This book presents a comprehensive account of Bose-Einstein Condensation (BEC), a fundamental phenomenon in quantum physics that reveals the collective behavior of bosonic systems at ultracold temperatures. Bridging theoretical models with experimental developments and applied perspectives, the book is structured to guide readers from foundational principles to advanced topics in the physics of quantum fluids. It is intended for researchers and graduate students in theoretical physics, particularly in condensed matter and ultracold atomic systems.Topics include the distinction between quantum gases and liquids, the role of the chemical potential in ultracold systems, and the historical and theoretical development of interacting Bose gases since the early work of Bogoliubov. The limitations of Bogoliubov theory-particularly its applicability to weakly interacting systems and neglect of finite-temperature effects-are discussed, along with recent advances such as the Lee-Huang-Yang (LHY) correction and the emergence of quantum droplets in two-component Bose mixtures.The book introduces field-theoretical methods for many-body systems, covering spontaneous symmetry breaking, Green functions, Matsubara formalism, and path integrals. A central focus is the application of Optimized Perturbation Theory (OPT), also known as the variational Gaussian approximation or linear delta expansion, to atomic BECs and quantum magnets (triplons). Theoretical developments are complemented by a brief overview of experimental methods.
