Nonlinear dynamics of mathematical models and proposed implementations in ultracold atoms
Abstract
Nonlinear effects are ubiquitous in nature. Many interesting phenomena are described by differential equations that are nonlinear. Even richer dynamics can be observed with additional long-range spatial coupling. For example, an interesting type of pattern discovered recently can form in systems with nonlocal coupling. The pattern, called chimera states, is composed of both phase coherent and incoherent regions coexisting in the same system. Through numerical studies of oscillatory media with nonlocal diffusive coupling, I show here for the first time that stable chimera knot structures can exist in 3D. Knots were not previously known to be stable in oscillatory media, nor were such non-trivial chimera patterns known to exist in 3D.
To realize different nonlinear phenomena in a controlled way in experiments, a flexible physical system is required. Ultracold atomic systems, specifically, Bose-Einstein condensates (BECs), are good candidates because of the high controllability of almost all parameters, including the nonlinearity, in real time. Hence, experimental studies can be carried out for a variety physical systems, including many classical and quantum field equations. In particular, in this thesis I study a setup of BECs with a third-order Kerr nonlinearity to generate Schr\"odinger cat states, which have applications in quantum metrology. I showed that cat states involving hundreds of atoms should be realizable in BECs. This requires careful optimization of the experimental parameters and analysis of the atom loss.
Inspired by the previous two projects, it is an interesting question if chimera states can exist in BECs. By analyzing the underlying mechanism of the effective nonlocal diffusive coupling, I establish here a new analogous mechanism to achieve mediated nonlocal spatial hopping for particles in BECs with two interconvertible states. By adiabatically eliminating the fast mediating channel, I obtain the mean-field of Bose-Hubbard model with fully tunable hopping strength, hopping range, and nonlinearity. This is the first known conservative system exhibiting chimera patterns. More importantly, I show that the model should be implementable in BECs with current technology.
Description
Keywords
Physics--Atomic, Physics--Theory
Citation
Lau, H. W. (2017). Nonlinear dynamics of mathematical models and proposed implementations in ultracold atoms (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca. doi:10.11575/PRISM/27119