1. Introduce soliton in quasi-one-dimensional time-dependent Gross-Pitaevskii equation. 2. Introduce the basic idea of topology and the homotopy groups. 3. Introduce vortex in spinless condensate as a typical example of topological defect. 4. Introduce the Berezinskii-Kosterlitz-Thouless transition as a topological defect driven phase transition, and emphasize the topological and energy requirements for such a transition. 5. Discuss the geometric configuration that minimizes the energy of a vortex lattice. 6. Introduce the Majorana stellar representation as a useful tool to visualise the symmetry of a high-spin wave function. 7. Introduce two different phases of spin-1 condensate. 8. Introduce the relation between a mean-field state and the singlet pair condensate. 9. Introduce spin vortex and half vortex in spinor condensate, and discuss various possibilities of spin vortex core. 10. Introduce two categories of topological excitations in a Bose condensate, and introduce monopole and skyrmion as the typical example of each category. 11. Introduce simulating Dirac monopole in synthetic magnetic field. 12. Discuss the symmetry of the Hamiltonian and the symmetry of various phases in a spin-orbit coupled BEC. 13. Discuss the relation between symmetry and phase transitions, using spin-orbit coupled condensate as an example. 14. Discuss the Galilean invariance and the superfluid critical velocity.
Introduce soliton in quasi-one-dimensional time-dependent Gross-Pitaevskii equation. 2. Introduce the basic idea of topology and the homotopy groups. 3. Introduce vortex in spinless condensate as a typical example of topological defect. 4. Introduce the Berezinskii-Kosterlitz-Thouless transition as a topological defect driven phase transition, and emphasize the topological and energy requirements for such a transition. 5. Discuss the geometric configuration that minimizes the energy of a vortex lattice. 6. Introduce the Majorana stellar representation as a useful tool to visualise the symmetry of a high-spin wave function. 7. Introduce two different phases of spin-1 condensate. 8. Introduce the relation between a mean-field state and the singlet pair condensate. 9. Introduce spin vortex and half vortex in spinor condensate, and discuss various possibilities of spin vortex core. 10. Introduce two categories of topological excitations in a Bose condensate, and introduce monopole and skyrmion as the typical example of each category. 11. Introduce simulating Dirac monopole in synthetic magnetic field. 12. Discuss the symmetry of the Hamiltonian and the symmetry of various phases in a spin-orbit coupled BEC. 13. Discuss the relation between symmetry and phase transitions, using spin-orbit coupled condensate as an example. 14. Discuss the Galilean invariance and the superfluid critical velocity.
Bose–Einstein condensation is a quantum statistical phase transition that occurs in a system consisting of bosons when a single-particle quantum state becomes macroscopically occupied. This peculiar state of matter was first predicted in 1925 and finally realized seventy years later in vapours of weakly-interacting alkali-metal atoms. Since then, Bose–Einstein condensates have been one of the most fascinating research fields in modern physics. \n \nThe gaseous condensates offer a robust platform to accurately study interacting many-particle systems from the first principles. Experimentally, the possibility to precisely control a condensate with external fields and directly image its order parameter provides unforeseen opportunities to obtain deep insight into phenomena across different subfields of physics. In particular, gaseous condensates can emulate complicated models that arise in atomic, condensed-matter, and even particle physics, allowing realizations of exotic phenomena that are elusive in their original contexts. An outstanding example is the existence of various topological defects in ultacold quantum gases with internal degrees of freedom. \n \nIn this thesis, we investigate the creation, stability, and dynamical properties of various topological defects in spinor Bose–Einstein condensates. The majority of the theoretical results is obtained by numerically solving the dynamics using Gross–Pitaevskii equations for spin-1 condensates. Many theoretical predictions are confirmed by the very good agreement with experiments. \n \nThe first experimental observations of a topological point defect in an order parameter describing the quantum gas are presented. Such a point defect is reminiscent to the magnetic monopole particle appearing in grand unified theories. Therefore, the discovery of monopoles in quantum gases further encourages the quest for finding magnetic monopoles in natural electromagnetic fields, a search largely initiated by Paul Dirac almost a century ago. \n \nSubsequently, the fine structure and decay dynamics of the point defect are studied numerically and verified in experiments. The created monopole is gradually destroyed during the polar-to-ferromagnetic quantum phase transition, which results in the spontaneous emergence of a Dirac monopole in synthetic magnetic field. In addition to the singular point defect, the first experimental realization of a knot soliton in the context of quantum field is reported. This thesis lays the foundation for studies of the dynamics and stability of three-dimensional topological structures in quantum systems.
Topology provides deep conceptual links between the various branches of physics. Bose–Einstein condensates with spin degrees of freedom are among the most accessible quantum systems available for studying topological structures. A wide range of topological defects and textures available in condensates, such as vortices, monopoles, knots, and skyrmions, are analogous to those predicted in electromagnetism, high-energy physics, and cosmology.
In this dissertation, we numerically and experimentally investigate novel creation methods for topological structures and study their dynamical properties. Specifically, we experimentally observe the evolution of an isolated monopole into a Dirac monopole in the presence of a quadrupole magnetic field. The Dirac monopole appears in the synthetic magnetic field of the condensate and is accompanied by spontaneously emerging nodal lines. We observe the decay of a quantum knot into a polar-core spin vortex in the presence of a uniform magnetic field. Furthermore, we observe that a decaying coreless-vortex state gives rise to a pair of singular SO(3) vortices.
Many of the studied creation methods for topological structures in the condensate rely on adiabatic control of the external magnetic field. The counterdiabatic protocol offers a way to speed up the otherwise slow magnetic field driving required for adiabatic dynamics with a correcting magnetic field. Using this method, we numerically implement a scheme for fast vortex pumping which leads to the vortex with highest angular momentum reported to date in Bose–Einstein condensates with experimentally feasible methods. We further use the counterdiabatic protocol in a novel way to create quantum knots in the condensate.
Another focal point of this dissertation is the study on different types of skyrmions in spinor condensates. Our simulations of two-dimensional skyrmions are in a quantitative agreement with an experiment carried out elsewhere, explaining the experimentally observed instabilities. We numerically analyze the exotic spin-2 skyrmions available in the cyclic and biaxial nematic phases. Importantly, we present the first experimental observations of Shankar skyrmions in spin-1 condensates and analyze their synthetic electromagnetic properties.
This dissertation addresses an extensive amount of topological structures available in the condensate, but many different structures await future studies. In addition, the precise computational characterization of the elementary excitations of monopoles, knots, and skyrmions is of great interest. The results of this dissertation form a sturdy basis for future experimental studies on the dynamics of topological structures in spinor condensates.; Topologia tarjoaa syvallisia kasitteellisia yhteyksia fysiikan eri haarojen valilla. Bosen–Einsteinin kondensaatit spin-vapausasteilla ovat yksia monipuolisimmista systeemeista topologisten rakenteiden tutkimukseen. Laaja kirjo kondensaatissa esiintyvia topologisia rakenteita, kuten kvanttipyorteet, monopolit, solmut ja skyrmionit, ovat…
1. Introduce the basic concepts of single-particle dispersion in a lattice. 2. Introduce maximally localized Wannier wave functions and their importance in constructing interacting models. 3. Introduce the basic idea of quantum simulation. 4. Introduce the Dirac point in the honeycomb lattice and the concept of a semimetal. 5. Illustrate the Dirac and the Weyl points as topological defects in the momentum space. 6. Introduce the concept of the symmetry protection for the topological stability of the Dirac point. 7. Introduce the Su-Schrieffer-Heeger model and the concept of symmetry-protected topology. 8. Introduce the Haldane model and the topological phase transition in the Haldane model. 9. Summarize the common mathematical structures between topological characters of a band theory and topological excitation in a Bose condensate. 10. Discuss the physical consequences of topological invariants for a band insulator in both near-equilibrium transport and the far-from-equilibrium quench dynamics. 11. Discuss the equivalence between neural atoms in a moving lattice and charged particles in an electric field. 12. Introduce the Floquet theory and the Floquet effective Hamiltonian from thehigh-frequency expansion. 13. Discuss various applications of periodical-driven optical lattices. 14. Introduce the Hamiltonian for atom-cavity system and highlight the role of the Langevin force term. 15. Introduce the superradiant quantum phase transition of a Bose condensate coupled to cavity light.
Quantum topological defects usually are excited on density. In this study, we extend the topological defect to another dimensionality: the spin. We indicate that the spin singularity, where the spin density |S| vanishes locally, occurs commonly in the two-dimensional spin-1 Bose-Einstein condensates. We assume a coreless half-quantum vortex solution in the spin-1 Bose-Einstein condensates according to the spin singularity. This half-quantum vortex can further induce a half-quantum spin defect, which is similar to the general half-quantum Skyrmion but its centre proposes singularity on the spin density. We use the variational method to obtain a possible stable solution under the combined interaction of the spin-orbit coupling and the external magnetic field. Our calculation shows that the isotropic spin-orbit coupling provides local energy minimum to stabilize the coreless half-quantum vortex and the corresponding singular half-quantum spin texture excitation. The combined restriction of the isotropic spin-orbit coupling and the perpendicular magnetic field is a key factor to obtain this kind of excitation. Our study also provides the stability phase diagram and the most possible size of the half-quantum vortex. Furthermore, our analysis shows that adjusting of the spin interaction hardly affects the stability of the half-quantum vortex.
Two-component Bose-Einstein condensate offers an ideal platform for investigating many intriguing topological defects due to the interplay between intraspecies and interspecies interactions. The recent realization of spin-orbit coupling in two-component Bose-Einstein condensate, owing to coupling between the spin and the centre-of-mass motion of the atom, provides possibly new opportunities to search for novel quantum states. In particular, the gradient magnetic field in the Bose-Einstein condensate has brought a new way to create topologically nontrivial structures including Dirac monopoles and quantum knots. Previous studies of the gradient magnetic field effect in the Bose-Einstein condensate mainly focused on the three-component case. However, it remains unclear how the gradient magnetic field affects the ground state configuration in the rotating two-component Bose-Einstein condensate with spin-orbit coupling. In this work, by using quasi two-dimensional Gross-Pitaevskii equations, we study the ground state structure of a rotating two-component Bose-Einstein condensate with spin-orbit coupling and gradient magnetic field. We concentrate on the effects of the spin-orbit coupling and the gradient magnetic field on the ground state. The numerical results show that increasing the strength of the spin-orbit coupling can induce a phase transition from skyrmion lattice to skyrmion chain in the presence of the gradient magnetic field. Unlike the study of skyrmion in rotating two-component Bose-Einstein condensate with only spin-orbit coupling, the skyrmion chain can occur under the isotropic spin-orbit coupling when the gradient magnetic field is considered. It is worth noting that the skyrmion chain here is arrayed along the diagonal direction. Next we examine the effect of the gradient magnetic field on spin-orbit coupled two-component Bose-Einstein condensate. For the case of weak spin-orbit coupling and the slow rotation, a phase transition from a single plane-wave to half-skyrmion is found through increasing magnetic field gradient strength. For the case of strong spin-orbit coupling and the fast rotation, the nature of the ground state is shown to support the formation of a hidden vortex as the gradient magnetic field is enhanced. These hidden vortices have no visible cores in density distributions but have phase singularities in phase distributions, which are arrayed along the diagonal direction. This result confirms a new method of creating the hidden vortices in the two-component Bose-Einstein condensate. These topological structures can be detected by using the time-of-flight absorption imaging technique. Our results illustrate that the gradient magnetic field not only provides an opportunity to explore the exotic topological structures in spin-orbit coupled spinor Bose-Einstein condensate, but also is crucial for realizing the phase transitions among different ground states. This work paves the way for the future exploring of topological defect and the corresponding dynamical stability in quantum systems subjected to a gradient magnetic field.
Spinor Bose gases may exhibit both phase and ferromagnetic order. Consequently, they may support superflow of both mass and spin currents. In this thesis we utilize a stochastic Gross-Pitaevskii model to investigate the superfluid properties of a two-dimensional spin-1 Bose gas with ferromagnetic interactions. Owing to the low dimensionality, the superfluidity of this system can not be attributed to the presence of a Bose-Einstein condensate, with thermal fluctuations precluding long-range order. Rather, mass and spin superfluidity emerge via respective Berezinskii-Kosterlitz-Thouless (BKT) transitions. Such transitions are characterized by the emergence of quasi-long-range order, driven by the binding of topological defect pairs. First, we consider an easy-plane ferromagnetic gas. We find the system superfluidity may be understood by direct application of existing BKT theory. In particular, mass and spin superfluidity are respectively associated with quasi-long-range order in the global phase and local magnetization. The former arises from the binding of mass vortex pairs, while the latter arises from the binding of polar-core spin vortex pairs. Following this, we introduce axial magnetization, and consider a gas in the broken-axisymmetric phase. We find axial magnetization acts to couple mass and spin superflows via a mechanism analogous to the superfluid drag exhibited by two-component superfluids. This coupling renders invalid the direct applicability of existing BKT theory: here mass and spin superfluidity may each be associated with both phase and magnetic order. We remedy this with a generalization making concrete the relation between superfluidity and quasi-long-range order in this system. In addition, through variations of the quadratic Zeeman energy and axial magnetization we identify three distinct superfluid phases, which serve as finite-temperature generalizations of the polar, broken-axisymmetric, and easy-axis ferromagnetic ground states.