Cosmology emerging as the gauge structure of a nonlinear quantum system

• Published in: arXiv.org 2016-07
• Main Authors: Chon-Fai Kam, Ren-Bao, Liu
• Format: Article
• Language: English
• Subjects: Big bang cosmology; Bosons; Cosmology; Curvature; Differential geometry; Energy levels; Harmonic oscillators; Nonlinear systems; Parameters; Phases; Quantum mechanics; Quantum physics; Quantum theory; Resonant frequencies; Superposition (mathematics); System effectiveness; Topological insulators; Topology; Universe
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1607.00555

Berry phases and gauge structures in parameter spaces of quantum systems are the foundation of a broad range of quantum effects such as quantum Hall effects and topological insulators. The gauge structures of interacting many-body systems, which often present exotic features, are particularly interesting. While quantum systems are intrinsically linear due to the superposition principle, nonlinear quantum mechanics can arise as an effective theory for interacting systems (such as condensates of interacting bosons). Here we show that gauge structures similar to curved spacetime can arise in nonlinear quantum systems where the superposition principle breaks down. In the canonical formalism of the nonlinear quantum mechanics, the geometric phases of quantum evolutions can be formulated as the classical geometric phases of a harmonic oscillator that represents the Bogoliubov excitations. We find that the classical geometric phase can be described by a de Sitter universe. The fundamental frequency of the harmonic oscillator plays the role of the cosmic scale factor and the classical geometric phase is an integral of a differential angle 2-form, which is half of the curvature 2-form of the associated de Sitter universe. While the gauge structure of a linear quantum system presents monopole singularity at energy level degeneracy points, nonlinear quantum systems, corresponding to their quantum critical surfaces in the parameter spaces, exhibits a conic singularity in their gauge structure, which mimics the casual singularity at the big bang of the de Sitter universe. This finding opens up a new approach to studying the gauge and topological structures of interacting quantum systems and sets up a new stage for quantum simulation of fundamental physics.