein in 1924, was finally realized in dilute, weakly interacting gases of alkali atoms in 1995. A few years later, quantized vortices - intimately related to the phenomenon of superfluidity - were created and observed in these novel quantum many-body systems.\n\nIn this Thesis, vortices in dilute atomic Bose-Einstein condensates are studied theoretically and computationally using microscopic mean-field theory formalisms. Special emphasis is on the structure and the stability of vortices at finite temperatures. The formation of quantized circulation in Bose-Einstein condensates is investigated, as well as multiply quantized vorticity and the adiabaticity of the precessing motion of a vortex.\n\nThis research strongly relies on numerically solving the Gross-Pitaevskii equation for the macroscopic condensate wavefunction and the coupled Bogoliubov equations and their variants for the collective quasiparticle excitation eigenmodes. The computed elementary excitation spectra are used, for instance, in determining the local stability of various condensate states. The vortex formation process is studied at finite temperatures in terms of the surface modes of the condensate providing the Landau critical velocities for the breakdown of superfluidity. Free energy calculations are used for determining the thermodynamic equilibrium states.\n\nSingly quantized, axisymmetric vortex states are found to be locally energetically stable even in the zero-temperature limit within the self-consistent finite-temperature approximations. This would imply for the vortex a precession direction opposite to that observed in the experiment. Adiabatic approximation of quantum mechanics is formulated for Bose-Einstein condensed systems and is applied to a case of a moving vortex line. The experimentally observed precessing motion of the vortex is found to be nonadiabatic. A method to create stable multiply quantized vortices using an additional pinning potential is presented. The computed critical trap rotation frequencies for vortex nucleation at finite temperatures are in agreement with experiments and the predictions of zero-temperature field theories.
The splitting process of quadruply charged vortices into less charged vortices is studied at zero temperature. We solve the Bogoliubov-de Gennes (BdG) equation numerically and classify the excitations by angular momentum l relative to the condensates. Unstable modes having complex eigenvalues appear when l = 2-6. It is found that two excitations are coupled with each other when their total energy and angular momentum are zero and yield a pair of complex eigenvalues. The decay process is also numerically simulated by solving the time-dependent Gross-Pitaevskii equation. The behavior of a condensate shows good agreement with the results of BdG. The quadruply charged vortex splits into four singly quantized vortices, which are arranged as a line, triangle, and square according to the angular momentum of the excited eigenmode. We also observe the effect of nonlinearity.
Multiquantum vortices in dilute atomic Bose-Einstein condensates confined in long cigar-shaped traps are known to be both energetically and dynamically unstable. They tend to split into single-quantum vortices even in the ultralow temperature limit with vanishingly weak dissipation, which has also been confirmed in the recent experiments [Y. Shin et al., Phys. Rev. Lett. 93, 160406 (2004)] utilizing the so-called topological phase engineering method to create multiquantum vortices. We study the stability properties of multiquantum vortices in different trap geometries by solving the Bogoliubov excitation spectra for such states. We find that there are regions in the trap asymmetry and condensate interaction strength plane in which the splitting instability of multiquantum vortices is suppressed, and hence they are dynamically stable. For example, the doubly quantized vortex can be made dynamically stable even in spherical traps within a wide range of interaction strength values. We expect that this suppression of vortex-splitting instability can be experimentally verified.
The condensation of weakly interacting alkali atom gases observed in 1995 is a manifestation of the macroscopic quantum phenomenon called Bose-Einstein condensation. In contrast to superconductivity and Helium superfluids, the dilute alkali atom condensates are directly observable using optical imaging techniques and they constitute highly controllable quantum systems. Hence, they offer an ideal test bench to investigate the applicability of quantum field theories and to predict the properties of the condensates starting from the first principles.\n\nSince the first experimental realizations of dilute Bose-Einstein condensates, a great interest has been focused on this field, resulting in numerous experimental studies concerning, for example, nonlinear dynamics and the elementary excitations of these many-particle quantum systems. The existence of topological phase singularities, quantized vortices, first predicted theoretically and finally observed experimentally in 1999, led to another series of pioneering theoretical and experimental studies.\n\nSeveral methods to create single-quantum vortices were realized already in 2001 but, however, the puzzle of creating multiply quantized vortices still remained. In this thesis, the so-called topological method to create two and four times quantized vortices is studied in detail. Since the instrumentation required for the experimental realization of the topological method had already been installed in many laboratories, the theoretical predictions of this thesis were experimentally verified shortly after their publication. Multiply quantized vortices are energetically unstable and tend to split into singly quantized vortices. However, the dissipation of energy in the condensates is minimal at low temperatures and, consequently, can lead to long lifetimes for the multiquantum vortices. Nevertheless, one prediction of this thesis is that the lifetimes of doubly quantized vortices are relatively short, owing to the known dynamical instability of the state. The splitting of the doubly quantized vortex has recently been observed experimentally.\n\nThe stability of vortices in dilute Bose-Einstein condensates is intimately related to the excitation spectrum of the stationary vortex state. However, developing an accurate, computationally feasible finite-temperature quantum field theory remains an open problem. In this thesis, we study the elementary excitations of an irrotational condensate within a recently developed systematic second order perturbation theory. The collapse and revival of certain elementary excitations is discovered.\n\nThe most recent studies of this thesis are devoted to stationary, but rotationally asymmetric vortex states. In weakly interacting condensates, a novel stationary state, a vortex tripole holding finite angular momentum is presented. Also the lowest elementary excitations of the so-called vortex dipole, tripole, and quadrupole states are studied.
We consider the splitting mechanism of a multiply charged vortex into singly charged vortices in a Bose-Einstein condensate confined in a harmonic potential at zero temperature. The Bogoliubov equations support unstable modes with complex eigenfrequencies (CE modes), which cause the splitting instability without the influence of thermal atoms. The investigation of the excitation spectra shows that the negative-energy (NE) mode plays an important role in the appearance of the CE modes. The configuration of vortices in splitting is determined by the angular momentum of the associated NE mode. This structure has also been confirmed by the numerical simulation of the time-dependent Gross-Pitaevskii equation.