Geometry of variational methods: dynamics of closed quantum systems
Abstract
We present a systematic geometric framework to study closed quantum systems based on suitably chosen variational families. For the purpose of (A) real time evolution, (B) excitation spectra, (C) spectral functions and (D) imaginary time evolution, we show how the geometric approach highlights the necessity to distinguish between two classes of manifolds: Kähler and non-Kähler. Traditional variational methods typically require the variational family to be a Kähler manifold, where multiplication by the imaginary unit preserves the tangent spaces. This covers the vast majority of cases studied in the literature. However, recently proposed classes of generalized Gaussian states make it necessary to also include the non-Kähler case, which has already been encountered occasionally. We illustrate our approach in detail with a range of concrete examples where the geometric structures of the considered manifolds are particularly relevant. These go from Gaussian states and group theoretic coherent states to generalized Gaussian states.
Document Details
- Document Type
- Pub Defense Publication
- Publication Date
- Oct 08, 2020
- Source ID
- 10.21468/scipostphys.9.4.048
Entities
People
- Eugene A. Demler
- Juan Ignacio Cirac Sasturáin
- Jutho Haegeman
- Lucas Hackl
- Tao Shi
- Tommaso Guaita
Organizations
- Air Force Office of Scientific Research
- Chinese Academy of Sciences
- European Research Council
- German Research Foundation
- Ghent University
- Harvard University
- Max Planck Institute of Quantum Optics
- Munich Center for Quantum Science and Technology
- National Natural Science Foundation of China
- University of Chinese Academy of Sciences
- University of Copenhagen
- Villum Foundation