K. Kustura, C. C. Rusconi, O. Romero-Isart Quadratic Quantum Hamiltonians: General Canonical Transformation to a Normal Form,
Phys. Rev. A 99 22130 (2019-02-28),
http://dx.doi.org/10.1103/PhysRevA.99.022130 doi:10.1103/PhysRevA.99.022130 (ID: 720073)
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.