A comparison between classical and Bohmian quantum chaos

Abstract

We study the emergence of chaos in a 2d system corresponding to a classical Hamiltonian system $V= \frac{1}{2}(\omega_x^2x^2+\omega_y^2y^2)+\epsilon xy^2$ consisting of two interacting harmonic oscillators and compare the classical and the Bohmian quantum trajectories for increasing values of $\epsilon$. In particular we present an initial quantum state composed of two coherent states in $x$ and $y$, which in the absence of interaction produces ordered trajectories (Lissajous figures) and an initial state which contains {both chaotic and ordered} trajectories for $\epsilon=0$. In both cases we find that, in general, Bohmian trajectories become chaotic in the long run, but chaos emerges at times which depend on the strength of the interaction between the oscillators.