Bose-Einstein Condensation Beyond Mean Field: A Partial Differential Equation Approach to Quantum Fluctuations

University Of Maryland, College Park, College Park MD

Investigators

Abstract

This research addresses a series of challenging questions for the physics of very cold atoms by analytical methods of applied mathematics. In Bose-Einstein condensation, a certain type of particles (bosons) occupy a single macroscopic quantum state called condensate at very low temperatures. This condensate is one of the most coherent states of matter known to date, allowing for a precise control of atomic systems in laboratory settings. The Bose-Einstein condensation has been observed experimentally in atomic gases that are trapped through magnetic and optical means, and has sparked active experimental research into attractive applications of fundamental importance in quantum information and computation, e.g., the design and construction of quantum atomic computers, as well as in precision measurements. The investigator of this project will develop models and carry out analysis that will lead to a better and more fundamental understanding of atomic effects in Bose-Einstein condensation. The impact of this research will be felt by the applied mathematics community as well as by communities of some of the applications, including atomic and condensed-matter physicists, and computer scientists and engineers with specialty in quantum information. The investigator will train one graduate student who will gain interdisciplinary research education and perspective in cutting-edge problems of modern science. The goal of this research is to develop a hierarchy of models and analytical tools in order to link the motion and interactions of individual atoms to the macroscopic properties of ultra-cold atomic gases in Bose-Einstein condensation. The methods invoked in this project single out the effect of pair excitations, which cause the many-body wave function of such systems to deviate from the usual mean-field tensor product of one-particle states. The methods of the investigator include: (i) perturbation theory for many-body operators, which links the microscopic Hamiltonian to systems of low-dimensional partial differential equations; (ii) homogenization of the derived systems of equations for settings with microstructures of experimental relevance; and (iii) the global existence and uniqueness of the resulting partial differential equations. This research effort will elucidate fundamental mechanisms of controlling the condensate through the many-body wave function, hence improving past predictions based solely on mean-field theory. Results obtained in this direction can guide simulations of and experiments related to complex condensed-matter systems, and the design of new devices for quantum information and precision metrology.