Mathematical Analysis of the Bose Polaron

• Applicant: Professor Dr. Peter Pickl
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 529797785

At very low temperature, a gas of Bosonic atoms undergoes a phase transition to a Bose-Einstein condensate: a macroscopic number of particles exhibit collective quantum behavior. Introducing an impurity particle into the condensate can reveal many of its fundamental properties, and such coupled systems are expected to yield numerous applications in quantum engineering. At zero temperature, all particles of a non-interacting Bose gas will occupy the one-particle ground state. For weakly interacting systems, this ground state is modified and some particles occupy states of higher energy. The theory behind these excitations goes back to a seminal work of Bogoliubov from 1947, who showed that excitations typically appear in pairs, and described them in terms of a quantum field. In this project we examine systems of many bosons with few impurities from a mathematical perspective. We will prove the validity of an effective model in which the impurities interact with Bogoliubov's excitation field. Recent results establish this model in the special case of a dense, weakly interacting, homogeneous condensate on a torus. In order to cover more physical situations, we aim at generalizing them in two directions: First, we will consider dilute gases, as they are typically used in experimental Bose-Einstein condensates. For such systems, interactions are very singular as they are short range and strong. They induce a non-neglibible correlation structure, requiring renormalization of Bogoliubov's theory. Second, we will treat the physical situation of non-periodic systems. For extended systems, a detailed analysis of the relevant scales in space and time will allow for an approximate reduction to the case of constant density. Combining insights on both questions will lead to a deeper understanding of Bogoliubov's approximation, which is an important model case for more general quasi-particle approximations in condensed-matter physics.

• DFG Programme: Research Grants
• International Connection: France
• Cooperation Partner: Professor Dr. Jonas Lampart