Quadratic Quantum Hamiltonians: General Canonical Transformation to a Normal Form
arXiv:1809.09499 · doi:10.1103/PhysRevA.99.022130
Abstract
A system of linearly coupled quantum harmonic oscillators can be diagonalized when the system is dynamically stable using a Bogoliubov canonical transformation. However, this is just a particular case of more general canonical transformations that can be performed even when the system is dynamically unstable. Specific canonical transformations can transform a quadratic Hamiltonian into a normal form, which greatly helps to elucidate the underlying physics of the system. Here, we provide a self-contained review of the normal form of a quadratic Hamiltonian as well as step-by-step instructions to construct the corresponding canonical transformation for the most general case. Among other examples, we show how the standard two-mode Hamiltonian with a quadratic position coupling presents, in the stability diagram, all the possible normal forms corresponding to different types of dynamical instabilities.
arXiv commentPublished version. 16 pages, 1 figure, 2 tables
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