An Investigation of the Collective Modes and Phases of Bose-Einstein Condensates

Das, Priyam (2011) An Investigation of the Collective Modes and Phases of Bose-Einstein Condensates. PhD thesis, Indian Institute of Science Education and Research Kolkata.

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Abstract

Bose-Einstein condensate (BEC) is a new state of matter, experimentally realized in a dilute alkali gas of weakly interacting bosons. The gas is cooled to very low temperatures, of the order of nano-Kelvin, at which point a large fraction of the atoms occupy the lowest quantum state. Under such conditions, the quantum effects become apparent on a macroscopic scale. Many fundamental ideas of quantum mechanics have been realized and proven to be true in experiments, by studying the Bose-Einstein condensates. Amongst them, the observation of macroscopic matter wave interference, the creation of vortices, dark and bright solitons etc, find special mention. The dynamics of this system in presence of an optical lattice reveals a superfluid to Mott insulator quantum phase transition. This has opened a new avenue for research on BEC and its applications. From a theoretical standpoint, the dynamics of BEC can be described by means of an effective mean-field theory. This formalism is much simpler than the many-body Schr¨odinger equation and based on a classical nonlinear evolution equation, known as Gross-Pitaevskii equation. The nonlinear nature of the GPE leads to several interesting phenomena like soliton propagation, vortices, four wave mixing etc., which are closely related to nonlinear fiber optics. The present thesis deals with nonlinear excitations and different phases of BEC and is organized as follows: after a brief introduction, in chapter two, we start with BEC in a one-dimensional shallow optical lattice, experiencing both cubic and quintic nonlinearity. A superfluid density wave, of twice the periodicity of lattice potential, is identified. This indicates matter redistribution, where atoms from neighbouring sites have been depleted leading to a density wave behaviour. Interestingly, it ceases to exist when only one of these interactions is operative. We predict the loss of superfluidity through a classical dynamical phase transition, where modulational instability leads to the loss of phase coherence. In presence of pure two-body interaction, the superfluid density shows a periodicity commensurate with the lattice potential. Apart from the superfluid state, a density wave insulating phase is also found to exist, possessing two frequency modulations commensurate with the lattice potential. A dynamical phase transition connects the superfluid phase with an insulating phase, which in turn, is connected with the density wave insulating phase. The solutions are found to be marginally stable, as per the Vakhitov-Kolokolov (VK) criterion. Chapter three describes the dynamics of the atomic-molecular Bose-Einstein condensate (AMBEC) by mean-field Gross-Pitaevskii equation (GPE), taking into account the atom-molecular interaction, as well as atom-molecule inter-conversion. The two-body atom-atom, atom-molecule and molecule-molecule interactions are analogous to the cubic nonlinearity of the standard GPE, whereas, the quadratic nonlinear term arising from the inter-conversion between atoms and molecules is identical to the scenario in strongly coupled BEC in one dimension. The presence of both weak and strong interactions adds further richness to this system. We find the molecular BEC possesses a complex dark (grey) soliton, while the atomic condensate is a bright one. The energy and momentum of this system is computed in order to see the effect of the both weak and strong interactions on Lieb mode profile and compare it with the pure atomic condensate profile. Another class of solutions exist in this system, where both atomic and molecular condensates possess bright solitons. In chapter four, we investigate the effect of a harmonic trap on the dynamics of grey solitons in Bose-Einstein condensates. It is observed that, these nonlinear excitations of the cigar-shaped Bose-Einstein condensate have strong coupling with the trap at low momenta and hence can be effectively isolated from the Bogoliubov sound modes, which respond weakly to harmonic confinement. This strong coupling with the trap also makes the grey soliton amenable for control and manipulation through trap modulation and temporal variation of the two-body interaction. We have shown that in presence of an expulsive oscillator trap, the grey soliton gets amplified, while having a compression in its width. Exact solutions are also obtained in case of the two component Bose-Einstein condensates (TBECs), where one component is a grey soliton and the other a bright one. The presence of the trap and inter-species interaction significantly affects the grey soliton dynamics. Similar to the single component, in the present case, we have shown that both grey and bright soliton can be compressed simultaneously. A detailed study of the energy and momentum of the soliton is reported, in order to find the effect of the trap on the Lieb mode. In chapter five, we study a BEC-defect complex in a quasi-one dimensional scenario, where the defect atom moves in the condensate with constant velocity. Explicit solutions of the coupled mean-field equations, describing defect-grey soliton dynamics are obtained, demonstrating the coexistence of grey soliton and a localized defect. Unlike the case of dark soliton, where the defect trapping center has vanishing superfluid density, the moving grey soliton necessarily possesses a finite superfluid component at the defect location. The presence of the defect can lower the grey soliton’s maximum velocity, as compared to the defect free case, where the maximum velocity is the sound velocity. We have also carried out the stability analysis of the obtained solutions using renormalized momentum formalism and Vakhitov-Kolokolov criteria and find that the obtained solutions are stable through both the methods. The grey soliton’s energy gets substantially modified through its interaction with the defect, opening up the possibility of its control through defect dynamics. Chapter six describes a parametrically forced Bose-Einstein condensate in the presence of a general moving optical lattice. The interaction between the atoms in the condensate and the time dependent lattice potential leads to a novel propagating superfluid matter wave, which can be controlled through chirp management. This system, when placed in a trap, accelerates and undergoes rapid nonlinear compression, controlled by the chirp. The density achieves its maximum, when the matter wave changes direction. It is also shown that this system exhibits dynamical superfluid insulator transition, where superfluidity breaks down and the condensate transits to an insulating phase. The stability has been investigated using Vakhitov-Kolokolov criterion and found that it depends on the chirp, as well as on the nature of the atom-atom interaction. The exact expression for energy is obtained and analyzed in detail to gain physical understanding of the chirp management of the sinusoidal excitations and also the dynamical phase transition.

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