In this thesis the features that arise from the interference of simple coherent waves are described, namely the topological defects of phase known as phase vortices permeating such wavefields. Phase vortices are associated with the rotation of energy and mass and arise naturally in regions containing coherent light and/or matter. The simplest possible system exhibiting these defects is that of three linearly interfering plane waves, described using the classical theory of monochromatic complex scalar wavefields. We explore the related case of three outgoing waves produced by point sources, such as a laser-illuminated screen that has been punctured by three pinholes. We develop a description of the resulting vortex structure in the optical far-field, and relate this to the source arrangement and parameters. This allows an estimate of the number and positions of the vortices, and leads to connections to particular results from information theory and number theory. Numerical simulations of the near-field defect structure are also presented. Findings from this study are then applied to the time evolution of a nonlinear Bose-Einstein condensate of atoms, which shares the property of coherence that enables its description by a related model of interfering Gaussian wavepackets. Numerical modelling of the nonlinear system is performed for both trapped and untrapped Bose-Einstein condensates (BECs) in two and three spatial dimensions, for BEC pieces initially positioned at the three corners of an equilateral triangle. We demonstrate the production of a distorted honeycomb vortex-antivortex lattice, due to wavepacket interference. The wavepacket expansion is halted by the presence of a magnetic trap, causing the lattice to melt and providing a population of mobile vortices and antivortices. The addition of nonlinearity in this system over the linear case leads to the formation of vortex clusters, whose dynamics we investigate. A model of a Laguerre-Gauss (LG) laser beam, which carries a phase vortex along its axis, is studied following its interaction with a ground-glass plate. We present preliminary results showing that the beam acquires interference-generated vortices whose number density evolves as a function of propagation distance and beam topological charge, due to interaction with the roughened glass. Beyond the nonlinear coherent wavefields applied to the description of Bose-condensed gases, a pseudo-spinor system described as a nonlinear coupled system of two complex scalar fields is studied. Such systems admit the existence of texture defects similar to those that exist in ferromagnetic systems. We numerically model a pseudo-spinor system that we arrange to generate carefully aligned honeycomb vortex-antivortex lattices in the coupled components. We find that this corresponds to a texture defect lattice and identify its constituent texture defects as merons and a type of planar defect. Following the analogy with the single component BEC, we also study texture defects and clusters in the trapped pseudo-spinor system.
In this thesis the features that arise from the interference of simple coherent waves are described, namely the topological defects of phase known as phase vortices permeating such wavefields. Phase vortices are associated with the rotation of energy and mass and arise naturally in regions containing coherent light and/or matter. The simplest possible system exhibiting these defects is that of three linearly interfering plane waves, described using the classical theory of monochromatic complex scalar wavefields. We explore the related case of three outgoing waves produced by point sources, such as a laser-illuminated screen that has been punctured by three pinholes. We develop a description of the resulting vortex structure in the optical far-field, and relate this to the source arrangement and parameters. This allows an estimate of the number and positions of the vortices, and leads to connections to particular results from information theory and number theory. Numerical simulations of the near-field defect structure are also presented. Findings from this study are then applied to the time evolution of a nonlinear Bose-Einstein condensate of atoms, which shares the property of coherence that enables its description by a related model of interfering Gaussian wavepackets. Numerical modelling of the nonlinear system is performed for both trapped and untrapped Bose-Einstein condensates (BECs) in two and three spatial dimensions, for BEC pieces initially positioned at the three corners of an equilateral triangle. We demonstrate the production of a distorted honeycomb vortex-antivortex lattice, due to wavepacket interference. The wavepacket expansion is halted by the presence of a magnetic trap, causing the lattice to melt and providing a population of mobile vortices and antivortices. The addition of nonlinearity in this system over the linear case leads to the formation of vortex clusters, whose dynamics we investigate. A model of a Laguerre-Gauss (LG) laser beam, which carries a phase vortex along its axis, is studied following its interaction with a ground-glass plate. We present preliminary results showing that the beam acquires interference-generated vortices whose number density evolves as a function of propagation distance and beam topological charge, due to interaction with the roughened glass. Beyond the nonlinear coherent wavefields applied to the description of Bose-condensed gases, a pseudo-spinor system described as a nonlinear coupled system of two complex scalar fields is studied. Such systems admit the existence of texture defects similar to those that exist in ferromagnetic systems. We numerically model a pseudo-spinor system that we arrange to generate carefully aligned honeycomb vortex-antivortex lattices in the coupled components. We find that this corresponds to a texture defect lattice and identify its constituent texture defects as merons and a type of planar defect. Following the analogy with the single component BEC, we also study texture defects and clusters in the trapped pseudo-spinor system.
Numerical simulations of the interference of a three-way segmented nonrotating Bose-Einstein condensate reveal the production of a honeycomb vortex lattice containing significant numbers of vortices and antivortices. If confined within a trap, the lattice subsequently melts, exhibiting a rich assortment of vortex-antivortex interactions. In contrast with nonlinear vortex production mechanisms previously described for Bose-Einstein condensates, the process here is shown to be primarily one of linear superposition, with initial vortex locations approximately described by a linear theory of wave packet interference.
We provide a detailed description of our previously proposed scheme for topological interface engineering with constructed defects and textures perforating across coherent interfaces between different broken symmetries [Borgh and Ruostekoski, Phys. Rev. Lett. 109, 015302 (2012)]. We consider a spin-1 Bose-Einstein condensate, in which polar and ferromagnetic phases are prepared in spatially separated regions. We show that a stable coherent interface is established between the two phases, allowing defects of different topology to connect continuously across the boundary. We provide analytic constructions of interface-crossing defect solutions that could be experimentally phase imprinted using existing technology. By numerically minimizing the energy, we calculate the core structures of interface-crossing defect configurations. We demonstrate nontrivial core deformations to considerably more complex structures, such as the formation of an arch-shaped half-quantum line defect, an Alice arch, at the interface, with the topological charge of a point defect, whose emergence may be understood by the ``hairy ball'' theorem. Another example of an energetically stable object is the connection of a coreless vortex to a pair of half-quantum vortices. We show that rotation leads to spontaneous nucleation of defects in which a coreless vortex continuously transforms to a half-quantum vortex across the interface.
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.
We investigate the dynamics of formation of topological defects in ultra-cold quantum gases. Using classical field methods, such as the truncated Wigner approximation and the stochastic Gross-Pitaevskii equation, we explore the connection between the rate of a phase transition and the number of topological defects formed as a consequence of it. We show that in various systems such defects form in number comparable with the predictions of the Kibble-Zurek theory, known for thirty years but as yet unverified experimentally. We also consider the spontaneous production of entangled vortices following a quantum phase transition in a multi-component spin-1 Bose-Einstein condensate (BEC). To begin, we provide a literature review and describe some of the background theory of multi-component condensates, stochastic methods and Kibble-Zurek theory. Part of this thesis focus on a two component Bose-Einstein condensates of the same atomic species. In particular on the quantum critical point separating the miscible from the immiscible phase of such systems. We show that such phase transition cannot only be achieved with the tuning of the interaction constants between the components, but also through changes in the intensity of the coupling between atoms in different hyperfine states. We first determine the value of the critical coupling, and also the critical exponents defining the universality class of the coupling—induced phase transition. We then proceed to simulate the dynamics of the transition in effective 1D ring and harmonically trapped condensates, and test the results against predictions of the Kibble-Zurek (KZ) mechanism. We find good agreement between the KZ theory and the numerical results in the ring geometry, and we motivate the origin of the discrepancies between the two in harmonically trapped BECs. We use the ”equivalence” of inhomogeneous phase transitions in anisotropic systems and spatially inhomogeneous quenches to engineer coupling quenches and systematically study key aspects of the inhomogeneous KZ mechanism. The coupling induced miscible—immiscible transition results in the formation of a stable pattern of domains that easily be detected and counted in current ultra-cold gases experiment, and the scheme is a viable candidate for the first experimental testing of the KZ predictions. We then examine the spontaneous formation of a superflow as result of the thermal Bose-Einstein phase transition in ring traps. According to the Kibble-Zurek theory, the fast production of a condensate in a ring trap should result in the formation of solitons, with the number depending on the rate of the temperature quench. Solitons are short-living topological defects and their observation in ultra-cold gases experiments remains challenging. We show here that even after they decay, solitons leave a signature in the variance of the final winding number of the condensate. Solitons fragment a 1D condensate into regions of random phase, hence the larger the number of solitons, the higher the probability that the resulting phase wraps around the ring by a multiple of 2π. We numerically simulate the BEC transition in this system with the stochastic Gross-Pitaevskii equation, and observe that the winding number scales with the quenching time as predicted by the KZ mechanism. We examine the effect of inhomogeneity on the expectation value of the winding number, and find that the tilting of the ring greatly suppress the spontaneous formation of vortices. This has implications for the necessary horizontal alignment of any experiment on this system. Finally, we consider the polar to anti-ferromagnetic phase transition in magnetically quenched spin-1 BEC. In this quantum phase transition a pure condensate of atoms in the mF = 0 Zeeman sublevel is converted into a three component BEC with atoms in mF = 0, ±1 through spin changing collisions. An analogous mechanism, four—wave mixing, in quantum optics produces entangled photons, and it has been proposed in ultra-cold gases as a method to obtain Einstein-Podolski-Rosen entanglement of macroscopic atomic ensembles. We begin with a rotating 1D spinor condensate of atoms in the mF = 0 component, and perform a quench of the magnetic field through the critical point. We show how the transition produces a pair of entangled components with different winding numbers, and quantify the entanglement as a function of the modes considered and of the type of magnetic quench, sudden or instantaneous. Lastly, we propose a method to use the original spin zero component as coherent source to perform homodyne detection and consider the two rotating entangled components as a resource for the sensing of rotations.
In 1995, atomic physics took a quantum leap with the experimental discovery of almost ideal Bose-Einstein condensation in dilute gases of alkali atoms. The gaseous condensates offer a rare chance to study an interacting many-particle system accurately from first-principles theories, and they can be used to simulate many seminal models that have proven elusive in their original context of, e.g., solid-state or high-energy physics. An important topic in the research has been the quantized vortex, which was experimentally realized in 1999 and whose existence is intimately related to the concepts of quantum phase coherence and superfluidity.\nIn this dissertation, unconventional vortex structures are investigated in the dilute condensates at ultralow temperatures. A combination of analytical and numerical methods is used to examine the structure, stability, and dynamical behavior of multiquantum vortices and vortex-antivortex pairs in spin-polarized condensates, unusual vortex lattices in two-species condensates, and spin textures in condensates with dipolar interactions. In studying the properties of the multiquantum vortices, particular emphasis is placed on exploring the practical limits of producing them through adiabatic pumping of vorticity. The majority of the research is conducted by solving the mean-field Gross-Pitaevskii and Bogoliubov equations.\nVarious original, experimentally verifiable results concerning the exotic vortex structures are presented. Novel splitting patterns of vortices with large quantum numbers are introduced, and it is shown that such vortices can be feasibly stabilized by piercing them with a focused laser beam. A recent experiment on vortex-antivortex pairs is simulated, and an excellent quantitative agreement with the experimental data is obtained. Unconventional ground-state vortex lattices, such as ones having a square geometry or consisting of two-quantum vortices, are shown to exist in rotating two-species condensates. Dipolar interactions are found to support helical spin-vortex states, and the energies of spin-wave excitations are observed to increase rapidly with the dipolar coupling strength.\nThis dissertation contributes to the understanding of superfluid phenomena in Bose-Einstein condensates and has significant implications for the prospects of detecting novel vortex structures in current experiments.
This thesis is concerned with the structure and stability of vortices in spin-1 atomic Bose-Einstein Condensates (BECs) in rotating, optical traps. We numerically study these vortex structures using a classical mean-field theory which allows atomic interactions to change the local expectation value of the atomic spin. Initially applying a model in which the atoms interact only via scattering which does not conserve an initial longitudinal magnetisation, we identify the energetically stable configurations of singular and nonsingular vortices via propagation in imaginary time in a rotating frame of reference. We find that the cores of singular vortices fill with atoms in the spinor BEC and show that this can be understood in terms of an energetic hierarchy of length scales. By refining the numerical model to explicitly conserve longitudinal magnetisation, we show that the conservation of a strong magnetisation can lead to a mixing of the two phases of the ground-state manifold (polar and ferromagnetic), which are characterised by the expectation value of the spin. This occurs as a result of the introduction of a new characteristic length scale determined by the longitudinal magnetisation. A surprising consequence is the stability of a ferromagnetic coreless vortex in the polar interaction regime, which otherwise is energetically unstable. We construct analytic spinor wavefunctions which parametrise the interpolation between the polar and ferromagnetic phases, exhibiting different vortex topologies in the respective phases. Finally by identifying stationary states of the system, we show how nonlocal dipoledipole interactions between atoms introduces an additional length scale determined by the strength of the dipolar interaction. The energetic hierarchy of length scales then determines whether the dipolar interactions have a significant effect on the stationary vortex structures. In particular we show how a BEC with polar interactions adopts the properties of a ferromagnetic condensate when the dipolar interactions dominate.