Time-periodic quantum states of weakly interacting bosons in a harmonic trap

Abstract

We consider identical quantum bosons with weak contact interactions in a two-dimensional isotropic harmonic trap, and focus on states at the Lowest Landau Level (LLL). At linear order in the coupling parameter $g$, we exploit the rich algebraic structure of the problem to give an explicit construction of a large family of quantum states with energies of the form $E_0+g{E}_1/4+O(g^2)$, where $E_0$ and $E_1$ are integers. As a result, any superposition of these states evolves periodically with a period of at most $8\pi /g$ until, at much longer time scales of order $1/g^2$, corrections to the energies of order $g^2$ become important and may upset this perfectly periodic behavior. We further construct coherent-like combinations of these states that naturally connect to classical dynamics in an appropriate regime, and explain how our findings relate to the known time-periodic features of the corresponding weakly nonlinear classical theory. We briefly comment on possible generalizations of our analysis to other numbers of spatial dimensions and other analogous physical systems.

arXiv comment14 pages, v2: published version

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• Exact solutions for a coherent phenomenon of condensation in conservative Hamiltonian systems
• Phase-space localization at the lowest Landau level
• Nonlinear dynamics of superpostion of wavepackets
• A superintegrable quantum field theory