Normal forms of quadratic bose operators

Shiroman Prakash, Amolak Ratan Kalra, Akalank Jain

Valeri V. Kucherenko

Quadratic boson operators have arise in the asymptotic approach to the N particle Schrödinger equation.

A. D. Jannussis, G. N. Brodimas, D. Sourlas +3 more

Katja Kustura, Cosimo C. Rusconi, Oriol Romero‐Isart

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.

Xu Xiu-Wei, MU Hai-feng, Shu-Yan Liu +1 more

By utilizing generalized linear quantum transformation theory, the evolution operator, normal and anti-normal Wigner characteristic functions, P- and Q-representations of a multi-mode coupled quadratic Boson system are presented. The squeezing properties of a time-dependent double-mode coupled quadratic Bose system are investigated as a specific example.