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arXiv · 1809.09499

Quadratic Quantum Hamiltonians: General Canonical Transformation to a Normal Form

Katja Kustura·Cosimo C. Rusconi·Oriol Romero-Isart
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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.

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Katja Kustura, Cosimo C. Rusconi, Oriol Romero-Isart. 2018-09-25. Quadratic Quantum Hamiltonians: General Canonical Transformation to a Normal Form. https://doi.org/10.1103/physreva.99.022130

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