Abstract Atom optics demonstrates optical phenomena with coherent matter waves, providing a foundational connection between light and matter. Significant advances in optics have followed the realization of structured light fields hosting complex singularities and topologically non-trivial characteristics. However, analogous studies are still in their infancy in the field of atom optics. Here, we investigate and experimentally create knotted quantum wavefunctions in spinor Bose–Einstein condensates which display non-trivial topologies. In our work we construct coordinated orbital and spin rotations of the atomic wavefunction, engineering a variety of discrete symmetries in the combined spin and orbital degrees of freedom. The structured wavefunctions that we create map to the surface of a torus to form torus knots, Möbius strips, and a twice-linked Solomon’s knot. In this paper we demonstrate close connections between the symmetries and underlying topologies of multicomponent atomic systems and of vector optical fields—a realization of topological atom-optics.
Abstract Atom optics demonstrates optical phenomena with coherent matter waves, providing a foundational connection between light and matter. Significant advances in optics have followed the realization of structured light fields hosting complex singularities and topologically non-trivial characteristics. However, analogous studies are still in their infancy in the field of atom optics. Here, we investigate and experimentally create knotted quantum wavefunctions in spinor Bose–Einstein condensates which display non-trivial topologies. In our work we construct coordinated orbital and spin rotations of the atomic wavefunction, engineering a variety of discrete symmetries in the combined spin and orbital degrees of freedom. The structured wavefunctions that we create map to the surface of a torus to form torus knots, Möbius strips, and a twice-linked Solomon’s knot. In this paper we demonstrate close connections between the symmetries and underlying topologies of multicomponent atomic systems and of vector optical fields—a realization of topological atom-optics.
Spinor Bose–Einstein condensates (BECs) and singular optical systems have both recently served as sandboxes to create and study analogs of phenomena from other fields of physics that are otherwise difficult to create and control experimentally. Interfacing singular optics and spinor BECs allows us to take advantage of and build on the foundations of singular optics to create and describe complex spin textures in BECs that serve as analogs of other systems. Here, the complete BEC wavefunctions are precisely engineered via a two-photon Raman interaction to contain π-symmetric (lemon, star) or 2π-symmetric (saddle, spiral) C-point singularities. The optical Raman beams are singular optical beams that contain these singularities and transfer them to the condensate, thereby creating vector-vortex spin textures—the spinor counterparts to scalar vortices—in pseudo-spin-1/2 BECs. With a version of atom-optic polarimetry, we can measure the Stokes parameters of the atomic cloud and characterize the singularities by the patterns present in their ellipse fields or by the C-point index. In the low density limit, these spin textures are analogs of optical vector vortices and should have dynamics driven by a matter-wave Gouy phase. With precise tuning of Raman beam parameters, we can create full Bloch BECs that contain every possible superposition between two states in the atomic cloud. Full Bloch BECs are similar to topologically stable magnetic skyrmions such as those created in thin metal films and nanowires, which may prove useful for atom-spintronics and topological quantum processes.
The formal correspondences between pseudo-spin-1/2 systems in optics and in atomic physics have provided fertile ground for exploring polarization in atom optics. Previous experimental results demonstrated atomic polarimetry techniques for two-level wave functions and explored wave-field singularities in the multicomponent wave function of a spinor Bose-Einstein condensate using visualization techniques developed in optics [J. T. Schultz, A. Hansen, and N. P. Bigelow, Opt. Lett. 39, 4271 (2014); A. Hansen, J. T. Schultz, and N. P. Bigelow, Optica 3, 355 (2016)]. Here we further this discussion, reexamining the atomic Stokes parameters and Stokes polarimetry in the context of spin-1 systems where the tensor moments of the higher-spin system enrich the physics considerably. We show that our atomic polarimetry methods provide tools to engineer the multipole tensor moments of the higher-spin atomic wave function, and to realize nontrivial couplings between these multipole moments and an atomic center-of-mass orbital angular momentum. The different forms of coupling between internal tensor moments and external angular momenta can be used to realize a variety of topological structures in the atomic wave field. We identify these $\ensuremath{\pi}$- and $2\ensuremath{\pi}$-symmetric features in the streamlines of the atomic Stokes fields constructed in analogy with singular optics.
Knots are familiar entities that appear at a captivating nexus of art, technology, mathematics, and science. As topologically stable objects within field theories, they have been speculatively proposed as explanations for diverse persistent phenomena, from atoms and molecules to ball lightning and cosmic textures in the universe. Recent experiments have observed knots in a variety of classical contexts, including nematic liquid crystals, DNA, optical beams, and water. However, no experimental observations of knots have yet been reported in quantum matter. We demonstrate here the controlled creation and detection of knot solitons in the order parameter of a spinor Bose-Einstein condensate. The experimentally obtained images of the superfluid directly reveal the circular shape of the soliton core and its accompanying linked rings. Importantly, the observed texture corresponds to a topologically non-trivial element of the third homotopy group and demonstrates the celebrated Hopf fibration, which unites many seemingly unrelated physical contexts. Our observations of the knot soliton establish an experimental foundation for future studies of their stability and dynamics within quantum systems.
A rich variety of order parameter manifolds of multicomponent Bose-Einstein condensates (BECs) admit various kinds of topological excitations, such as fractional vortices, monopoles, skyrmions, and knots. In this paper, we discuss two topological excitations in spinor BECs: non-Abelian vortices and knots. Unlike conventional vortices, non-Abelian vortices neither reconnect themselves nor pass through each other, but create a rung between them in a topologically stable manner. We discuss the collision dynamics of non-Abelian vortices in the cyclic phase of a spin-2 BEC. In the latter part, we show that a knot, which is a unique topological object characterized by a linking number or a Hopf invariant [π3 (S2) = Z], can be created using a conventional quadrupole magnetic field in a cold atomic system.
The physics of optical vortices, also known as twisted light, is now a well-established and a growing branch of optical physics with a number of important applications and significant interdisciplinary connections.Optical vortex fields of widely varying forms and degrees of complexity can be realised in the laboratory by a host of different means.The interference between such beams with designated orbital angular momenta and optical spins (the latter is associated with wave polarisations) can be structured to conform to various geometrical arrangements.The focus of this review is on how such tailored forms of light can exert a controllable influence on atoms with which they interact.The main physical effects involve atoms in motion due to application of optical forces.The now mature area of atom optics has had notable successes both of fundamental nature and in applications such as atom lasers, atom guides and Bose-Einstein condensates.The concepts in atom optics encompass not only atomic beams interacting with light, but atomic motion in general as influenced by optical and other fields.Our primary concern in this review is on atoms in structured light where, in particular, the twisted nature of the light is made highly complex with additional features due to wave polarisation.These features bring to the fore a variety of physical phenomena not realisable in the context of atomic motion in more conventional forms of laser light.Atoms near resonance with such structured light fields become subject to electromagnetic fields with complex polarisation and phase distributions, as well as intricately structured intensity gradients and radiative forces.From the combined effect of optical spin and orbital angular momenta, atoms may also experience forces and torques involving an interplay between the internal and centre of mass degrees of freedom.Such interactions lead to new forms of processes including scattering, trapping and rotation and, as a result, they exhibit characteristic new features at the micro-scale and below.A number of distinctive properties involving angular momentum exchange between the light and the atoms are highlighted, and prospective applications are discussed.Comparison is made between the theoretical predictions in this area and the corresponding experiments that have been reported to date.
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…