Collaborative Research: From Quantum Droplets & Spinor Solitons to Vortex Knots & Topological States: Beyond the Standard Mean-Field in Atomic BECs

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The realm of Bose-Einstein condensates (BECs) was originally proposed as a curious feature of the statistical properties of atomic particles with integer spin by Bose and Einstein in the 1920's. This consisted of the condensation of the excited states particles into the ground state of the system and the formation of a macroscopic, coherent “super-wave” therein, allowing the study and observation of quantum mechanical properties beyond microscopic scales. However, the temperatures needed for its experimental realization were so low that it took about 70 years for E.A. Cornell, W. Ketterle, and C.E. Wieman to realize BECs in the lab. The importance of this feat was recognized only a few years later via the 2001 Nobel Prize in Physics. This has, in turn, enabled a pristine platform where numerous exciting features of nonlinear dynamics of waves and coherent structures can be studied and experimentally observed. Importantly, these coherent structures are also of wide applicability in numerous other areas of physics including, most notably, nonlinear optics, plasma physics, and water waves. Within atomic physics, BECs have also been fundamental toward the study of remarkable quantum features such as superconductivity and superfluidity and, in that capacity, they have been front and center toward the experimental discoveries connected to the vortices and their lattices cited in the 2003 Nobel Prize in Physics and the topological phases and their transitions associated with the 2016 Nobel Prize in Physics. The aim of this project is to advance the state-of-the-art at this exciting nexus of atomic physics theory, physical BEC experiments, applied mathematical analysis, and the forefront of scientific computing, while at the same time training a new generation of scientists and mathematicians at this scientific interface and transcending disciplinary boundaries. In line with the past trajectory of the PIs, an emphasis on the diversity, equity and inclusion of under-represented groups will be sought within this research effort.More concretely, the principal thrust of the present project consists of the study of non-trivial extensions of standard BEC settings. In particular, the main axes of the proposal consider the following themes. (1) Two-component mutually attractive BECs that allow, through quantum corrections and the famous Lee-Huang-Yang (LHY) contribution, for the highly timely formation of so-called quantum droplets. The key realization for such droplets is that their emergence stems from the interplay between repulsive mean-field and attractive beyond-mean-field contributions. (2) Three (F =1) and five (F=2) spin component settings supporting symbiotic (dark-antidark and dark-bright) solitary wave structures with unprecedented integrable or weakly non-integrable properties. (3) 3D vortex knot structures in one and multi-component/spinor settings. Vortex knots constitute one of the most elusive types of vortical structures for which limited experimental and theoretical analysis exists. The PIs will also explore in the spinor settings complex non-trivial topological patterns such as Alice rings and Dirac monopoles. (4) Topologically nontrivial toroidal trapping settings, where the interplay of the intrinsic metric and curvature of the system with the effective nonlinearity can yield unprecedented coherent structures and dynamics thereof. More broadly within this theme, the PIs will study nonlinear waves such as solitons and vortices confined on different types of curved surfaces. This ambitious program should push the boundaries of the state-of-the-art mean-field-theoretic understanding, offering numerous beyond-mean-field insights and elucidating their range of validity as well as the interplay of nonlinearity with quantum, as well as thermodynamic effects.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.