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Two species of mutually interacting ultracold bosonic atoms are studied in a ring-shaped trap with a species-selective azimuthal lattice which may rotate. We examine the spectrum and the states in a collective spin formalism. The system can be modeled as a pair of coupled Lipkin-Meshkov-Glick Hamiltonians, and can be used to generate a high degree of entanglement. The Hamiltonian has two components: a linear part that can be controlled by manipulating the azimuthal lattice, and an interaction-dependent quadratic part. Exact solutions are found for the quadratic part for equal strengths of intraspecies and interspecies interactions. In different regimes the Hamiltonian can emulate a beam splitter or a two-mode squeezer of quantum optical systems. We study entanglement properties of the ground state of the Hamiltonian in dependence on various parameters with the prospect of possible quantum information and metrology applications. 

We present a comprehensive study of stationary states in a coherent medium with a quadratic or Kerr nonlinearity in the presence of localized potentials in one dimension for both positive and negative signs of the nonlinear term as well as for barriers and wells. The description is in terms of the nonlinear Schrödinger equation and hence applicable to a variety of systems, including interacting ultracold atoms in the mean field regime and light propagation in optical fibers. We determine the full landscape of solutions in terms of a potential step and build solutions for rectangular barrier and well potentials. It is shown that all the solutions can be expressed in terms of a Jacobi elliptic function with the inclusion of a complex-valued phase shift. Our solution method relies on the roots of a cubic polynomial associated with a hydrodynamic picture, which provides a simple classification of all the solutions, both bounded and unbounded, while the boundary conditions are intuitively visualized as intersections of phase space curves. We compare solutions for open boundary conditions with those for a barrier potential on a ring, and also show that numerically computed solutions for smooth barriers agree qualitatively with analytical solutions for rectangular barriers. A stability analysis of solutions based on the Bogoliubov equations for fluctuations shows that persistent instabilities are localized at sharp boundaries and are predicated by the relation of the mean density change across the boundary to the value of the derivative of the density at the edge. We examine the scattering of a wave packet by a barrier potential and show that at any instant the scattered states are well described by the stationary solutions we obtain, indicating applications of our results and methods to nonlinear scattering problems. 

We consider ultracold atoms trapped in a toroidal trap with an azimuthal lattice for utility as a macroscopic simulator of quantum optics phenomena. We examine the dynamics induced by the adiabatic introduction of the lattice that serves to couple the normal modes as an analog of a laser field coupling electronic states. The system is found to display two distinct behaviors, manifest in the angular momentum—coherent oscillation and self-trapping—reminiscent of nonlinear dynamics yet not requiring interatomic interactions. The choice is set by the interplay of discrete parameters, the specific initial mode, and the periodicity of the lattice. However, rotation can cause continuous transition between the two regimes, causing periodic quenches and revivals in the oscillations as a function of the angular velocity. Curiously, the impact of rotation is determined entirely by the energy spectrum in the absence of the lattice, a feature that can be attributed to adiabaticity. We assess the effects of varying the lattice parameters and consider applications in rotation sensing. 

The degree of localization of the Harper-Hofstadter model is shown to display striking periodic dependence on phase degrees of freedom, which can depend on the nature of the boundary condition, reminiscent of the Aharonov-Bohm effect. In the context of implementation in a finite ring-shaped lattice structure, this phase dependence can be utilized as a fundamentally different principle for precision sensing of rotation and magnetic fields based on localization rather than on interferometry. 

Emergence of fundamental forces from gauge symmetry is among our most profound insights about the physical universe. In nature, such symmetries remain hidden in the space of internal degrees of freedom of subatomic particles. Here we propose a way to realize and study gauge structures in real space, manifest in external degrees of freedom of quantum states. We present a model based on a ring-shaped lattice potential, which allows for both Abelian and non-Abelian constructs. Non trivial Wilson loops are shown possible via physical motion of the system. The underlying physics is based on the close analogy of geometric phase with gauge potentials that has been utilized to create synthetic gauge fields with internal states of ultracold atoms. By scaling up to an array with spatially varying parameters, a discrete gauge field can be realized in position space, and its dynamics mapped over macroscopic size and time scales.