Quadratic Quantum Hamiltonians: General Canonical Transformation to a Normal Form

TL;DR

This paper reviews how to transform quadratic quantum Hamiltonians into a normal form using canonical transformations, aiding the understanding of stability and instability in quantum systems.

Contribution

It provides a comprehensive, step-by-step method for constructing canonical transformations to normalize quadratic Hamiltonians, including unstable cases.

Findings

Normal forms reveal different dynamical instability types.

Canonical transformations can be applied beyond stable systems.

Illustrative examples demonstrate the method's effectiveness.

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.